VOLUME
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, orplasma) or shape occupies or contains.[1] Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. The volumes of more complicated shapes can be calculated by integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive.[2]
In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.
Contents [hide]
Units[edit]
Volume measurements from the 1914The New Student's Reference Work.
Approximate conversion to millilitres:[3]ImperialU.S. liquidU.S. dryGill142 ml118 ml138 mlPint568 ml473 ml551 mlQuart1137 ml946 ml1101 mlGallon4546 ml3785 ml4405 ml
Any unit of length gives a corresponding unit of volume, namely the volume of a cube whose side has the given length. For example, a cubic centimetre(cm3) would be the volume of a cube whose sides are one centimetre (1 cm) in length.
In the International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres,so
1 cubic metre = 1000 litres.Small amounts of liquid are often measured in millilitres, where
1 millilitre = 0.001 litres = 1 cubic centimetre.Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, and the hogshead.
Related terms[edit]Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litres or its derived units), and volume being how much space an object displaces (commonly measured in cubic metres or its derived units).
Volume and capacity are also distinguished in capacity management, where capacity is defined as volume over a specified time period. However in this context the term volume may be more loosely interpreted to mean quantity.
The density of an object is defined as mass per unit volume. The inverse of density is specific volume which is defined as volume divided by mass. Specific volume is a concept important inthermodynamics where the volume of a working fluid is often an important parameter of a system being studied.
The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s-1]).
Volume formulas[edit]ShapeVolume formulaVariablesCubea = length of any side (or edge)Cylinderr = radius of circular face, h = heightPrismB = area of the base, h = heightRectangular prisml = length, w = width, h = heightSpherer = radius of sphere
which is the integral of the surface area of a sphereEllipsoida, b, c = semi-axes of ellipsoidPyramidB = area of the base, h = height of pyramidConer = radius of circle at base, h = distance from base to tip or heightTetrahedron[4]edge length Parallelepiped
a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edgesAny volumetric sweep
(calculus required)h = any dimension of the figure,
A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. a and b are the limits of integration for the volumetric sweep.
(This will work for any figure if its cross-sectional area can be determined from h).Any rotated figure (washer method)
(calculus required) and are functions expressing the outer and inner radii of the function, respectively.Volume ratios for a cone, sphere and cylinder of the same radius and height[edit]
A cone, sphere and cylinder of radius r and height h
The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.
Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is
the volume of the sphere is
while the volume of the cylinder is
The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.[5]
Volume formula derivations[edit]Sphere[edit]The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.
The surface area of the circular disk is .
The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is;
or
where y or z can be taken to represent the radius of a disk at a particular x value.
Using y as the disk radius, the volume of the sphere can be calculated as
Now
Combining yields gives
This formula can be derived more quickly using the formula for the sphere's surface area, which is . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to
=
Cone[edit]The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well.
However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0,0,0) with radius r, is as follows.
The radius of each circular disk is r if x = 0 and 0 if x = h, and varying linearly in between—that is,
The surface area of the circular disk is then
The volume of the cone can then be calculated as
and after extraction of the constants:
Integrating gives us
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, orplasma) or shape occupies or contains.[1] Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. The volumes of more complicated shapes can be calculated by integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive.[2]
In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.
Contents [hide]
- 1 Units
- 2 Related terms
- 3 Volume formulas
- 4 Volume formula derivations
- 5 See also
- 6 References
- 7 External links
Units[edit]
Volume measurements from the 1914The New Student's Reference Work.
Approximate conversion to millilitres:[3]ImperialU.S. liquidU.S. dryGill142 ml118 ml138 mlPint568 ml473 ml551 mlQuart1137 ml946 ml1101 mlGallon4546 ml3785 ml4405 ml
Any unit of length gives a corresponding unit of volume, namely the volume of a cube whose side has the given length. For example, a cubic centimetre(cm3) would be the volume of a cube whose sides are one centimetre (1 cm) in length.
In the International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres,so
1 cubic metre = 1000 litres.Small amounts of liquid are often measured in millilitres, where
1 millilitre = 0.001 litres = 1 cubic centimetre.Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, and the hogshead.
Related terms[edit]Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litres or its derived units), and volume being how much space an object displaces (commonly measured in cubic metres or its derived units).
Volume and capacity are also distinguished in capacity management, where capacity is defined as volume over a specified time period. However in this context the term volume may be more loosely interpreted to mean quantity.
The density of an object is defined as mass per unit volume. The inverse of density is specific volume which is defined as volume divided by mass. Specific volume is a concept important inthermodynamics where the volume of a working fluid is often an important parameter of a system being studied.
The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s-1]).
