Area of the Circular Base: Graded Practice Set

Grade level: Grade 8 (BC/Ontario-aligned)

How to use this record

This record is for practice, not first instruction. If you need the concept, notation, or worked teaching examples first, use these linked records:

• Overview: rea.m08.geometry-measurement.volume.cylinder-derivation.base-area
• Lesson: rea.m08.geometry-measurement.volume.cylinder-derivation.base-area-lesson
• Broader cylinder derivation practice: rea.m08.geometry-measurement.volume.cylinder-volume-derivation-practice-set
• Error repair: rea.m08.geometry-measurement.volume.cylinder-volume-derivation-misconceptions

In every problem here, you are finding the area of one circular base of a cylinder or a cylinder-like situation. Use:

• A = pi r^2

Unless a question says otherwise, leave answers in terms of pi. If a decimal is requested, round to the nearest tenth.

Difficulty tags

• [Foundational] direct use of radius in A = pi r^2
• [Developing] radius must be found from diameter, wording, or context
• [Challenge] multi-step, reverse, comparison, or error-analysis problem

Practice Set

1. Direct radius practice

1. [Foundational] A cylinder has base radius 2 cm. Find the area of one circular base.
2. [Foundational] A cylinder has base radius 5 m. Find the area of one circular base.
3. [Foundational] A cylinder has base radius 9 mm. Find the area of one circular base.
4. [Foundational] A cylinder has base radius 1.5 cm. Find the area of one circular base.
5. [Foundational] A circular base has radius 12 in. Find its area.
6. [Foundational] A circular base has radius 0.8 m. Find its area as a decimal to the nearest tenth.

2. Diameter to radius

7. [Developing] A cylinder has base diameter 8 cm. Find the area of one circular base.
8. [Developing] A cylinder has base diameter 14 m. Find the area of one circular base.
9. [Developing] A cylinder has base diameter 3.6 cm. Find the area of one circular base.
10. [Developing] A circular base has diameter 25 mm. Find its area in terms of pi.
11. [Developing] A soup can has circular base diameter 10 cm. Find the area of its base.
12. [Developing] A water tank has base diameter 1.2 m. Find the area of its base to the nearest tenth.

3. Mixed interpretation and comparison

13. [Developing] One cylinder has base radius 4 cm. Another has base radius 6 cm. Find the area of each base.
14. [Developing] A student says, “If the diameter doubles, the base area doubles.” Test this claim using circles with diameters 6 cm and 12 cm.
15. [Developing] A cylinder has radius 7 cm. Find the combined area of its two circular bases.
16. [Developing] A cylinder and a circle have the same radius, 3.5 m. What is the area of the cylinder’s circular base?

4. Harder applications and reasoning

17. [Challenge] A cylindrical candle has diameter 11 cm. Find the area of its circular base to the nearest tenth.
18. [Challenge] The area of a cylinder’s circular base is 64pi cm^2. Find the radius and the diameter.
19. [Challenge] The area of a cylinder’s circular base is about 153.9 cm^2. Using pi ≈ 3.14, find the radius.
20. [Challenge] A student computes the base area of a cylinder with diameter 18 cm as pi(18)^2 = 324pi cm^2. Identify the error and find the correct base area.
21. [Challenge] Two cylinders have base radii 3 cm and 9 cm. How many times as large is the larger base area compared with the smaller one?
22. [Challenge] A circular base has area 49pi m^2. Without finding a decimal, determine its radius. Then state whether its diameter is greater than, less than, or equal to 10 m.
23. [Challenge] A metal tin has base diameter 16 cm. The label covers only the side, not the top or bottom. A student mistakenly uses the base area formula to estimate label material. Explain why that is the wrong surface, then still find the area of one circular base.
24. [Challenge] A cylinder’s base radius increases from 4 cm to 10 cm. By how much does the circular base area increase? Give your answer in terms of pi and as a decimal to the nearest tenth.

Answer Key

1. 4pi cm^2
2. 25pi m^2
3. 81pi mm^2
4. 2.25pi cm^2
5. 144pi in^2
6. 0.64pi m^2 ≈ 2.0 m^2
7. 16pi cm^2
8. 49pi m^2
9. 3.24pi cm^2
10. 156.25pi mm^2
11. 25pi cm^2
12. 0.36pi m^2 ≈ 1.1 m^2
13. 16pi cm^2 and 36pi cm^2
14. Areas are 9pi cm^2 and 36pi cm^2; the area becomes 4 times as large, not 2 times.
15. 98pi cm^2
16. 12.25pi m^2
17. 30.25pi cm^2 ≈ 95.0 cm^2
18. Radius 8 cm, diameter 16 cm
19. Radius 7 cm
20. Error: used diameter instead of radius; correct area 81pi cm^2
21. 9 times as large
22. Radius 7 m; diameter is greater than 10 m
23. The label covers the curved side, not the circular base; base area 64pi cm^2
24. Increase 84pi cm^2 ≈ 263.9 cm^2

Full Solutions for the Hardest Third

17. Cylindrical candle with diameter 11 cm

We need the area of a circular base.

