Area of the Circular Base: Worked Examples

This record is for Grade 8 Geometry and Measurement in the cylinder-volume derivation unit. It focuses only on the area of the circular base. For the full teaching sequence, see rea.m08.geometry-measurement.volume.cylinder-derivation.base-area-lesson. For how this base area is then used inside the full cylinder formula, see rea.m08.geometry-measurement.volume.cylinder-volume-derivation-worked-examples. For frequent errors and repairs, see rea.m08.geometry-measurement.volume.cylinder-volume-derivation-misconceptions. Independent follow-up practice belongs in rea.m08.geometry-measurement.volume.cylinder-volume-derivation-practice-set.

Problem-solving frame

When a question asks for the area of the circular base, the core decision is:

1. Identify whether the given measure is a radius or a diameter.
2. If needed, convert diameter to radius using r = d/2.
3. Use A = pi r^2.
4. Decide whether the answer should stay in exact form (with pi) or be approximated as a decimal.
5. Keep the units as square units.

A reliable self-check: area should grow quickly as the radius grows, because the radius is squared.

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Worked Example 1: Direct radius given

Problem. A cylinder has base radius 4 cm. Find the area of one circular base.

Solution.

Use the circle area formula:

A = pi r^2

Substitute r = 4:

A = pi(4)^2

A = 16pi

So the area of the base is 16pi cm^2.

As a decimal,

16pi ≈ 50.3

So the area is about 50.3 cm^2.

Commentary. The simplest version of the problem gives the radius directly, so there is no conversion step. This is the cleanest place to build the habit that the radius gets squared, not the whole formula and not pi.

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Worked Example 2: Diameter given instead of radius

Problem. A cylinder has base diameter 10 m. Find the area of the circular base.

Solution.

The formula uses radius, not diameter.

r = d/2 = 10/2 = 5

Now use A = pi r^2:

A = pi(5)^2

A = 25pi

So the exact area is 25pi m^2.

As a decimal,

25pi ≈ 78.5

So the area is about 78.5 m^2.

Commentary. The key choice is noticing that 10 is not the value to square directly. A common error is writing A = pi(10)^2; that would use the diameter as if it were the radius and make the answer four times too large.

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Worked Example 3: Decimal radius

Problem. The radius of a cylinder's base is 2.5 cm. Find the base area.

Solution.

Use A = pi r^2:

A = pi(2.5)^2

A = pi(6.25)

A = 6.25pi

So the exact area is 6.25pi cm^2.

As a decimal,

6.25pi ≈ 19.6

So the area is about 19.6 cm^2.

Commentary. With decimals, it helps to square carefully before multiplying by pi. This also shows that exact form is not only for whole numbers; 6.25pi cm^2 is still an exact answer.

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Worked Example 4: Fraction radius

Problem. A small cylinder has base radius 1/2 in. Find the area of the base.

Solution.

Use A = pi r^2:

A = pi(1/2)^2

A = pi(1/4)

A = pi/4

So the exact area is pi/4 in^2.

As a decimal,

pi/4 ≈ 0.785

So the area is about 0.785 in^2.

Commentary. Fractions are a good check on whether the learner really understands the formula. Since the radius is less than 1, the area should also be a small positive number. That estimate matches the result.

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Worked Example 5: Unit conversion before using the formula

Problem. A cylinder has base diameter 30 mm. Find the area of the base in cm^2.

Solution.

Because the answer must be in cm^2, convert the measurement first.

30 mm = 3 cm

So the diameter is 3 cm, and the radius is:

r = 3/2 = 1.5 cm

Now use the formula:

A = pi r^2

A = pi(1.5)^2

A = pi(2.25)

A = 2.25pi

So the exact area is 2.25pi cm^2.

As a decimal,

2.25pi ≈ 7.07

So the area is about 7.07 cm^2.

Commentary. The important choice here is to convert the linear unit before calculating area. If someone mixes mm and cm carelessly, the square units at the end become meaningless.

