E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

Radius, Diameter and Circumference

E2.3 A1About 50 minutes

Learning goals

I can name and measure the radius, diameter and circumference of a circle.
I can explain that the circumference of every circle is a little more than 3 times its diameter, and that this number is called π.
I can use C = π × d or C = 2 × π × r to find a circumference.
I can work backwards from a circumference to find the diameter or radius.

Words to know

centre
The point in the middle of a circle that is the same distance from every point on the circle.

radius
A line segment from the centre to the circle, or its length. The plural is radii.

diameter
A line segment through the centre from one side of the circle to the other, or its length. d = 2 × r.

circumference
The distance around a circle: its perimeter.

π (pi)
The number you get when you divide any circle's circumference by its diameter: about 3.14. Its decimal digits go on forever without repeating.

Let's start with a bike wheel

Put a chalk mark on the bottom of a bike tire. Roll the bike forward until the mark comes back to the bottom. The bike has moved exactly one circumference of the wheel. If you know the wheel's size, can you predict how far it goes in one turn? By the end of this lesson, you can!

A bicycle wheel on a paved path with a white chalk mark on the tire where it touches the ground. — When the wheel turns once and the chalk mark comes back to the ground, the bike has travelled one circumference. AI-generated image

Parts of a circle

A circle is all the points that are the same distance from a centre.

The radius (r) goes from the centre to the circle.
The diameter (d) goes straight across, through the centre. It is made of two radii, so d = 2 × r and r = d ÷ 2.
The circumference (C) is the distance all the way around the circle.

A circle with its centre, a radius r, a diameter d and the circumference C (the distance around). The diameter is drawn exactly twice as long as the radius.

Discovering π

Students measured round objects with a string and a ruler:

Object	Diameter d	Circumference C	C ÷ d
can	7.5 cm	23.6 cm	about 3.15
dinner plate	26 cm	81.7 cm	about 3.14
coin	2.4 cm	7.5 cm	about 3.13

Sample measurements. Every circle gives a ratio a little more than 3.

For every circle, big or small, the circumference divided by the diameter is the same number: about 3.14. Mathematicians call it π (the Greek letter pi, said like "pie"). Its decimal digits go on forever and never fall into a repeating pattern: 3.14159 ... In this unit, use π ≈ 3.14.

Because C ÷ d = π:

C = π × d or, since d = 2 × r, C = 2 × π × r

And working backwards: d = C ÷ π.

Roll a wheel through one full turn and the distance it covers is its circumference: 3 diameters and a little bit more, about 3.14 × d.

Worked example

Example 1: Bike wheel

A bike wheel has a diameter of 60 cm. How far does the bike go in one full turn of the wheel?

One turn = one circumference.
C = π × d ≈ 3.14 × 60 = 188.4 cm.
Estimate check: 3 × 60 = 180, a bit less than 188.4 ✓

About 188.4 cm (almost 1.9 m).

Worked example

Example 2: Starting from the radius

A round wall clock has a radius of 15 cm. What is its circumference?

d = 2 × 15 = 30 cm.
C = π × d ≈ 3.14 × 30 = 94.2 cm. (Or C = 2 × π × r = 2 × 3.14 × 15 = 94.2 cm.)

About 94.2 cm.

Worked example

Example 3: Working backwards

A string wrapped once around a maple tree trunk measures 157 cm. What are the trunk's diameter and radius? (Pretend the trunk is a perfect circle.)

d = C ÷ π ≈ 157 ÷ 3.14 = 50 cm.
r = d ÷ 2 = 25 cm.
Check: 3.14 × 50 = 157 ✓

Diameter about 50 cm, radius about 25 cm.

Worked example

Example 4: Half a circle

A window is shaped like a semicircle (half a circle) with a straight bottom edge 80 cm long. What is the perimeter of the window?

The straight edge is a diameter: d = 80 cm.
Curved part: half of a circumference = 3.14 × 80 ÷ 2 = 125.6 cm.
Add the straight edge: 125.6 + 80 = 205.6 cm.

About 205.6 cm. (Don't forget the straight side!)

Before you calculate

Check: were you given the radius or the diameter? Double the radius to get the diameter.
Estimate with 3: the circumference is a bit more than 3 times the diameter. If d = 20 cm, C should be a bit more than 60 cm.
π ≈ 3.14 gives an approximate answer. That's why we say about.

Hands-on activity

Hunt for π

You need: a piece of string (or a strip of paper), a ruler or tape measure, and 4 or more round objects: a can, a plate, a roll of tape, a jar lid, a bowl.

Measure the diameter of each object: measure straight across the widest part, through the centre.
Wrap the string once around the object, mark where it meets, then straighten the string and measure it. That's the circumference.
Make a table: object, d, C, C ÷ d.
Calculate C ÷ d for each object (a calculator is fine).

What to look for: every answer should be close to 3.1. Ask: "Why aren't they all exactly 3.14?" (Small measuring errors, and the string can stretch or slip.)

Just for fun

Pi Day is celebrated on March 14 (3/14), because π ≈ 3.14. Plenty of math fans celebrate by eating pie. Mathematicians can't resist a good pun.

Your turn

A circle has a radius of 7 cm. What is its diameter?

Find the circumference of a circle with a diameter of 10 cm.

Find the circumference of a circle with a radius of 5 m.

Find the circumference of a circle with a diameter of 3.5 cm.

A circle has a circumference of 62.8 cm. Find its diameter.

A circle has a circumference of 47.1 m. Find its radius.

Why is the circumference always a bit more than 3 times the diameter?

More practice: warm-up

d = 18 cm. What is r?

r = 12 cm. What is d?

What is π to two decimal places?

Find C when d = 2 cm.

True or false: for a tiny circle, C ÷ d is smaller than for a huge circle.

More practice: on your own

A bike wheel has a diameter of 66 cm. How far does it roll in one turn? Give your answer in cm and in m.

How far does the same wheel roll in 100 turns?

A round table has a radius of 0.75 m. How much ribbon goes once around its edge?

The edge of a round pool cover is 18.84 m long. What are the diameter and radius of the cover?

Find the perimeter of a quarter circle with a radius of 10 cm (the curved edge plus the two straight edges).

Challenge

Which is longer: the circumference of a circle with a diameter of 10 cm, or the perimeter of a square with sides of 8 cm? By how much?

A bike wheel has a diameter of 70 cm. About how many full turns does it make in 1 km?

If you double the diameter of a circle, what happens to its circumference? Explain.

Check your understanding

Quiz

A circle has a radius of 6 cm. What is its diameter?

What is the circumference of a circle with a diameter of 20 cm? (π ≈ 3.14)

What is π?

A circle has a circumference of 31.4 cm. What is its diameter?

What is the circumference of a circle with a radius of 4 cm?

If the radius of a circle doubles, its circumference ...

For parents and teachers

Big idea: E2.3 asks students to use the relationships between the radius, diameter and circumference of a circle to explain the circumference formula and to solve problems. The key discovery is that C ÷ d is the same for every circle: π.

How to help at home: Do the π hunt with string and household objects. It's fine to use a calculator for the division. Talk about why the answers are close to, but not exactly, 3.14.

Common difficulties:

Using the radius as if it were the diameter. Always ask, "Is this the radius or the diameter?"
Forgetting the straight edges when finding the perimeter of a semicircle or quarter circle.
Expecting an exact answer. With π ≈ 3.14, answers are approximate.

This lesson builds A1 skills: investigating, finding a pattern in data, and checking answers by estimating with 3.