Measuring circles

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1. Circumference of a circle

Warm-up

Three quick questions from earlier lessons.

Discover

Wrap a string around a can and compare it with the can's diameter. About how many diameters long is the string?

  1. A bit more than 3
  2. Exactly 2

Key idea

Circumference is about 3.14 diameters.

For every circle, circumferencediameter=π≈3.14\dfrac{\text{circumference}}{\text{diameter}} = \pi \approx 3.14. So C=πd=2πrC = \pi d = 2\pi r, where the diameter dd is twice the radius rr.

Worked example

A bike wheel of radius 13 in.

d=26d = 26 in
d = 2r.
C=26π≈81.7C = 26\pi \approx 81.7 in
C = πd.

Your turn, guided

Your turn: circumference 18π18\pi. The diameter.

1818
C = πd.

Practice ★

  1. A circle has a diameter of 26 cm. Find its circumference (exact, in terms of π).Not to scale
  2. A circle has a diameter of 24 cm. Find its circumference (exact, in terms of π).Not to scale
  1. A wheel has a radius of 16 in. How far does it roll in one full turn (exact, in terms of π)?Not to scale

Watch out!

Practice ★★

  1. A circle has a circumference of 26π26\pi cm. Find its radius (cm).
  2. A circle has a circumference of 33π33\pi cm. Find its radius (cm).

Exit ticket

Three questions to check you have got it.

  1. A circle has a diameter of 26 cm. Find its circumference (exact, in terms of π).Not to scale
  2. A circle has a diameter of 24 cm. Find its circumference (exact, in terms of π).Not to scale
  3. A circle has a diameter of 11 cm. Find its circumference (exact, in terms of π).Not to scale

Summary card

Circumference

C=πd=2πrC = \pi d = 2\pi r.

Watch out: radius or diameter? Check before using π.

2. Area of a circle

Warm-up

Three quick questions from earlier lessons.

Discover

Cut a circle of radius rr into thin wedges and line them up: they form almost a rectangle, half the circumference long and rr tall. What is its area?

  1. πr⋅r=πr2\pi r \cdot r = \pi r^2
  2. 2πr2\pi r

Key idea

Area counts square units inside.

The area of a circle is A=πr2A = \pi r^2 (square units). The circumference C=2πrC = 2\pi r is a length (units). If you know the diameter, halve it first.

Worked example

A pizza with diameter 12 in.

r=6r = 6
Halve.
A=36π≈113A = 36\pi \approx 113 in²
πr².

Your turn, guided

Your turn: the area of a circle with radius 10 (in terms of π, type pi).

100π100\pi
π · 100.

Practice ★

  1. Find the area of a circle with radius 3 cm (exact, in terms of π).Not to scale
  2. Find the area of a circle with radius 12 cm (exact, in terms of π).Not to scale
  1. Find the area of a half-circle with radius 14 m (exact, in terms of π).Not to scale

Watch out!

Practice ★★

  1. A round table has a diameter of 6 ft. Find its area (exact, in terms of π).Not to scale
  2. A round table has a diameter of 9 ft. Find its area (exact, in terms of π).Not to scale
  1. Boss problem. A round garden has a diameter of 18 m. Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

  1. Find the area of a circle with radius 3 cm (exact, in terms of π).Not to scale
  2. Find the area of a circle with radius 12 cm (exact, in terms of π).Not to scale
  3. Find the area of a circle with radius 7 cm (exact, in terms of π).Not to scale

Summary card

Area of a circle

A=πr2A = \pi r^2; around is C=2πrC = 2\pi r.

Watch out: square the radius, not the diameter.

Answers: Measuring circles

1. Circumference of a circle

A

  1. 26π26 \pi
  2. 24π24 \pi

B

  1. 32π32 \pi

C

  1. 1313
  2. 332\frac{33}{2}

D

  1. 26π26 \pi
  2. 24π24 \pi
  3. 11π11 \pi

2. Area of a circle

A

  1. 9π9 \pi
  2. 144π144 \pi

B

  1. 98π98 \pi

C

  1. 9π9 \pi
  2. 81π4\frac{81 \pi}{4}

D

  1. 81π81 \pi

E

  1. 9π9 \pi
  2. 144π144 \pi
  3. 49π49 \pi