Unit 5, Lesson 19
Estimating a Hemisphere
Grade 8
19.1
Warm-up
Notice and Wonder: Two Shapes
Here are two shapes. What do you notice? What do you wonder?
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Here are two shapes. What do you notice? What do you wonder?
A hemisphere with radius 5 units fits snugly into a cylinder of the same radius and height.
A cone fits snugly inside a hemisphere, and they share a radius of 5.
We can estimate the volume of a hemisphere by comparing it to other shapes for which we know the volume. For example, a hemisphere of radius 1 unit fits inside a cylinder with a radius of 1 unit and height of 1 unit.
Since the hemisphere is inside the cylinder, it must have a smaller volume than the cylinder, making the cylinder's volume a reasonable overestimate for the volume of the hemisphere.
The volume of this particular cylinder is about 3.14 cubic units since
Using similar logic, a cone of radius 1 unit and height 1 unit fits inside of the hemisphere of radius 1 unit.
Since the cone is inside the hemisphere, the cone must have a smaller volume than the hemisphere, making the cone's volume a reasonable underestimate for the volume of the hemisphere.
The volume of this particular cone is about 1.05 cubic units since
Averaging the volumes of the cylinder and the cone, we can estimate the volume of the hemisphere to be about 2.10 cubic units since