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Worked Example 6: Reverse problem from area to radius

Problem. The area of a circular base is 49pi cm^2. Find the radius.

Solution.

Start with the formula:

A = pi r^2

Substitute the given area:

49pi = pi r^2

Divide both sides by pi:

49 = r^2

Take the positive square root:

r = 7

So the radius is 7 cm.

Commentary. In geometry, a radius is a length, so we take the positive root. This example is useful because it runs the formula backward instead of forward.

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Worked Example 7: Reverse problem from area to diameter

Problem. A cylinder's circular base has area 81pi m^2. Find the diameter.

Solution.

Use A = pi r^2.

81pi = pi r^2

Divide by pi:

81 = r^2

r = 9

Now convert radius to diameter:

d = 2r = 18

So the diameter is 18 m.

Commentary. There are two decisions: first solve for radius because the formula is written in terms of r, then convert to diameter because that is what the question asks for. Stopping at 9 m would answer the wrong question.

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Worked Example 8: Comparing two bases

Problem. Cylinder A has radius 3 cm. Cylinder B has radius 6 cm. How many times as large is the area of B's base compared with A's base?

Solution.

Find each area.

For A:

A_A = pi(3)^2 = 9pi

For B:

A_B = pi(6)^2 = 36pi

Now compare:

A_B / A_A = 36pi / 9pi = 4

So B's base area is 4 times A's base area.

Commentary. This is a very important pattern: doubling the radius does not double the area. Because area depends on r^2, doubling the radius multiplies area by 2^2 = 4.

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Worked Example 9: Find a missing radius from a verbal condition

Problem. The base area of a cylinder is 64pi cm^2. Another cylinder has a base area that is one-fourth as large. What is the radius of the second cylinder?

Solution.

First find the second area:

(1/4) x 64pi = 16pi

So the second cylinder has base area 16pi cm^2.

Now solve for the radius:

A = pi r^2

16pi = pi r^2

Divide by pi:

16 = r^2

r = 4

So the radius of the second cylinder is 4 cm.

Commentary. This problem combines comparison and reverse solving. It is easy to confuse “one-fourth the area” with “one-fourth the radius,” but those are not the same statement.

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Worked Example 10: Challenge problem with exact and approximate reasoning

Problem. A manufacturer is choosing between two can bases.

• Base P has diameter 12 cm.
• Base Q has radius 5 cm.

Which base has the greater area, and by how much?

Solution.

Find each area in exact form first.

Base P

Diameter is 12 cm, so radius is:

r = 12/2 = 6 cm

Area:

A_P = pi(6)^2 = 36pi cm^2

Base Q

Radius is already given as 5 cm.

A_Q = pi(5)^2 = 25pi cm^2

Compare

Since 36pi > 25pi, base P has the greater area.

Difference:

36pi - 25pi = 11pi

So the exact difference is 11pi cm^2.

As a decimal,

11pi ≈ 34.6

So base P is larger by about 34.6 cm^2.

Commentary. Exact form makes the comparison cleaner because both answers contain the same factor pi. It is better to compare 36pi and 25pi directly before rounding.

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Quick error checks

Use these checks after every problem:

• Did I use radius, not diameter, in A = pi r^2?
• Did I square only the radius?
• Are my units square units like cm^2, m^2, or in^2?
• If I rounded, did I keep enough precision until the end?
• Does the answer size make sense? A bigger radius should give a much bigger area.

Mini practice with answers

1. A cylinder has base radius 8 cm. Find the base area. Answer: 64pi cm^2.

2. A cylinder has base diameter 14 m. Find the base area. Answer: 49pi m^2.

3. A circular base has area 100pi in^2. Find the radius. Answer: 10 in.

4. One cylinder base has radius 2 cm and another has radius 4 cm. How many times as large is the second area? Answer: 4 times.

Connection back to cylinder volume

The whole reason this skill matters is that a cylinder's volume is built from base area x height. Once the circular base area is known, it becomes the key input in the larger derivation and problem solving. This record isolates that one skill so it can be practiced cleanly before returning to full cylinder-volume problems.