223. Total Surface Area of Cylinders

Learning Intentions

• Identify circular bases and curved surface area.
• Apply surface area formulas for cylinders.
• Calculate total surface area Use appropriate rounding.

Total Surface Area of Cylinders

Learning Intentions

• Identify circular bases and curved surface area.
• Apply surface area formulas for cylinders.
• Calculate total surface area using appropriate rounding.

Pre-requisite Summary

• A cylinder has two congruent circular bases.
• The curved surface of a cylinder wraps around the circular bases.
• The area of one circular base is .
• The curved surface area of a cylinder is .
• The total surface area of a cylinder is .
• The radius is half the diameter, so .
• Surface area is measured in square units such as , and .
• Rounding should match the instruction in the question, such as nearest whole number or one decimal place.

Worked Examples

Worked Example 1

A cylinder has radius and height .

a) Identify the two circular bases.

b) Identify the curved surface.

c) State which measurements are needed to find total surface area.

Worked Example 2

A cylinder has diameter and height .

a) Find the radius.

b) Identify the height.

c) State the formula needed for total surface area.

Worked Example 3

Find the curved surface area of a cylinder with radius and height .

Use:

Worked Example 4

Find the total surface area of a cylinder with radius and height .

Use:

Worked Example 5

A cylindrical container has radius and height .

Find the total surface area, rounded to the nearest square centimetre.

Worked Example 6

A cylindrical water tank has diameter and height .

Find the total surface area, rounded to one decimal place.

Problems

Problem 1

A cylinder has radius and height .

a) Identify the two circular bases.

b) Identify the curved surface.

c) State which measurements are needed to find total surface area.

Problem 2

A cylinder has diameter and height .

a) Find the radius.

b) Identify the height.

c) State the formula needed for total surface area.

Problem 3

Find the curved surface area of a cylinder with radius and height .

Use:

Problem 4

Find the total surface area of a cylinder with radius and height .

Use:

Problem 5

A cylindrical container has radius and height .

Find the total surface area, rounded to the nearest square centimetre.

Problem 6

A cylindrical water tank has diameter and height .

Find the total surface area, rounded to one decimal place.

Exercises

Understanding and Fluency

Exercise 1

A cylinder has radius and height .

a) Identify the radius.

b) Identify the height.

c) Identify the two circular bases.

Exercise 2

A cylinder has diameter and height .

a) Find the radius.

b) Identify the curved surface.

c) State the formula for total surface area.

Exercise 3

Find the area of one circular base for a cylinder with radius .

Use:

Exercise 4

Find the combined area of the two circular bases for a cylinder with radius .

Use:

Exercise 5

Find the curved surface area of a cylinder with radius and height .

Use:

Exercise 6

Find the curved surface area of a cylinder with radius and height .

Use:

Exercise 7

Find the total surface area of a cylinder with radius and height .

Use:

Exercise 8

Find the total surface area of a cylinder with radius and height .

Use:

Exercise 9

A cylinder has diameter and height .

a) Find the radius.

b) Substitute the values into .

c) Calculate the total surface area.

Exercise 10

A cylinder has diameter and height .

a) Find the radius.

b) Substitute the values into .

c) Round the total surface area to one decimal place.

Reasoning

Exercise 11

Explain why the formula for total surface area of a cylinder includes .

Exercise 12

Explain why the formula for curved surface area of a cylinder is:

Exercise 13

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

Explain what part of the surface area is missing.

Exercise 14

A student uses the diameter instead of the radius in:

Explain why this gives an incorrect answer.

Exercise 15

Explain why total surface area is measured in square units instead of cubic units.

Problem-solving

Exercise 16

A cylindrical can has radius and height .

a) Find the area of the two circular bases.

b) Find the curved surface area.

c) Find the total surface area, rounded to the nearest square centimetre.

Exercise 17

A metal pipe is closed at both ends. It has diameter and length .

a) Find the radius.

b) Find the total surface area.

c) Round the answer to the nearest square centimetre.

Exercise 18

A cylindrical water tank has radius and height .

a) Identify the measurements needed for total surface area.

b) Find the total surface area.

c) Round the answer to one decimal place.

Exercise 19

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

a) Explain why the two circular bases are not included.

b) Find the area of the label.

c) State the correct square unit.

Potential Misunderstandings

• Confusing curved surface area with total surface area.
• Forgetting to include both circular bases.
• Including the circular bases when only the label or curved surface is required.
• Substituting diameter instead of radius.
• Forgetting that .
• Using instead of for curved surface area.
• Using cubic units instead of square units.
• Rounding too early before the final answer.
• Forgetting to square the radius in .
• Thinking a cylinder has rectangular faces instead of one curved surface.

Next: 224. Solving Problems Involving Volume and Total Surface Area