Volume formulas[edit]ShapeVolume formulaVariablesCubea = length of any side (or edge)Cylinderr = radius of circular face, h = heightPrismB = area of the base, h = heightRectangular prisml = length, w = width, h = heightSpherer = radius of sphere
which is the integral of the surface area of a sphereEllipsoida, b, c = semi-axes of ellipsoidPyramidB = area of the base, h = height of pyramidConer = radius of circle at base, h = distance from base to tip or heightTetrahedron[4]edge length Parallelepiped
a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edgesAny volumetric sweep
(calculus required)h = any dimension of the figure,
A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. a and b are the limits of integration for the volumetric sweep.
(This will work for any figure if its cross-sectional area can be determined from h).Any rotated figure (washer method)
(calculus required) and are functions expressing the outer and inner radii of the function, respectively.Volume ratios for a cone, sphere and cylinder of the same radius and height[edit]
A cone, sphere and cylinder of radius r and height h
The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.
Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is
the volume of the sphere is
while the volume of the cylinder is
The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.[5]
Volume formula derivations[edit]Sphere[edit]The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.
The surface area of the circular disk is .
The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is;
or
where y or z can be taken to represent the radius of a disk at a particular x value.
Using y as the disk radius, the volume of the sphere can be calculated as
Now
Combining yields gives
This formula can be derived more quickly using the formula for the sphere's surface area, which is . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to
=
Cone[edit]The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well.
However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0,0,0) with radius r, is as follows.
The radius of each circular disk is r if x = 0 and 0 if x = h, and varying linearly in between—that is,
The surface area of the circular disk is then
The volume of the cone can then be calculated as
and after extraction of the constants:
Integrating gives us
VOLUME of SOLIDS
Volumes cube = a3
rectangular prism = a b c
irregular prism = b h
cylinder = b h = r2 h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 r2 h
sphere = (4/3) r3
ellipsoid = (4/3) pi r1 r2 r3
Volumes cube = a3
rectangular prism = a b c
irregular prism = b h
cylinder = b h = r2 h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 r2 h
sphere = (4/3) r3
ellipsoid = (4/3) pi r1 r2 r3
Year 9 Interactive Maths - Second Edition
Volume of a CylinderA cylinder with radius r units and height h units has a volume of Vcubic units given by
Example 27Find the volume of a cylindrical canister with radius 7 cm and height of 12
Volume enclosed by a cylinder
Definition: The number of cubic units that will exactly fill a cylinder
Try this Drag the orange dot to resize the cylinder. The volume is calculated as you drag.
See also: Surface area of a cylinder
How to find the volume of a cylinder Although a cylinder is technically not a prism, it shares many of the properties of a prism. Like prisms, the volume is found by multiplying the area of one end of the cylinder (base) by its height.
Since the end (base) of a cylinder is a circle, the area of that circle is given by
Multiplying by the height h we get
where:
π is Pi, approximately 3.142
r is the radius of the circular end of the cylinder
h height of the cylinder
Calculator
Some notes on the volume of a cylinderRecall that a cylinder is like an empty soup can. It has nothing inside, and the walls of the can have zero thickness. So strictly speaking, the cylinder has zero volume.
The strictly correct way of saying it is "the volume enclosed by a cylinder" - the amount of soup it holds. But many textbooks simply say "the volume of a cylinder" to mean the same thing. What they usually mean when they say this is the volume enclosed by the cylinder.
Oblique cylindersRecall that an is one that 'leans over' - where the top center is not over the base center point. In the figure above check "allow oblique' and drag the top orange dot sideways to see an oblique cylinder.
It turns out that the volume formula works just the same for these. You must however use the perpendicular height in the formula. This is the vertical line to left in the figure above. To illustrate this, check 'Freeze height'. As you drag the top of the cylinder left and right, watch the volume calculation and note that the volume never changes.
See Oblique Cylinders for a deeper discussion on why this is so.
UnitsRemember that the radius and the height must be in the same units - convert them if necessary. The resulting volume will be in those cubic units. So, for example if the height and radius are both in centimeters, then the volume will be in cubic centimeters.
Sample problems:Volume of a Cylinder formula:
V = Π r2h
(let pi = 3.14)
Find the volume of a cylinder having a radius of 5 cm and a height of 12 cm.
Solution:
Given:
r = 5 cm; h = 12 cm
The formula:
V = Π r2h
V = (3.14)(52)(12)
V = (3.14)(25)(12)
V = 942 cubic cm!
The volume of a cylinder is 1632 ft3. If the height of the cylinder is
24 ft., what is the area of it's base?
Solution:
This one is a little different.
We have been given the volume and the height,
and from this information we must determine
the area of the base.
