Surface area is the total area of the faces and curved surface of a solid figure. Mathematical description of the surface area is considerably more involved than the definition of arc length or polyhedra (objects with flat polygonal faces) the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of the surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
http://en.wikipedia.org/wiki/Surface_area
Surface area is the area of a given surface. Roughly speaking, it is the "amount" of a surface (i.e., it is proportional to the amount of paint needed to cover it), and has units of distance squared. Surface area is commonly denoted for a surface in three dimensions, or for a region of the plane (in which case it is simply called "the" area
http://mathworld.wolfram.com/SurfaceArea.html
http://en.wikipedia.org/wiki/Surface_area
Surface area is the area of a given surface. Roughly speaking, it is the "amount" of a surface (i.e., it is proportional to the amount of paint needed to cover it), and has units of distance squared. Surface area is commonly denoted for a surface in three dimensions, or for a region of the plane (in which case it is simply called "the" area
http://mathworld.wolfram.com/SurfaceArea.html
Surface area of a cube
To derive the formula of the surface area of a cube, do the followings:
Start with a cube as shown below and call the length of one side a:
In order to make a cube like the one shown above, you basically use the following cube template:
Looking at the cube template, it is easy to see that the cube has six sides and each side is a square
The area of one square is a × a = a2
Since there are six sides, the total surface area, call it SA is:
SA = a2 + a2 + a2 + a2 + a2 + a2
SA = 6 × a2
Example #1:
Find the surface area if the length of one side is 3 cm
Surface area = 6 × a2
Surface area = 6 × 32
Surface area = 6 × 3 × 3
Surface area = 54 cm2
Example #2:
Find the surface area if the length of one side is 5 cm
Surface area = 6 × a2
Surface area = 6 × 52
Surface area = 6 × 5 × 5
Surface area = 150 cm2
Example #3:
Find the surface area if the length of one side is 1/2 cm
Surface area = 6 × a2
Surface area = 6 × (1/2)2
Surface area = 6 × 1/2 × 1/2
Surface area = 6 × 1/4
Surface area = 6/4 cm2
Surface area = 3/2 cm2
Surface area = 1.5 cm2
Surface area of a square pyramid
It is not complicated to derive the formula of the surface area of a square pyramid.
Start with a square pyramid as shown below and call the length of the base s and the height of one triangle l
l is the slant height. It is not for no reason this height is called slant height!
The word slant refers also to something that is oblique or bent, or something that is not vertical or straight up. Basically, anything that is not horizontal or vertical!
The area of the square is s2
The area of one triangle is (s × l)/2
Since there are 4 triangles, the area is 4 × (s × l)/2 = 2 × s × l
Therefore, the surface area, call it SA is:
SA = s2 + 2 × s × l :
Example #1:
Find the surface area of a square pyramid with a base length of 5 cm, and a slant height of 10 cm
SA = s2 + 2 × s × l
SA = 52 + 2 × 5 × 10
SA = 25 + 100
SA = 125 cm2
Example #2:
Find the surface area with a base length of 3 cm, and a slant height of 2 cm
SA = s2 + 2 × s × l
SA = 32 + 2 × 3 × 2
SA = 9 + 12
SA = 21 cm2
Example #3:
Find the surface area with a base length of 1/2 cm, and a slant height of 1/4 cm
SA = s2 + 2 × s × l
SA = (1/2)2 + 2 × 1/2 × 1/4
SA = 1/4 + 1 × 1/4
SA = 1/4 + 1/4
SA = 2/4
SA = 1/2 cm2
http://www.basic-mathematics.com/surface-area-of-a-square-pyramid.html
It is not complicated to derive the formula of the surface area of a square pyramid.
Start with a square pyramid as shown below and call the length of the base s and the height of one triangle l
l is the slant height. It is not for no reason this height is called slant height!
The word slant refers also to something that is oblique or bent, or something that is not vertical or straight up. Basically, anything that is not horizontal or vertical!
The area of the square is s2
The area of one triangle is (s × l)/2
Since there are 4 triangles, the area is 4 × (s × l)/2 = 2 × s × l
Therefore, the surface area, call it SA is:
SA = s2 + 2 × s × l :
Example #1:
Find the surface area of a square pyramid with a base length of 5 cm, and a slant height of 10 cm
SA = s2 + 2 × s × l
SA = 52 + 2 × 5 × 10
SA = 25 + 100
SA = 125 cm2
Example #2:
Find the surface area with a base length of 3 cm, and a slant height of 2 cm
SA = s2 + 2 × s × l
SA = 32 + 2 × 3 × 2
SA = 9 + 12
SA = 21 cm2
Example #3:
Find the surface area with a base length of 1/2 cm, and a slant height of 1/4 cm
SA = s2 + 2 × s × l
SA = (1/2)2 + 2 × 1/2 × 1/4
SA = 1/4 + 1 × 1/4
SA = 1/4 + 1/4
SA = 2/4
SA = 1/2 cm2
http://www.basic-mathematics.com/surface-area-of-a-square-pyramid.html
Hexagonal Pyramid
Definition of Hexagonal Pyramid
How many edges does a hexagonal pyramid have?
Choices:
A. 12
B. 6
C. 8
D. 10
Correct Answer: A
Solution:
Step 1: There are 12 edges that are formed by joining 7 vertices in the hexagonal pyramid.