1. Start with the diameter: 11 cm
2. Find the radius: r = 11/2 = 5.5 cm
3. Use A = pi r^2
4. Substitute: A = pi(5.5)^2
5. Square the radius: (5.5)^2 = 30.25
6. So A = 30.25pi cm^2
7. Decimal form: 30.25 x 3.14 = 94.985
8. Rounded to the nearest tenth: 95.0 cm^2

Answer: 30.25pi cm^2 ≈ 95.0 cm^2

18. Base area is 64pi cm^2

We are told the area and must work backward.

1. Use A = pi r^2
2. Set the given area equal to the formula: 64pi = pi r^2
3. Divide both sides by pi: 64 = r^2
4. Take the positive square root: r = 8 The radius is positive because it is a length.
5. Find diameter: d = 2r = 16

Answer: radius 8 cm, diameter 16 cm

19. Base area is about 153.9 cm^2

We use pi ≈ 3.14.

1. Start with A = pi r^2
2. Substitute the approximate area: 153.9 ≈ 3.14r^2
3. Divide both sides by 3.14: r^2 ≈ 153.9/3.14
4. Compute: 153.9/3.14 = 49.0127...
5. So r^2 ≈ 49
6. Take the positive square root: r ≈ 7

Check:

• 3.14 x 7^2 = 3.14 x 49 = 153.86, which rounds to 153.9

Answer: radius 7 cm

20. Student used pi(18)^2

The diameter is 18 cm, but the formula needs the radius.

1. Identify the error: the student substituted the diameter into A = pi r^2
2. Find the correct radius: r = 18/2 = 9 cm
3. Use the formula correctly: A = pi(9)^2
4. Square the radius: 9^2 = 81
5. So the correct area is 81pi cm^2

Why the original answer is too large:

• Using 18 instead of 9 squares a number that is twice as big.
• Squaring doubles? No. Squaring makes the result 4 times as large.

Answer: Error: used diameter instead of radius. Correct base area: 81pi cm^2

21. Compare radii 3 cm and 9 cm

We compare the two base areas.

1. Smaller base: A1 = pi(3)^2 = 9pi
2. Larger base: A2 = pi(9)^2 = 81pi
3. Find the ratio: A2 / A1 = 81pi / 9pi = 9

So the larger base area is 9 times the smaller base area.

This shows an important pattern: when the radius is multiplied by 3, the area is multiplied by 3^2 = 9.

Answer: 9 times as large

22. Base area is 49pi m^2

We work backward without decimals.

1. Use A = pi r^2
2. Set 49pi = pi r^2
3. Divide by pi: 49 = r^2
4. Take the positive square root: r = 7 m
5. Find the diameter: d = 2r = 14 m
6. Compare 14 m with 10 m: 14 > 10

Answer: radius 7 m; diameter is greater than 10 m

23. Tin label versus base area

This is a meaning question first, then a calculation.

1. The label wraps around the curved side of the tin.
2. The base area formula A = pi r^2 finds the area of a flat circular end, not the curved side.
3. So using base area to estimate label material is the wrong surface.
4. The question still asks for the area of one circular base.
5. Diameter is 16 cm, so radius is 8 cm.
6. Use A = pi r^2 = pi(8)^2 = 64pi cm^2

Answer: The label covers the curved side, not the base; one circular base has area 64pi cm^2

24. Radius increases from 4 cm to 10 cm

We need the increase in base area.

1. Original area: A1 = pi(4)^2 = 16pi
2. New area: A2 = pi(10)^2 = 100pi
3. Increase: A2 - A1 = 100pi - 16pi = 84pi cm^2
4. Decimal form: 84 x 3.14 = 263.76
5. Rounded to the nearest tenth: 263.8 cm^2

Answer: increase 84pi cm^2 ≈ 263.8 cm^2

Quick self-check prompts

Use these if your answers do not match the key.

• Did you use the radius, not the diameter, in A = pi r^2?
• Did you square the radius before multiplying by pi?
• Did you keep the units as square units, such as cm^2 or m^2?
• If the problem gave a diameter, did you divide by 2 first?
• If the problem asked for two bases, did you multiply one-base area by 2?

Common error alerts

For fuller misconception repair, see rea.m08.geometry-measurement.volume.cylinder-volume-derivation-misconceptions.

Frequent mistakes in this topic:

• using diameter directly in place of radius
• forgetting to square the radius
• writing linear units instead of square units
• doubling area when the diameter doubles, instead of recognizing the area scales by the square factor

Next step

After this set, move to rea.m08.geometry-measurement.volume.cylinder-volume-derivation-practice-set to combine base area with height in full cylinder-volume problems.