The area of the base is the circle (Π r2)
V = Π r2h
Plug in the height:
1632 = (Π r2)(24)
If we divide both sides by the height (24)
it will leave the area of the base:
1632/24 = (Π r2)(24)/24
68 = Π r2
The area of the base is 68 sq. ft.!
A cylindrical swimming pool is 4.5 feet high and has a diameter of 24 feet. If the pool is filled right up to the top edge how many cubic feet of water can the pool hold?
Solution:
First, change the diameter (given) to a radius:
d = 24 ft....therefore.... r = 12 ft.
Plug in the values:
V = Π r2h
V = (3.14)(122)(4.5)
V = (3.14)(144)(4.5)
V = 2034.72 cubic feet of water!
A small can of soup has a radius of 3.5 cm and a height of 12 cm. A family size can has a radius of 5 cm and is 15 cm high. Which contains more soup, one family sized can, or two small cans?
Solution:
The small can:
r = 3.5 cm, h = 12 cm
V = (3.14)(3.52)(12)
V = (3.14)(12.25)(12)
V = 461.58 cubic cm
2 cans would have 2(461.58)
V = 923.16 cubic cm)
Family sized can:
r = 5 cm, h = 15 cm
V = (3.14)(52)(15)
V = (3.14)(25)(15)
V = 1177.5 cubic cm
Now subtract:
1177.5 - 923.16
Difference = 254.34
1 family size can is 254.34 cu. cm larger
than 2 small cans!! Surprised??!!
A coffee grower is going to package his coffee in cylindrical cans which will hold exactly 785 cubic inches of his product. If the cans are to be 10 inches in height, what must the radius of the can be?
Solution:
Plug in the information we know:
785 = (3.14)(r2)(10)
Since multiplication is commutative, reorder the factors:
785 = (3.14)(10(r2)
785 = (31.4)(r2)
Divide both sides by 31.4 to isolate the r2
785/31.4 = (31.4)(r2)/31.4
25 = r2
If r2 = 25, then r = 5
The radius of the can must be 5 cm!
http://www.studyzone.org/mtestprep/math8/g/volcylprac.cfm
Sphere:
A sphere is 3-dimensional (solid) figure in which all of it's points are the same distance from one fixed point in space, which is called it's center.
Don't get confused! A sphere is not a circle. A circle is 2 - dimensional, or flat. A circle is also known as a plane figure.
A line segment which joins any point on the surface of the sphere with it's center is called it's radius.
The Volume formula:
Unlike many other formulas for finding the volume of a 3-dimensional, or solid figure, we need only one dimension to calculate the volume of a sphere....it's radius!
The formula for the volume of a sphere in which it's radius is "r" is:
V = 4/3 Π r3
For all of the model problems below let Π = 3.14
Model ProblemsLet's start with an easy example.
An official handball has a diameter of 4.8 centimeters. Find it's volume.
First you must recognize that the dimension you have been given is the diameter of the sphere. However, as was discussed earlier in this lesson, the dimension
needed is the radius.
That's easy to fix!
Simply cut the diameter in half (or divide by 2).
If d = 4.8, then
r = (4.8)/2 = 2.4!
Then, plug the value of "r" into the formula:
V = 4/3 Π r3
V = 4/3 (3.14)(2.4)3
V = 4/3(3.14)(13.824)
V = 4/3(43.40736)
The easiest way to multiply a number like this
by a fraction is simply multiply the number (43.40736)
by 4 (the numerator), and then divide
the result by 3 (the denominator).
First multiply by 4
4(43.40736)
173.62944
Then divide by 3
(173.62944)/3
57.87648
V = 57.87648 cubic centimeters!
Got the idea? Let's try another..
A red rubber ball needs air before the class plays kickball. After it has been properly inflated for the game, it's radius has increased from 8 inches to 11 inches in length.
What was the change in the ball's volume, to the the nearest cubic inch?
To answer this problem we will need to find the volume twice. Once for the ball in it's deflated form (radius = 8"),
then a second time after it has been filled
with air (radius = 11 inches).
Then to find the change we will subtract the 2 volumes.
Deflated ball:
V = 4/3Π r3
V = 4/3 (3.14)(8)3
V = 4/3(3.14)512)
V = 4/3(1607.68)
V = 2143.573333
V = 2144 cubic inches.
Inflated ball:
V = 4/3(3.14)(11)3
V = 4/3(3.14)(1331)
V = 4/3(4179.34)
V = 5572.453333
V = 5572 cubic inches
The change in volume:
5572 - 2144
3428 cubic inches!
Now let's look at one more..
