Lesson 19

Estimating a Hemisphere

Lesson Narrative

The purpose of this lesson is to introduce students to working with spheres by using shapes they are now familiar with—cubes, cones, and cylinders—to estimate the volume of a hemisphere. In the previous lesson, students saw that changing the radius of a cone by a factor of $$a$$ scales the volume by a factor of $$a^2$$. Here, the connection between spheres and cubes is made in the first activity to help them build understanding for why changing the radius of a sphere (or, in the case of this lesson, hemisphere) by a factor of $$a$$ changes the volume by a factor of $$a^3$$. For example, think about a cube with side length 2 units that has a volume of $$\displaystyle 2^3 = 8.$$ If the length, width, and height are all scaled by a factor of $$a$$, then each edge length would be $$2a$$ units and the new volume would be $$\displaystyle (2a)^3=8a^3$$ cubic units, which is $$a^3$$ times the original volume. The first activity sets the expectation that spheres work the same way, so that the $$r^3$$ in the formula for the volume of a sphere of radius $$r$$, given in the next lesson, makes sense.

In the second activity, students fit a hemisphere inside a cylinder, and use the volume of the cylinder to make an estimate of the volume of the hemisphere. Then they do the same thing with a cone that fits inside the hemisphere. The volume of the hemisphere has to be between the volume of the cone and the volume of the cylinder, both of which students can calculate from work in previous lessons. So this activity gives a range of possibilities for volume of the hemisphere. In the next lesson, students will see the exact formula.

Learning Goals

Teacher Facing

• Calculate the volume of a cylinder and cone with the same radius and height, and justify (orally and in writing) that the volumes are an upper and lower bound for the volume of a hemisphere of the same radius.
• Estimate the volume of a hemisphere using the formulas for volume of a cone and cylinder, and explain (orally) the estimation strategy.

Student Facing

Let’s estimate volume of hemispheres with figures we know.

Required Preparation

If possible, have some physical examples of hemispheres on hand for students to see. Examples could be glass paperweights or dome lids. Alternatively, have a sphere, such as a globe or basketball, with a marked equator to clearly divide it into two hemispheres.

Student Facing

• I can estimate the volume of a hemisphere by calculating the volume of shape I know is larger and the volume of a shape I know is smaller.