Inroduction to Volume Word Problems
… treat Nature by the sphere, the cylinder and the cone …
—PAUL CÉZANNE (1839–1906)
This lesson will explain each of the volume formulas for common geometric solids, and how to apply these formulas to word problems with these figures.
Volume
The volume of a three-dimensional figure is the amount of space in the figure.
Many volume formulas can be summarized as the area of the base multiplied by the height, or V = Bh.
Tip:The volume of any figure is expressed in cubic units, or units^{3}. |
Volume of a Rectangular Prism
A rectangular prism is a three-dimensional solid whose faces are all rectangles. The formula for the volume of a rectangular solid is Volume = length × width× height, or V = l × w × h.
Example
What is the volume of a rectangular prism with a height of 10 m, a width of 14 m, and a height of 20 m?
Read and understand the question. This question is looking for the volume of a rectangular prism when the three dimensions are given.
Make a plan. Use the formula V = l × w × h, and substitute the given values. Then, evaluate the formula.
Carry out the plan. The volume is V = 10 × 14 × 20 = 2,800 m^{3}
Check your answer. To check this solution, divide the volume by two of the dimensions to check if the result is the third dimension: 2,800 ÷ 10 = 280; 280 ÷ 14 = 20, which is the remaining dimension. This answer is checking.
Volume of a Cube
A cube is a special type of rectangular prism where each face is the same shape and size. A cube has faces that are all congruent squares, so each edge is the same length. The volume of a cube can be found by using the formula for rectangular prisms, but it can also be found by using the formula V = e^{3}, where e is the measure of an edge of the cube.
Example
The measure of the edge of a cube is 7 cm. What is the volume of the cube?
Read and understand the question. This question is looking for the volume of a cube when the measure of an edge of the cube is known.
Make a plan. Use the formula V = e^{3}, and substitute the given value of e. Then, evaluate the formula.
Carry out the plan. The volume formula becomes V= (7)^{3}= 7 × 7 × 7 = 343 cm^{3}.
Check your answer. To check this solution, divide the volume by the measure of the edge of the cube two times to see if the result is also the measure of the edge:
- 343 ÷ 7 = 49
- 49 ÷ 7 = 7
which is the measure of the edge of the cube. This answer is checking.
Volume of a Triangular Prism
A triangular prism is a three-dimensional solid with triangles as the bases and rectangles as the lateral faces. An example of a triangular prism is shown next.
The formula for the volume of a triangular prism can be found by using the formula V = Bh, where B is the area of the base and h is the height of the prism.
Example
What is the volume of a triangular prism with a base area of 12 m^{2}and a height of 4 m?
Read and understand the question. This question is asking for the volume of a triangular prism when the area of each base and the height of the prism are given.
Make a plan. Use the formula V = Bh, where B is the area of the base and h is the height of the prism. Substitute the known values and evaluate the formula.
Carry out the plan. The formula becomes V= 12 × 4 = 48. The volume is 48 m^{3}.
Check your answer. To check this problem, divide the volume of the prism by the height, and check to see if the result is the area of the base: 48 ÷ 4 = 12, which is the area of the base. This answer is checking.
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