A ball having a diameter of 10 cm is being shipped in a box shaped like a cube whose height, width, and length are also exactly 10 cm.. How many cubic centimeters of packing material are needed to hold the ball firmly in place inside the box?
Can you picture the problem? Can you see a ball inside a cube, so that when the top is closed the ball fits snugly in the box, but there are these areas of open space where the ball and box don't touch. That's the volume we will be solving for.
First let's find the volume of the cube.
A cube is a rectangular prism in which all of the dimensions are the same. In this case l = 10, w = 10 and h = 10.
V = (l)(w)(h)
V = (10)(10)(10)
V = 1000 cubic cm.
Now for the volume of the ball:
d = 10, so r = 5
V = 4/3 Π r3
V = 4/3(3.14)(5)3
V = 4/3(3.14)(125)
V = 4/3(392.5)
V = 523.333... cubic cm
The amount of "empty space" is the difference
between these 2 volumes.
Empty space = 1000 - 523 (to nearest cubic cm)
Empty space = 477 cubic cm!
Did that answer surprise you??
If you think about it, the ball only fills up a little bit more than half of the volume of the cube!
http://www.studyzone.org/mtestprep/math8/g/volsphereprac.cfm
How to find the volume of a cubeRecall that a cube has all edges the same length (See Cube definition). The volume of a cube is found by multiplying the length of any edge by itself twice. So if the length of an edge is 4,the volume is 4 x 4 x 4 = 64
Or as a formula:
volume = s3where:
s is the length of any edge of the cube. Calculator
In the figure above, drag the orange dot to resize the cube. From the edge length shown, calculate the volume of the cube and verify that it agrees with the calculation in the figure.
When we write volume = s3, strictly speaking this should be read as "s to the power 3", but because it is used to calculate the volume of cubes it is usually spoken as "s cubed".
Some notes on the volume of a cubeRecall that a cube is like an empty box. It has nothing inside, and the walls of the box have zero thickness. So strictly speaking, the cube has zero volume. When we talk about the volume of a cube, we really are talking about how much liquid it can hold, or how many unit cubes would fit inside it.
Think of it this way: if you took a real, empty metal box and melted it down, you would end up with a small blob of metal. If the box was made of metal with zero thickness, you would get no metal at all. That is what we mean when we say a cube has no volume.
The strictly correct way of saying it is "the volume enclosed by a cube" - the amount space there is inside it. But many textbooks simply say "the volume of a cube" to mean the same thing. However, this is not strictly correct in the mathematical sense. What they usually mean when they say this is the volume enclosed by the cube.
UnitsRemember that the length of an edge and the volume will be in similar units. So if the edge length is in miles, then the volume will be in cubic miles, and so on.
Volume of a cube
Given the length of one side, call it a, the volume of a cube can be found by using the following formula:
Vcube = a3 = a × a × a
Example #1:
Find the volume if the length of one side is 2 cm
Vcube = 23
Vcube = 2 × 2 × 2
Vcube = 8 cm3
Example #2:
Find the volume if the length of one side is 3 cm
Vcube = 33
Vcube = 3 × 3 × 3
Vcube = 27 cm3
Example #2:
Find the volume if the length of one side is 3/2 cm
Vcube = (3/2)3
Vcube = 3/2 × 3/2 × 3/2
Vcube = (3 × 3 × 3)/(2 × 2 × 2) Vcube = 27/8 Vcube = 3.375 cm3
http://www.basic-mathematics.com/volume-of-a-cube.html
Volume of a Triangular Prism - Lesson
Topic Index | Grade 7 Math | Intermediate Test Prep |StudyZone
When finding the volume of a right triangular prism we need to use the formula:
Remember... "w" stands for width, "h" stands for height and "l" stands for length.
Let's try some!!!
Find the volume of this triangular prism. Use the formula:
You may use the calculator to help with the math!
(Remember that 1/2 is the same as .5)
0789/456*123-0+/-.+C
=
Answer...
V = 864 cm³
Volume of a Pyramid Formula
A pyramid is a polyhedron with a polygonal base and triangular faces equal to the number of sides in the base. All the triangular faces meet at a single point called the apex. The faces of the pyramid connect the bases with the apex. Volume of a Pyramid is the measure of the number of units occupied by the pyramid.
The Volume of a Pyramid Formula is given as,
Where
a - apothem length of the pyramid.
b - base length of the pyramid.
h - height of the pyramid.
Volume of a Pyramid ProblemsBack to Top
Some solved problems on the volume of a pyramid are given below:Solved ExamplesQuestion 1: Find the volume of a pentagonal pyramid of apothem length 5 cm, base length 7 cm and height 11 cm ?
