Unit 5, Lesson 9
Cylinder Volumes
Geometry
9.1
Warm-up
Here are two solids.
- What information would you need to calculate the volume of each solid?
- What is the same and different about how you would find the volume of each solid?
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Here are two solids.
Here are two containers. All measurements are in centimeters.
Suppose each two-dimensional figure is rotated around the vertical axis shown. Each small square in the grid represents 1 square centimeter.
AA grid. A vertical line is drawn 3 units to the right. A six sided figure is shaded with points at 3 units right, 1 unit up. Another point at 8 units right, 1 unit up. Another point at 8 units right, 4 units up. Another point at 5 units right, 4 units up. Another point at 5 units right, 7 units up. Another point at 3 units right, 7 units up.
BA grid. A vertical line is drawn 3 units to the right. A four sided figure is shaded with points at 6 units right, 1 unit up. Another point 8 units right, 1 unit up. Another point 8 units right, 7 units up. Another point 6 units right, 7 units up.
For each solid:
Cylinder and prism volumes can be found by multiplying the area of the figure’s base by its height. The formula , where represents volume, is the area of the base, and is height, captures this concept. Consider the solid formed by rotating this rectangle around the horizontal axis shown. The result is a hollow cylinder of height 5 units with inner radius 1 unit and outer radius 4 units.
A grid. A horizontal line is drawn 1 unit up. A four sided figure is shaded with points at 4 units right, 2 units up. Another point 4 units right, 5 units up. Another point 9 units right, 2 units up. Another point 9 units right and 5 units up.
A cylinder with the middle cylinder missing. The height is 5. The distance between the outside of the cylinder and interior cylinder is 3. The radius of the interior cylinder is 1.
To calculate the volume of the outer cylinder, start by finding the area of the circular base. The circle’s radius measures 4 units, so its area is square units because . Multiply that by the cylinder’s height of 5 units to get cubic units.
For the inner cylinder, the area of the base is square units, because . The volume is therefore cubic units. Now subtract the volume of the inner, hollow part from the volume of the outer cylinder to get the volume of the solid: cubic units because .