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The volume of a solid is the amount the shape contains. Volume is a measure of capacity and is measure in cubic units. Volume of a Rectangular Prism. To ...
Typology: Lecture notes
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In this unit, you will use different formulas and procedures to find the volume and surface area of many solids. It may be helpful to draw some of the shapes as nets and write the formula or procedure to solve them below the drawing. You may use a calculator to help with the computations. Use the calculator carefully and double check all calculations.
Surface Area
Volume
Using Precision When Calculating Measurements
When asked to find surface area, find the areas of the faces of the solid by using formulas previously discovered, and then add the areas of the faces together to find the total. Surface is measured in square units.
Surface Area of a Rectangular Prism
Example 1 : Compute the surface area of a rectangular prism with a width of four feet, a length of seven feet, and a height of five feet.
Square Unit
Surface Area of a Cube
A cube is a special rectangular prism with all of its edges measuring the same and all of its faces having the same area. The surface area of a cube is the total area of all of the square faces measured in square units.
The area of one face of a cube is a square.
Let e represent the length of one edge.
Then, the area of one face can be represent by e × e or e^2.
Since a cube has six faces, the total surface area of a cube is 6 × e^2.
“T” will be used to represent surface area in the formulas developed in this unit.
The formula for the surface area of a cube is:
The surface area ( T ) of a cube is:
T = 6 e^2
e
Square Unit
Example 2 : Find the surface area of a cube with an edge measuring seven inches.
2 2
T e T T T
Check: Area of one face is 7 × 7 or 49 square inches.
Area of six faces is 49 × 6 or 294 square inches.
The formula to find the area of a rectangle is A = lw.
Rectangle 1 : The largest rectangular face measures 15 inches by 9 inches.
135 sq in
A l w A A
Rectangle 2 : The rectangular face on which the prism is resting measures 15 inches by 7 inches.
105 sq in
A l w A A
Rectangle 3 : The back rectangular face measures 15 inches by 5 inches.
75 sq in
A l w A A
Step 3 : Find the total area of all five faces.
2
2 Triangular Areas 3 Rectangular Areas 2(17.5) 135 105 75 350 in
The surface area of the triangular prism is 350 square inches.
15 in
12 in
Height of Pyramid Slant Height
12 in
Surface Area of a Pyramid
To find the surface area of a pyramid, again, just add the areas of the faces of the solid. Notice that the sides of a pyramid are triangles.
Please pay particular close attention to the difference between the height of a pyramid and the height along the side of the pyramid which is called the slant height.
*The slant height is used when calculating surface area, not the height of the pyramid itself.
Example 4: Find the surface area of a pyramid with a square base measuring 12 inches on one edge and a slant height of 15 inches.
Step 1 : Find the area of the base.
144 sq in
A bh A A
Step 2 : Find the area of the triangular faces.
Since each triangular face has a base of 12 inches and a height of 15 inches, the areas of all four triangles are the same.
Find the area of one triangle, and then multiply by four.
1 Area of a triangle 2 1 (12)(15) Substitute (Use slant height as height.) 2 90 Simplify
A bh
Surface Area of a Cylinder
To find the surface area of a cylinder, a little more thinking is involved.
The top and bottom of a cylinder are circles.
The side of a cylinder is one continuous curved surface. When it is laid flat, it is shaped like a rectangle.
Notice that the length of the rectangle is the same as the circumference (distance around) of the circular base.
The area of the rectangular face is determined by multiplying the circumference of the base circle (length of the rectangle) by the height of the cylinder (width of the rectangle).
The formula for the surface area of a cylinder can be developed as follows:
2 2 2 2 2
Circular Face Circular Face Curved Surface (Rectangle)
T r r C h T r r r h T r rh
h (^) =
r
h
r = 3 in
8 in
Example 5: Find the surface area of a cylinder that has a radius of three inches and a height of eight inches.
Step 1 : Write the formula for the surface area of a cylinder.
Step 2 : Substitute the given information in the formula and simplify.
2 2
T r rh T T T T
The surface area of the cylinder is 207.24 square inches.
The surface area ( T ) of a cylinder is:
T = 2πr^2 + 2 πrh
Step 2 : Substitute the given information in the formula and simplify.
2
(3.14)(8 )^2 (3.14)(8)(12) (3.14)(64) (3.14)(8)(12) 200.96 301.
T r rl T T T T
The surface area of the cone is 502.4 square centimeters.
12 cm
8 cm
Surface Area of a Sphere
A sphere is a three-dimensional figure with all points the same distance from a fixed point, the center.
A hemisphere is half of a sphere that is created by a plane that intersects a sphere through its center.
The edge of a hemisphere is a great circle.
The surface area of a sphere is four times the base area , the area of the circle created by the great circle.
2
Surface area 4 Base Area T 4 π r
Base Area
A = πr^2 r
r
Great Circle
The surface area ( T ) of a sphere is:
T = 4πr^2
The volume of a solid is the amount the shape contains. Volume is a measure of capacity and is measure in cubic units.
Volume of a Rectangular Prism
To calculate the volume of a rectangular prism , multiply the area of the base (length × width) times height.
Example 1 : Compute the volume of a square prism with a base area of 25 square feet and a height of 9 feet.
*A square prism is a prism with a square base.
Since the base area is given, simply multiply the area of the base times the height to calculate the volume.
The volume of the square prism is 225 cubic feet.
V = Area of base × Height V = 25 × 9 V = 225
25 ft 2
9 ft
V = Area of Base × Height V = (length × width) × height V = l × w × h
h
l
w
Example 2 : Calculate the volume of a rectangular prism with a length of 11 feet, width of 8 feet, and a height of 23 feet.
Since all dimensions of the prism are given, use the volume formula for a rectangular prism to calculate the volume.
The volume of the rectangular prism is 2024 cubic feet.
V = l × w × h V = 11 × 8 × 23 V = 2024
The volume of a cylinder is the amount a cylinder can hold and is measured in cubic units. To calculate the volume of a cylinder, multiply the area of the base times the height.
Example 4 : Find the volume of a cylinder with a radius of 12 centimeters and the height is 23 centimeters. (Use 3.14 for “pi”.)
V = π × r^2 × h
V = 3.14 × 122 × 23
V = 3.14 × 144 × 23
The volume of the cylinder is 10,399.68 cubic centimeters (cm^3 ).
V = Area of Base × Height
The base of a cylinder is a circle. To calculate the area of a circle, use A = π × r^2.
V = (π × r^2 ) × h
h
Area of Base
Example 5 : Compute the volume of a cylinder with a base area of 420 square centimeters and a height of 14 centimeters.
Since the base area is given, simply multiply the area of the base times the height to calculate the volume.
The volume of the cylinder is 5880 cubic centimters.
V = Area of base × Height V = 420 × 14 V = 5880
14 cm
420 cm^2