Solution:
Given,
a = 5 cm
b = 7 cm
h = 11 cm
Volume of a pentagonal pyramid
= 56abh
= 56 * 5 cm * 7 cm * 11 cm
= 320.833 cm3
Question 2: Find the volume of a triangular pyramid of apothem length 7 cm, base length 13 cm and height 19 cm ?
Solution:
Given,
a = 7 cm
b = 13 cm
h = 19 cm
Volume of a triangular pyramid
= 16abh
= 16 * 7 cm * 13 cm * 19 cm
= 288.167 cm3
http://formulas.tutorvista.com/math/volume-of-a-pyramid-formula.html
Volume is measured in cubic units. Two common units of volume are the cubic centimeter () and the cubic inch (). The volume of a rectangular prism is a measure of how much space the object takes up.Volume is found using three dimensions: length, width, and height.
A cubic unit is 1 unit by 1 unit by 1 unit ().
A cubic centimeter is 1 cm by 1 cm by 1 cm ().
A cubic inch is 1 in by 1 in by 1 in ().
Example 1
A box measures 10 cm. long x 2 cm. wide x 4 cm. high.
The dimensions of a rectangular prism.
Its volume is the amount of space inside. By finding the product of the length, width, and height, you will find the number of 1cm cubes that will fit in the box.10 x 2 x 4 = 80 cubic centimeters, or 80 .
So, 80 1 cm. cubes will fit in the box.
Example 2
A box measures 6 in. long x 3 in. wide x 6 in. high.
Its volume is the amount of space inside. By finding the product of the length, width, and height, you will find the number of 1 in cubes that will fit in the box.6 x 3 x 6 = 108 cubic inches, or 108 .
So, 108 1 in. cubes will fit in the box.
Capacity
Liquids, sugar, salt, and so on can be measured by being poured into or out of a container. The volume of a container that is filled with a liquid or a solid that can be transferred is often its capacity.
Capacity is often measured in units such as fluid ounces, cups, pints, quarts, gallons, milliliters, and liters.
Remember: The formula for volume is V = l x w x h.
http://www.studyzone.org/mtestprep/math8/e/volumecapacity6l.cfm
Volume of a cone
Given the radius and h, the volume of a cone can be found by using the formula:
Formula: Vcone = 1/3 × b × h
b is the area of the base of the cone. Since the base is a circle, area of the base = pi × r2
Thus, the formula is Vcone = 1/3 × pi × r2 × h
Use pi = 3.14
Example #1:
Calculate the volume if r = 2 cm and h = 3 cm
Vcone = 1/3 × 3.14 × 22 × 3
Vcone = 1/3 × 3.14 × 4 × 3
Vcone = 1/3 × 3.14 × 12
Vcone = 1/3 × 37.68
Vcone = 1/3 × 37.68/1
Vcone = (1 × 37.68)/(3 × 1)
Vcone = 37.68/3
Vcone = 12.56 cm3
Example #2:
Calculate the volume if r = 4 cm and h = 2 cm
Vcone = 1/3 × 3.14 × 42 × 2
Vcone = 1/3 × 3.14 × 16 × 2
Vcone = 1/3 × 3.14 × 32
Vcone = 1/3 × 100.48
Vcone = 1/3 × 100.48/1
Vcone = (1 × 100.48)/(3 × 1)
Vcone = 100.48/3
Vcone = 33.49 cm3
http://www.basic-mathematics.com/volume-of-a-cone.html
Volume of a CylinderA cylinder with radius r units and height h units has a volume of Vcubic units given by
Example 27Find the volume of a cylindrical canister with radius 7 cm and height of 12
Volume enclosed by a cylinder
Definition: The number of cubic units that will exactly fill a cylinder
Try this Drag the orange dot to resize the cylinder. The volume is calculated as you drag.
See also: Surface area of a cylinder
How to find the volume of a cylinder Although a cylinder is technically not a prism, it shares many of the properties of a prism. Like prisms, the volume is found by multiplying the area of one end of the cylinder (base) by its height.
Since the end (base) of a cylinder is a circle, the area of that circle is given by
Multiplying by the height h we get
where:
π is Pi, approximately 3.142
r is the radius of the circular end of the cylinder
h height of the cylinder
Calculator
Some notes on the volume of a cylinderRecall that a cylinder is like an empty soup can. It has nothing inside, and the walls of the can have zero thickness. So strictly speaking, the cylinder has zero volume.
The strictly correct way of saying it is "the volume enclosed by a cylinder" - the amount of soup it holds. But many textbooks simply say "the volume of a cylinder" to mean the same thing. What they usually mean when they say this is the volume enclosed by the cylinder.
