Back to Know the formulas for the area and circumference of a circle and use them to solve problems

Area and Circumference of Circles

Show your work. For exact answers, leave in terms of π. For approximate answers, use π ≈ 3.14 unless told otherwise.

Grade 7·~45 min·Common Core Math - Grade 7·standard·7-g-b-4

Work through problems with immediate feedback

Recall / Warm-Up

$A circle with center O and a horizontal diameter labeled 14 cm.$

The diagram shows a circle with center O. The full width of the circle (the diameter) is labeled 14 cm. What is the radius of this circle?

28 cm

7 cm

14 cm

3.14 cm

Evaluate $9^2$ .

Which value is the best approximation for π (pi)?

3.14

3.41

31.4

0.314

Fluency Practice

A circle has a radius of 7 cm. Find its exact circumference. Leave your answer in terms of π.

A circle has a diameter of 20 m. Find its approximate circumference. Use π ≈ 3.14.

A circle has a radius of 5 cm. Find its exact area. Leave your answer in terms of π.

A circle has a diameter of 12 ft. Find its exact area. Leave your answer in terms of π.

A circle has a radius of 10 m. Which of the following correctly gives its circumference?

$20\pi$ m

$100\pi$ m²

$20\pi$ m²

$10\pi$ m

Varied Practice

A circle has a circumference of 31.4 cm. Find its radius. Use π ≈ 3.14.

A circle has an area of $64\pi$ m². Find its exact radius.

$A semicircle with its flat base at the bottom and radius labeled 6 cm.$

The figure shows a semicircle with radius 6 cm. What is the area of the semicircle? Give an exact answer in terms of π.

$18\pi$ cm²

$36\pi$ cm²

$72\pi$ cm²

$6\pi$ cm²

A wheel has radius 5 ft. Lin wants to know the distance around the wheel. Mia wants to know how much rubber is needed to cover the flat face of the wheel. Explain which formula each person should use and why. Include the formula, the calculated answer, and the correct units for each person.

Word Problems

$Top-down view of a circular garden with radius labeled r = 9 m and a dashed fence outline.$

A circular garden has a radius of 9 m. A gardener wants to put a fence all the way around the edge of the garden.

How many meters of fencing are needed? Use π ≈ 3.14 and round to the nearest tenth.

A circular pizza has a diameter of 16 inches.

What is the area of the pizza? Give an exact answer in terms of π.

$Top-down view of a circular swimming pool with diameter labeled 20 ft.$

A circular swimming pool has a diameter of 20 ft.

How many feet of rope are needed to rope off the entire edge of the pool? Give an exact answer in terms of π.

A cover for the pool is sold by the square foot. How many square feet does the cover need to be? Use π ≈ 3.14.

$A 12 m by 10 m rectangle with a circular rug of diameter 4 m centered inside. The shaded region shows the floor not covered by the rug.$

A rectangular room measures 12 m by 10 m. A circular rug with a diameter of 4 m is placed in the center of the room.

What is the area of the floor NOT covered by the rug? Use π ≈ 3.14.

Error Analysis

$Two-column contrast card. Left (yellow): student's work shows A = π × 8² = 64π cm². Right (teal): correct method shows r = 4 cm first, then A = π × 4² = 16π cm².$

A student solved this problem: "Find the area of a circle with diameter 8 cm."

Student's work:
$A = \pi \times 8^2 = 64\pi \text{ cm}^2$

What error did the student make?

Used the diameter instead of the radius in the area formula

Forgot to multiply by π

Should have used the circumference formula instead

Used the wrong value for π

$Two-column contrast card. Left (yellow): Priya's incorrect work using area formula (36π in²). Right (teal): correct circumference C = 2π(6) = 12π in.$

Priya is asked: "How much framing material is needed to put a border around a circular clock with radius 6 in?"

Priya's work:
$A = \pi(6^2) = 36\pi \text{ in}^2$

What error did Priya make?

She used the area formula instead of the circumference formula

She squared the radius incorrectly

She forgot to multiply by 2 in the formula

She used the radius when she should have used the diameter

Challenge / Extension

$A composite figure: a 10 cm by 6 cm rectangle with a semicircle attached to the right end.$

The figure shows a composite shape: a rectangle that is 10 cm long and 6 cm wide, with a semicircle attached to its right end. The semicircle's diameter equals the width of the rectangle. What is the total area of the composite figure? Use π ≈ 3.14.

A circle has radius $r$ . If the radius is doubled to $2r$ , by what factor does the circumference change? By what factor does the area change? Show your work using the formulas, then explain what this difference tells you about why circumference uses linear units and area uses square units.

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