Step 2: So, the hexagonal pyramid has 12 edges.
Related Terms for Hexagonal Pyramid
Definition of Hexagonal Pyramid
- A pyramid with a hexagonal base is called to be a Hexagonal Pyramid.
- A hexagonal pyramid has six triangular lateral faces and a hexagonal base.
- Hexagonal pyramid is also called as Heptahedron.
- Surface Area (SA) of a regular hexagonal pyramid is given by SA = 3as+ 3sl, where a is the apothem length of the hexagon, s is the side length of the hexagon, and l is the slant height.
- Volume (V) of a regular hexagonal pyramid is given by V = h, where s is the side length of the hexagon and h is its height.
- There is one hexagonal base and 6 triangular faces for the figure shown. Therefore, it is a hexagonal pyramid.
How many edges does a hexagonal pyramid have?
Choices:
A. 12
B. 6
C. 8
D. 10
Correct Answer: A
Solution:
Step 1: There are 12 edges that are formed by joining 7 vertices in the hexagonal pyramid.
Step 2: So, the hexagonal pyramid has 12 edges.
Related Terms for Hexagonal Pyramid
- Base
- Face
- Hexa
- Lateral Faces
- Pyramid
- Triangle
- www.icoachmath.com/math_dictionary/Hexagonal_Pyramid.html
Surface Area of a CylinderSince a cylinder is closely related to a prism, the formulas for their surface areas are related.
Remember the formulas for the lateral surface area of a prism is ph and the total surface area is ph + 2B. Since the base of a cylinder is a circle, we substitute 2πr for p and πr2 for B where r is the radius of the base of the cylinder.
So, the formula for the lateral surface area of a cylinder is L. S. A. = 2πrh.
Example 1:
Find the lateral surface area of a cylinder with a base radius of 3 inches and a height of 9 inches.
L. S. A. = 2π(3)(9) = 54π inches2
≈ 169.64 inches2
The general formula for the total surface area of a cylinder is T. S. A. = 2πrh + 2πr2.
Example 2:
Find the total surface area of a cylinder with a base radius of 5 inches and a height of 7 inches.
T. S. A. = 2π(5)(7) + 2π(5)2 = 70π + 50π
= 120π inches2
≈ 376.99 inches2
http://hotmath.com/hotmath_help/topics/surface-area-of-a-cylinder.html
Remember the formulas for the lateral surface area of a prism is ph and the total surface area is ph + 2B. Since the base of a cylinder is a circle, we substitute 2πr for p and πr2 for B where r is the radius of the base of the cylinder.
So, the formula for the lateral surface area of a cylinder is L. S. A. = 2πrh.
Example 1:
Find the lateral surface area of a cylinder with a base radius of 3 inches and a height of 9 inches.
L. S. A. = 2π(3)(9) = 54π inches2
≈ 169.64 inches2
The general formula for the total surface area of a cylinder is T. S. A. = 2πrh + 2πr2.
Example 2:
Find the total surface area of a cylinder with a base radius of 5 inches and a height of 7 inches.
T. S. A. = 2π(5)(7) + 2π(5)2 = 70π + 50π
= 120π inches2
≈ 376.99 inches2
http://hotmath.com/hotmath_help/topics/surface-area-of-a-cylinder.html
Surface Area of a ConeThe total surface area of a cone is the sum of the area of its base and the lateral (side) surface.
The lateral surface area of a cone is the area of the lateral or side surface only.
Since a cone is closely related to a pyramid, the formulas for their surface areas are related.
Remember, the formulas for the lateral surface area of a pyramid is and the total surface area is .
Since the base of a cone is a circle, we substitute 2πr for p and πr2 for B where r is the radius of the base of the cylinder.
So, the formula for the lateral surface area of a right cone is L. S. A. = πrl, where l is the slant height of the cone.
Example 1:
Find the lateral surface area of a right cone if the radius is 4 cm and the slant height is 5 cm.
L. S. A. = π(4)(5) = 20π ≈ 62.82 cm2
The formula for the total surface area of a right cone is T. S. A. = πrl + πr2.
Example 2:
Find the total surface area of a right cone if the radius is 6 inches and the slant height is 10 inches.
T. S. A. = π(6)(10) + π(6)2
= 60π + 36π
= 96π inches2
≈ 301.59 inches2
http://hotmath.com/hotmath_help/topics/surface-area-of-a-cone.html
The lateral surface area of a cone is the area of the lateral or side surface only.
Since a cone is closely related to a pyramid, the formulas for their surface areas are related.
Remember, the formulas for the lateral surface area of a pyramid is and the total surface area is .
Since the base of a cone is a circle, we substitute 2πr for p and πr2 for B where r is the radius of the base of the cylinder.
So, the formula for the lateral surface area of a right cone is L. S. A. = πrl, where l is the slant height of the cone.
Example 1:
Find the lateral surface area of a right cone if the radius is 4 cm and the slant height is 5 cm.
L. S. A. = π(4)(5) = 20π ≈ 62.82 cm2
The formula for the total surface area of a right cone is T. S. A. = πrl + πr2.
Example 2:
Find the total surface area of a right cone if the radius is 6 inches and the slant height is 10 inches.
T. S. A. = π(6)(10) + π(6)2
= 60π + 36π
= 96π inches2
≈ 301.59 inches2
http://hotmath.com/hotmath_help/topics/surface-area-of-a-cone.html