Oblique cylindersRecall that an is one that 'leans over' - where the top center is not over the base center point. In the figure above check "allow oblique' and drag the top orange dot sideways to see an oblique cylinder.
It turns out that the volume formula works just the same for these. You must however use the perpendicular height in the formula. This is the vertical line to left in the figure above. To illustrate this, check 'Freeze height'. As you drag the top of the cylinder left and right, watch the volume calculation and note that the volume never changes.
See Oblique Cylinders for a deeper discussion on why this is so.
UnitsRemember that the radius and the height must be in the same units - convert them if necessary. The resulting volume will be in those cubic units. So, for example if the height and radius are both in centimeters, then the volume will be in cubic centimeters.
Sample problems:Volume of a Cylinder formula:
V = Π r2h
(let pi = 3.14)
Find the volume of a cylinder having a radius of 5 cm and a height of 12 cm.
Solution:
Given:
r = 5 cm; h = 12 cm
The formula:
V = Π r2h
V = (3.14)(52)(12)
V = (3.14)(25)(12)
V = 942 cubic cm!
The volume of a cylinder is 1632 ft3. If the height of the cylinder is
24 ft., what is the area of it's base?
Solution:
This one is a little different.
We have been given the volume and the height,
and from this information we must determine
the area of the base.
The area of the base is the circle (Π r2)
V = Π r2h
Plug in the height:
1632 = (Π r2)(24)
If we divide both sides by the height (24)
it will leave the area of the base:
1632/24 = (Π r2)(24)/24
68 = Π r2
The area of the base is 68 sq. ft.!
A cylindrical swimming pool is 4.5 feet high and has a diameter of 24 feet. If the pool is filled right up to the top edge how many cubic feet of water can the pool hold?
Solution:
First, change the diameter (given) to a radius:
d = 24 ft....therefore.... r = 12 ft.
Plug in the values:
V = Π r2h
V = (3.14)(122)(4.5)
V = (3.14)(144)(4.5)
V = 2034.72 cubic feet of water!
A small can of soup has a radius of 3.5 cm and a height of 12 cm. A family size can has a radius of 5 cm and is 15 cm high. Which contains more soup, one family sized can, or two small cans?
Solution:
The small can:
r = 3.5 cm, h = 12 cm
V = (3.14)(3.52)(12)
V = (3.14)(12.25)(12)
V = 461.58 cubic cm
2 cans would have 2(461.58)
V = 923.16 cubic cm)
Family sized can:
r = 5 cm, h = 15 cm
V = (3.14)(52)(15)
V = (3.14)(25)(15)
V = 1177.5 cubic cm
Now subtract:
1177.5 - 923.16
Difference = 254.34
1 family size can is 254.34 cu. cm larger
than 2 small cans!! Surprised??!!
A coffee grower is going to package his coffee in cylindrical cans which will hold exactly 785 cubic inches of his product. If the cans are to be 10 inches in height, what must the radius of the can be?
Solution:
Plug in the information we know:
785 = (3.14)(r2)(10)
Since multiplication is commutative, reorder the factors:
785 = (3.14)(10(r2)
785 = (31.4)(r2)
Divide both sides by 31.4 to isolate the r2
785/31.4 = (31.4)(r2)/31.4
25 = r2
If r2 = 25, then r = 5
The radius of the can must be 5 cm!
http://www.studyzone.org/mtestprep/math8/g/volcylprac.cfm
Sphere:
A sphere is 3-dimensional (solid) figure in which all of it's points are the same distance from one fixed point in space, which is called it's center.
Don't get confused! A sphere is not a circle. A circle is 2 - dimensional, or flat. A circle is also known as a plane figure.
A line segment which joins any point on the surface of the sphere with it's center is called it's radius.
The Volume formula:
Unlike many other formulas for finding the volume of a 3-dimensional, or solid figure, we need only one dimension to calculate the volume of a sphere....it's radius!
The formula for the volume of a sphere in which it's radius is "r" is:
V = 4/3 Π r3
For all of the model problems below let Π = 3.14
Model ProblemsLet's start with an easy example.
An official handball has a diameter of 4.8 centimeters. Find it's volume.
First you must recognize that the dimension you have been given is the diameter of the sphere. However, as was discussed earlier in this lesson, the dimension
needed is the radius.
That's easy to fix!
Simply cut the diameter in half (or divide by 2).
If d = 4.8, then
r = (4.8)/2 = 2.4!
Then, plug the value of "r" into the formula:
V = 4/3 Π r3
V = 4/3 (3.14)(2.4)3
V = 4/3(3.14)(13.824)
V = 4/3(43.40736)
The easiest way to multiply a number like this
by a fraction is simply multiply the number (43.40736)
by 4 (the numerator), and then divide
the result by 3 (the denominator).
