Unit 6, Lesson 19
Estimating a Hemisphere
Accelerated 7
Preparation
Lesson Narrative
The purpose of this lesson is to introduce students to working with spheres by using shapes they are now familiar with—prisms, cones, and cylinders—to estimate the volume of a hemisphere. The work done in this lesson prepares students to reason about the formula for the volume of a sphere in a future lesson.
The Warm-up primes students to think about the structure of the volume formulas for cones and cylinders when the height and radius have the same value.
Next, they consider the relationship between the volume of a hemisphere and a prism that have matching dimensions. By comparing two different situations where the dimensions of one are double that of the other, this activity sets the expectation that when the radius of a sphere doubles, the volume increases by a factor of 8 since the length, width, and height of the sphere also double.
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, and students consider what this means by studying the volume equations for cylinders and cones whose radius and height are the same value (MP7).
- Calculate the volumes 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 by using the formulas for volume of a cone and volume of a cylinder, and explain (orally) the estimation strategy.
Let’s estimate volumes of hemispheres with figures we know.
Required Materials
Activity 3
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.