Which, if either, of these solids has the same volume as the pyramid?
A
B
16.2
Activity
Calculate the volume of each solid. Round your answers to the nearest tenth if necessary.
A
B
C
D: height 12 cm; area of base 32 cm2
A particular cone has radius and height .
Write an expression for the volume of this cone in terms of and .
What is the height of a cone whose volume is cubic units and whose radius is 3 units?
What is the radius of a cone whose volume is cubic units and whose height is 3 units?
16.3
Activity
A caterer is making an ice sculpture in the shape of a pyramid for a party. The caterer wants to use 11 liters of water, which makes about 720 cubic inches of ice. The sculpture must fit on a table with space around it for the food. The caterer needs to decide how large to make the base, which can be any shape. Draw and label the dimensions of 2 different pyramids that would work.
Student Lesson Summary
We can work backward from a given volume to find possible dimensions for a cone or pyramid.
Suppose we want to find dimensions for a cone so it has a volume of cubic inches. Start by substituting the volume into the pyramid volume formula to get . The base of a cone is a circle, so we can write . Multiply both sides of the equation by 3 and divide both sides by to get .
Now consider different possible values for and . If we can find a perfect square that divides evenly into 2,700, we can set the square root of that number to be the radius. The number 25 is a perfect square and divides into 2,700, so choose . Now . This tells us that if the pyramid’s radius is 5 inches, its height is 108 inches because .
These aren’t the only possible values. Suppose we set the radius to be 20 inches. Substitute this into the original equation and rearrange to find the value of .
A height of 6.75 inches together with a radius of 20 inches gives the cone a volume of cubic inches.