First multiply by 4
4(43.40736)
173.62944
Then divide by 3
(173.62944)/3
57.87648
V = 57.87648 cubic centimeters!
Got the idea? Let's try another..
A red rubber ball needs air before the class plays kickball. After it has been properly inflated for the game, it's radius has increased from 8 inches to 11 inches in length.
What was the change in the ball's volume, to the the nearest cubic inch?
To answer this problem we will need to find the volume twice. Once for the ball in it's deflated form (radius = 8"),
then a second time after it has been filled
with air (radius = 11 inches).
Then to find the change we will subtract the 2 volumes.
Deflated ball:
V = 4/3Π r3
V = 4/3 (3.14)(8)3
V = 4/3(3.14)512)
V = 4/3(1607.68)
V = 2143.573333
V = 2144 cubic inches.
Inflated ball:
V = 4/3(3.14)(11)3
V = 4/3(3.14)(1331)
V = 4/3(4179.34)
V = 5572.453333
V = 5572 cubic inches
The change in volume:
5572 - 2144
3428 cubic inches!
Now let's look at one more..
A ball having a diameter of 10 cm is being shipped in a box shaped like a cube whose height, width, and length are also exactly 10 cm.. How many cubic centimeters of packing material are needed to hold the ball firmly in place inside the box?
Can you picture the problem? Can you see a ball inside a cube, so that when the top is closed the ball fits snugly in the box, but there are these areas of open space where the ball and box don't touch. That's the volume we will be solving for.
First let's find the volume of the cube.
A cube is a rectangular prism in which all of the dimensions are the same. In this case l = 10, w = 10 and h = 10.
V = (l)(w)(h)
V = (10)(10)(10)
V = 1000 cubic cm.
Now for the volume of the ball:
d = 10, so r = 5
V = 4/3 Π r3
V = 4/3(3.14)(5)3
V = 4/3(3.14)(125)
V = 4/3(392.5)
V = 523.333... cubic cm
The amount of "empty space" is the difference
between these 2 volumes.
Empty space = 1000 - 523 (to nearest cubic cm)
Empty space = 477 cubic cm!
Did that answer surprise you??
If you think about it, the ball only fills up a little bit more than half of the volume of the cube!
http://www.studyzone.org/mtestprep/math8/g/volsphereprac.cfm
How to find the volume of a cubeRecall that a cube has all edges the same length (See Cube definition). The volume of a cube is found by multiplying the length of any edge by itself twice. So if the length of an edge is 4,the volume is 4 x 4 x 4 = 64
Or as a formula:
volume = s3where:
s is the length of any edge of the cube. Calculator
In the figure above, drag the orange dot to resize the cube. From the edge length shown, calculate the volume of the cube and verify that it agrees with the calculation in the figure.
When we write volume = s3, strictly speaking this should be read as "s to the power 3", but because it is used to calculate the volume of cubes it is usually spoken as "s cubed".
Some notes on the volume of a cubeRecall that a cube is like an empty box. It has nothing inside, and the walls of the box have zero thickness. So strictly speaking, the cube has zero volume. When we talk about the volume of a cube, we really are talking about how much liquid it can hold, or how many unit cubes would fit inside it.
Think of it this way: if you took a real, empty metal box and melted it down, you would end up with a small blob of metal. If the box was made of metal with zero thickness, you would get no metal at all. That is what we mean when we say a cube has no volume.
The strictly correct way of saying it is "the volume enclosed by a cube" - the amount space there is inside it. But many textbooks simply say "the volume of a cube" to mean the same thing. However, this is not strictly correct in the mathematical sense. What they usually mean when they say this is the volume enclosed by the cube.
UnitsRemember that the length of an edge and the volume will be in similar units. So if the edge length is in miles, then the volume will be in cubic miles, and so on.
Volume of a cube
Given the length of one side, call it a, the volume of a cube can be found by using the following formula:
Vcube = a3 = a × a × a
Example #1:
Find the volume if the length of one side is 2 cm
Vcube = 23
Vcube = 2 × 2 × 2
Vcube = 8 cm3
Example #2:
Find the volume if the length of one side is 3 cm
Vcube = 33
Vcube = 3 × 3 × 3
Vcube = 27 cm3
Example #2:
Find the volume if the length of one side is 3/2 cm
Vcube = (3/2)3
Vcube = 3/2 × 3/2 × 3/2
Vcube = (3 × 3 × 3)/(2 × 2 × 2) Vcube = 27/8 Vcube = 3.375 cm3
http://www.basic-mathematics.com/volume-of-a-cube.html
Volume of a Triangular Prism - Lesson
Topic Index | Grade 7 Math | Intermediate Test Prep |StudyZone
When finding the volume of a right triangular prism we need to use the formula:
Remember... "w" stands for width, "h" stands for height and "l" stands for length.
