GM Lesson 032 Surface Area of Spheres and Composite Solids

Learning Intentions

By the end of this lesson, students will be able to:

• Calculate the surface area of a sphere using .
• Identify exposed surfaces in composite solids.
• Calculate surface areas of composite three-dimensional objects by adding only the visible outside surfaces.

Prerequisites

Students should already be able to:

• Substitute values into measurement formulae.
• Calculate the area of circles using .
• Calculate the surface area of rectangular prisms, cylinders, pyramids and cones.
• Round answers appropriately using square units.
• Distinguish between radius, diameter, height and slant height.

Key Idea Summary

Surface area measures the total outside area of a three-dimensional object.

For a sphere:

where is the radius.

For a composite solid, not every surface is exposed. When two solids are joined together, the touching surfaces are internal and should not be included in the total surface area.

A useful strategy is:

1. Identify the solids used.
2. Identify which surfaces are exposed.
3. Write the surface area expression.
4. Substitute measurements.
5. Calculate and round with square units.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Sphere Surface Area Formula

For a sphere with radius :

This formula gives the entire outside surface of the sphere.

Worked Example 1: Surface Area of a Sphere

A basketball is approximately spherical with radius cm. Calculate its surface area to the nearest square centimetre.

Use:

Substitute :

The surface area is approximately .

Worked Example 2: Surface Area of a Hemisphere

A solid hemisphere has radius cm. Calculate its total surface area, including the flat circular base.

A hemisphere has:

• half the surface area of a sphere
• one circular base

Half the surface area of a sphere is:

The flat circular base is:

So the total surface area is:

Substitute :

The total surface area is approximately .

Worked Example 3: Cylinder with a Hemisphere on Top

A toy rocket is made from a cylinder with a hemisphere on top. The cylinder has radius cm and height cm. The hemisphere has the same radius. Calculate the exposed surface area, to the nearest square centimetre.

The exposed surfaces are:

• curved surface of the cylinder
• bottom circular base of the cylinder
• curved surface of the hemisphere

The top circle of the cylinder is not exposed because the hemisphere is attached there.

Curved surface area of the cylinder:

Bottom circular base:

Curved surface area of the hemisphere:

Total exposed surface area:

Substitute and :

The exposed surface area is approximately .

Worked Example 4: Composite Solid with a Sphere and Cylinder

A decorative object is made by placing a sphere of radius cm on top of a cylinder of radius cm and height cm. The sphere touches the cylinder at one circular face. Calculate the exposed surface area to the nearest square centimetre.

The exposed surfaces are:

• the surface area of the sphere, except the circular contact area
• the curved surface area of the cylinder
• the bottom circular base of the cylinder

Sphere surface area:

Circular contact area to subtract from the sphere:

Curved surface area of cylinder:

Bottom base of cylinder:

Total exposed surface area:

The subtracted circular area and the bottom base have the same area, so this simplifies to:

Substitute and :

The exposed surface area is approximately .

Understanding Checks

Check 1

A sphere has radius cm.

Write the correct formula for its surface area.

Check 2

A sphere has diameter cm.

What value should be used for ?

Check 3

A hemisphere has radius cm.

Which expression represents the curved surface area only?

A.
B.
C.
D.

Check 4

A cylinder has a cone attached to its top. The circular surfaces where they touch are not visible.

Should the top circular base of the cylinder be included in the exposed surface area?

Check 5

A composite solid is made from two shapes joined together.

Explain why it is usually incorrect to simply add the total surface areas of both original shapes.

Exercises

Simple Familiar Exercises

Exercise 1

Calculate the surface area of a sphere with radius cm. Give your answer in exact form using .

Exercise 2

Calculate the surface area of a sphere with radius m. Give your answer to the nearest square metre.

Exercise 3

A sphere has diameter cm. Calculate its surface area to the nearest square centimetre.

Exercise 4

A solid hemisphere has radius cm. Calculate its curved surface area only.

Exercise 5

A solid hemisphere has radius cm. Calculate its total surface area including the flat circular base.

Complex Familiar Exercises

Exercise 6

A water tank is shaped like a cylinder with a hemisphere on top. The radius is m and the cylindrical section has height m.

Calculate the exposed surface area, including the bottom circular base.

Exercise 7

A capsule-shaped object is made from a cylinder with a hemisphere attached to each end. The cylinder has radius cm and length cm.

Calculate the total surface area of the capsule.

Exercise 8

A cone with radius cm and slant height cm is attached to the top of a cylinder with radius cm and height cm.

Calculate the exposed surface area of the composite solid, including the bottom base of the cylinder.

Use:

Exercise 9

A sphere of radius cm is placed on top of a cylinder of radius cm and height cm.

Assume the circular contact area is not exposed. Calculate the total exposed surface area to the nearest square centimetre.

Exercise 10

A composite solid is made from a rectangular prism and a half-cylinder roof.

The rectangular prism is m long, m wide and m high. The half-cylinder roof sits on the m by m top face, so its diameter is m and its length is m.

Calculate the exposed surface area, excluding the bottom of the prism.

Homework Problems

Homework 1

Calculate the surface area of a sphere with radius cm. Give your answer to the nearest square centimetre.

Homework 2

A sphere has diameter cm. Calculate its surface area to the nearest square centimetre.

Homework 3

A solid hemisphere has radius m. Calculate its total surface area including the flat circular base.

Homework 4

A capsule is made from a cylinder with a hemisphere on each end. The cylinder has radius cm and length cm.

Calculate the total surface area of the capsule to the nearest square centimetre.

Homework 5

A cylinder of radius cm and height cm has a hemisphere attached to the top.

Calculate the exposed surface area, including the bottom circular base.

Homework 6

A sphere has surface area .

Find its radius.

Homework 7

A composite solid is made from a cone and a hemisphere joined at their circular bases. Both have radius cm. The cone has slant height cm.

Calculate the total exposed surface area.

Homework 8

A storage tank is made from a cylinder with a hemisphere on each end. The radius is m and the cylindrical section is m long.

Calculate the outside surface area to the nearest square metre.

Next: GM Lesson 033 Volumes of Prisms and Cylinders