Unit 1, Lesson 15
Building a Volume Formula for a Pyramid
Integrated Math 3
15.1
Warm-up
Two solids are shown.
A
B
For each solid, draw and label a prism or cylinder that has a base congruent to the solid’s and a height equal to the solid’s.
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Two solids are shown.
A
B
For each solid, draw and label a prism or cylinder that has a base congruent to the solid’s and a height equal to the solid’s.
Here is a triangular prism.
Suppose we split the prism into pyramids like the ones you built earlier. The first pyramid is split off by slicing through points , , and . The remaining part of the prism is sliced through , , and .
P1Triangular based pyramid on
axis scale negative 1 to 5 by 1’s. Points E(0 comma 4 comma 0), D(0 comma 4 comma 1) and F(3 comma 4 comma 0) form the triangular base. Point C(3 comma 0 comma 0) is the vertex of the pyramid.
P2Triangular based pyramid on x y z coordinate plane, origin O. x, y and z axis scales negative 1 to 5 by 1’s. Points A(0 comma 0 comma 2), O(0 comma 0 comma 0) and C(3 comma 0 comma 0) form the triangular base. Point D(0 comma 4 comma 1) is the vertex of the pyramid.
P3Triangular based pyramid on
axis scale negative 1 to 5 by 1’s. Points A(0 comma 0 comma 2), O(0 comma 0 comma 0) and C(3 comma 0 comma 0) form the triangular base. Point D(0 comma 4 comma 1) is the vertex of the pyramid.
Each solid in the image has a height of 6 units. The area of each solid’s base is 10 square units. A cross-section has been created in each by dilating the base using the apex as a center with a scale factor of .
Calculate the volume of each of the solids.
We can use relationships between pyramids and prisms to build a formula for the volume of a pyramid. The image shows three square pyramids assembled into a cube. All three pyramids have the same volume so their volume must be one-third the volume of the cube.
More generally, a triangular pyramid has one-third the volume of the prism that has the same height and a base congruent to the pyramid’s.
Suppose we have a pyramid that doesn’t have a triangular base. Call the area of its base and its height . To understand how to find the volume of such a solid, think about a triangular pyramid that also has an area of square units and a height of .
Now find a cross-section of each solid by dilating the solid’s base using the apex as a center with some scale factor between 0 and 1. The area of the cross-section in both solids will be . This is true for any value of . Because the cross-sections at all heights have equal area, the solids have the same volume. The same idea would apply to a cone. This means that the expression gives the volume of any pyramid or cone.