Let's try some!!!
Find the volume of this triangular prism. Use the formula:
You may use the calculator to help with the math!
(Remember that 1/2 is the same as .5)
0789/456*123-0+/-.+C
=
Answer...
V = 864 cm³
Volume of a Pyramid Formula
A pyramid is a polyhedron with a polygonal base and triangular faces equal to the number of sides in the base. All the triangular faces meet at a single point called the apex. The faces of the pyramid connect the bases with the apex. Volume of a Pyramid is the measure of the number of units occupied by the pyramid.
The Volume of a Pyramid Formula is given as,
Where
a - apothem length of the pyramid.
b - base length of the pyramid.
h - height of the pyramid.
Volume of a Pyramid ProblemsBack to Top
Some solved problems on the volume of a pyramid are given below:Solved ExamplesQuestion 1: Find the volume of a pentagonal pyramid of apothem length 5 cm, base length 7 cm and height 11 cm ?
Solution:
Given,
a = 5 cm
b = 7 cm
h = 11 cm
Volume of a pentagonal pyramid
= 56abh
= 56 * 5 cm * 7 cm * 11 cm
= 320.833 cm3
Question 2: Find the volume of a triangular pyramid of apothem length 7 cm, base length 13 cm and height 19 cm ?
Solution:
Given,
a = 7 cm
b = 13 cm
h = 19 cm
Volume of a triangular pyramid
= 16abh
= 16 * 7 cm * 13 cm * 19 cm
= 288.167 cm3
http://formulas.tutorvista.com/math/volume-of-a-pyramid-formula.html
Volume is measured in cubic units. Two common units of volume are the cubic centimeter () and the cubic inch (). The volume of a rectangular prism is a measure of how much space the object takes up.Volume is found using three dimensions: length, width, and height.
A cubic unit is 1 unit by 1 unit by 1 unit ().
A cubic centimeter is 1 cm by 1 cm by 1 cm ().
A cubic inch is 1 in by 1 in by 1 in ().
Example 1
A box measures 10 cm. long x 2 cm. wide x 4 cm. high.
The dimensions of a rectangular prism.
Its volume is the amount of space inside. By finding the product of the length, width, and height, you will find the number of 1cm cubes that will fit in the box.10 x 2 x 4 = 80 cubic centimeters, or 80 .
So, 80 1 cm. cubes will fit in the box.
Example 2
A box measures 6 in. long x 3 in. wide x 6 in. high.
Its volume is the amount of space inside. By finding the product of the length, width, and height, you will find the number of 1 in cubes that will fit in the box.6 x 3 x 6 = 108 cubic inches, or 108 .
So, 108 1 in. cubes will fit in the box.
Capacity
Liquids, sugar, salt, and so on can be measured by being poured into or out of a container. The volume of a container that is filled with a liquid or a solid that can be transferred is often its capacity.
Capacity is often measured in units such as fluid ounces, cups, pints, quarts, gallons, milliliters, and liters.
Remember: The formula for volume is V = l x w x h.
http://www.studyzone.org/mtestprep/math8/e/volumecapacity6l.cfm
Volume of a cone
Given the radius and h, the volume of a cone can be found by using the formula:
Formula: Vcone = 1/3 × b × h
b is the area of the base of the cone. Since the base is a circle, area of the base = pi × r2
Thus, the formula is Vcone = 1/3 × pi × r2 × h
Use pi = 3.14
Example #1:
Calculate the volume if r = 2 cm and h = 3 cm
Vcone = 1/3 × 3.14 × 22 × 3
Vcone = 1/3 × 3.14 × 4 × 3
Vcone = 1/3 × 3.14 × 12
Vcone = 1/3 × 37.68
Vcone = 1/3 × 37.68/1
Vcone = (1 × 37.68)/(3 × 1)
Vcone = 37.68/3
Vcone = 12.56 cm3
Example #2:
Calculate the volume if r = 4 cm and h = 2 cm
Vcone = 1/3 × 3.14 × 42 × 2
Vcone = 1/3 × 3.14 × 16 × 2
Vcone = 1/3 × 3.14 × 32
Vcone = 1/3 × 100.48
Vcone = 1/3 × 100.48/1
Vcone = (1 × 100.48)/(3 × 1)
Vcone = 100.48/3
Vcone = 33.49 cm3
http://www.basic-mathematics.com/volume-of-a-cone.html