GM Lesson 030 Surface Area of Cylinders

Learning Intentions

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

• Identify the two circular bases and the curved surface of a cylinder.
• Calculate the surface area of a cylinder.
• Apply cylinder surface area calculations in practical contexts.

Prerequisites

Students should already be able to:

• Calculate the area of a circle using .
• Calculate the circumference of a circle using .
• Substitute values into a formula.
• Round answers to an appropriate number of decimal places.
• Interpret area using square units such as , and .

Key Idea Summary

A cylinder has three surfaces:

• two identical circular bases
• one curved rectangular surface when unwrapped

For a closed cylinder:

Since the area of one circular base is , the two circular bases have total area:

The curved surface unwraps into a rectangle.

• The length of the rectangle is the circumference of the circular base:
• The height of the rectangle is the height of the cylinder:

So the curved surface area is:

Therefore, the surface area of a closed cylinder is:

where:

• is the total surface area
• is the radius of the circular base
• is the perpendicular height of the cylinder

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

Direct Instruction

A cylinder is made from circular and rectangular parts.

When a closed cylinder is unwrapped into a net, it contains:

• one circle for the top
• one circle for the bottom
• one rectangle for the curved surface

The rectangle has:

• length equal to the circumference of the circle
• height equal to the height of the cylinder

So:

The total surface area of a closed cylinder is:

This can also be factorised as:

Both forms give the same answer.

Worked Example 1: Calculating the Surface Area of a Cylinder

A closed cylinder has radius and height . Calculate its total surface area, correct to decimal place.

Using:

Substitute and :

Therefore, the total surface area is:

Worked Example 2: Separating the Bases and Curved Surface

A closed cylindrical tin has radius and height .

Calculate:

• the area of the two circular bases
• the curved surface area
• the total surface area

Area of the two circular bases:

Curved surface area:

Total surface area:

Therefore:

• area of the two bases is
• curved surface area is
• total surface area is

Worked Example 3: Practical Context

A cylindrical water tank has radius and height . The outside of the tank, including the top and bottom, is painted.

Calculate the surface area to be painted, correct to decimal places.

Using:

Substitute and :

Therefore, the area to be painted is:

Worked Example 4: Open Cylinder

A cylindrical bin has no lid. It has radius and height . Calculate the outside surface area of the bin, including the base but not the lid.

Since there is no lid, only one circular base is included.

Use:

Substitute and :

Therefore, the outside surface area is:

Understanding Checks

Check 1

A closed cylinder has radius and height .

State the values of and .

Check 2

For a cylinder with radius and height , write the correct substitution into:

Check 3

A student calculates the surface area of a closed cylinder using only:

Explain what part of the cylinder they have forgotten.

Check 4

A cylindrical can has radius and height .

Calculate the curved surface area only.

Check 5

A cylinder has diameter and height .

What value should be used for ?

Exercises

Simple Familiar Exercises

Exercise 1

A closed cylinder has radius and height .

Calculate its total surface area, correct to decimal place.

Exercise 2

A closed cylinder has radius and height .

Calculate its total surface area, correct to decimal place.

Exercise 3

A closed cylinder has radius and height .

Calculate its total surface area, correct to decimal place.

Exercise 4

A closed cylinder has radius and height .

Calculate its total surface area, correct to decimal places.

Exercise 5

A closed cylinder has diameter and height .

Calculate its total surface area, correct to decimal place.

Complex Familiar Exercises

Exercise 6

A cylindrical drink can has radius and height .

Calculate the total surface area of aluminium used to make the can, correct to decimal place.

Exercise 7

A cylindrical candle has diameter and height .

Calculate the total outside surface area, including the top and bottom.

Exercise 8

A cylindrical pipe has radius and length .

Calculate the curved surface area only.

Exercise 9

A cylindrical bin has no lid. It has radius and height .

Calculate the surface area of the bin, including the base but excluding the lid.

Exercise 10

A closed cylindrical tank has radius and height .

Paint covers per litre.

Calculate the number of litres of paint needed to paint the outside of the tank. Round up to the nearest whole litre.

Homework Problems

Problem 1

A closed cylinder has radius and height .

Calculate its total surface area, correct to decimal place.

Problem 2

A cylindrical can has diameter and height .

Calculate its total surface area, correct to decimal place.

Problem 3

A cylindrical label wraps around the curved surface of a tin. The tin has radius and height .

Calculate the area of the label.

Problem 4

A cylindrical bucket has no lid. It has radius and height .

Calculate the outside surface area, including the base but excluding the lid.

Problem 5

A closed cylindrical water tank has radius and height .

Calculate the total surface area, correct to decimal places.

Problem 6

A closed cylinder has diameter and height .

Calculate:

• the total area of the two circular bases
• the curved surface area
• the total surface area

Problem 7

A metal company makes closed cylindrical containers with radius and height .

Each container needs an extra of metal to allow for joins and waste.

Calculate the total amount of metal needed for one container, correct to the nearest square centimetre.

Problem 8

A cylindrical tank has radius and height .

Paint costs $ per square metre.

Calculate the cost of painting the outside of the closed tank, correct to the nearest dollar.

Next: GM Lesson 031 Surface Area of Pyramids and Cones