Circumference and Area of a Circle, With Worked Examples

The distance around a circle is C = 2πr (or πd), and the space inside is A = πr². Here r is the radius, the distance from the center to the edge, and d is the diameter, which is 2r. Students are expected to know these two formulas by Grade 7: the Common Core standard 7.G.B.4 asks students to know the formulas for the area and circumference of a circle, use them to solve problems, and give an informal derivation of the relationship between the two.

1. What pi is

Pi (π) is the ratio of any circle's circumference to its diameter. It is the same for every circle and is about 3.14159. Because it never ends, homework usually tells you to use 3.14, to use a calculator's π key, or to leave the answer "in terms of π." Follow the instruction on your sheet, since the last digits will differ. For scale, NASA's Jet Propulsion Laboratory says its highest-accuracy calculations, for interplanetary navigation, use 3.141592653589793, so 3.14 is a rounded classroom value, not the full number.

2. The formulas and when to use which

Circumference is a length, measured in units like inches. Area is a surface, measured in square units like square inches. The squared r in the area formula is the reason the units are squared.

3. Example 1: radius 5 cm

Circumference: C = 2 × π × 5 = 10π ≈ 31.42 cm. Area: A = π × 5² = 25π ≈ 78.54 square cm.

Check: the circumference should be a bit more than 3 times the diameter (3 × 10 = 30). It is 31.42, so that is reasonable. The area should be a bit more than 3 times a 5 by 5 square (3 × 25 = 75). It is 78.54.

4. Example 2: diameter 14 inches

The formulas need the radius for area, so halve the diameter first: r = 7.

Circumference: C = πd = 14π ≈ 43.98 inches. Area: A = π × 7² = 49π ≈ 153.94 square inches.

The classic mistake here is plugging 14 into πr² and getting 196π ≈ 615.75, which is four times too large.

5. Example 3: working backward from circumference

A circle has circumference 37.7 feet. Find the radius and area.

Rearrange C = 2πr to r = C ÷ (2π). So r = 37.7 ÷ 6.2832 ≈ 6.0 feet. Area: A = π × 6² = 36π ≈ 113.10 square feet.

Check: 2π × 6 = 37.70, which matches the given circumference.

6. Example 4: a half circle

A semicircular garden has a straight edge of 10 meters, which is the diameter, so r = 5 m.

Area: half of πr², so (25π) ÷ 2 ≈ 39.27 square meters. Perimeter: the curved part is half the circumference, 5π ≈ 15.71 m, plus the straight edge of 10 m, for a total of about 25.71 m.

Students often forget to add the straight edge when asked for the perimeter. The curved part alone is 15.71 m.

7. Example 5: how much bigger is a bigger circle?

Compare a pizza with an 8-inch diameter to one with a 12-inch diameter. The radii are 4 and 6.

Areas: π × 4² = 16π ≈ 50.27 square inches, and π × 6² = 36π ≈ 113.10 square inches.

The 12-inch pizza is 36 ÷ 16 = 2.25 times as big, not 1.5 times, even though the diameter is only 1.5 times larger. Area grows with the square of the radius. Another comparison: two 10-inch circles have a combined area of about 157.08 square inches, while one 14-inch circle has about 153.94 square inches, so one 14-inch circle is slightly less than two 10-inch ones.

8. Using 3.14 versus the calculator's π

For Example 1, using 3.14 gives C = 10 × 3.14 = 31.4 and A = 25 × 3.14 = 78.5. The calculator gives 31.42 and 78.54. Both are acceptable if the problem says which to use. Do not round π before the last step if the instructions say to use the calculator value, because rounding early can shift the final digit.

9. Word problem: a bike wheel

A wheel has a diameter of 26 inches. How far does it roll in one full turn?

One turn equals one circumference: 26π ≈ 81.68 inches. To roll 100 feet (1,200 inches) it turns 1,200 ÷ 81.68 ≈ 14.7 times.

10. Common mistakes

• Using the diameter as the radius. Read the problem for the word "diameter" and halve it.
• Mixing up the formulas. Circumference has r to the first power; area has r squared.
• Forgetting units. Circumference is in plain units; area is in square units.
• Squaring wrong. In πr², square only the r, then multiply by π. For r = 5 that is 25π, not (5π)².
• Stopping at the curve. Half circles and "pac-man" shapes need the straight edges added to the perimeter.

11. Key takeaways

• Circumference: C = 2πr = πd. Area: A = πr².
• Find the radius first; halve the diameter if you are given it.
• Check circumference against "about 3 times the diameter" and area against "about 3 times r squared."
• Doubling the radius makes the area four times as large.
• Follow the π instruction on your assignment, 3.14 or the calculator value.