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

Constructing Circles

E2.4About 45 minutes

Learning goals

I can use a compass to construct a circle with a given radius or diameter.
I can construct a circle when I am given its circumference.
I can construct circles without a compass, using a string and a pencil.

Words to know

compass
A tool for drawing circles. One leg has a sharp point that stays at the centre; the other holds a pencil.

construct
To draw accurately with tools such as a compass and a ruler.

concentric circles
Circles that have the same centre but different radii, like the rings of a target.

radius
The distance from the centre to the circle. It is the width you open your compass to.

Let's start with a target

You're making a target for a beanbag toss: three circles with the same centre. The rules say the outside circle must have a circumference of about 1.9 m. How wide do you open your measuring tool? You need the radius. This lesson is all about turning any information about a circle into the radius, then drawing it accurately.

Constructing a circle with a compass

Safety first: the compass point is sharp. Keep it pointing down, put a piece of cardboard under your paper, and never point it at anyone.

Set the radius. Put the compass point on the 0 mark of a ruler. Open the compass until the pencil tip is on the radius mark (for r = 3 cm, on the 3).
Mark the centre. Draw a small dot where you want the centre.
Place the point on the centre dot and press gently.
Turn. Hold the compass by its top handle and turn it in one smooth motion, tilting it slightly in the direction you're turning.
Check. Measure a diameter through the centre. It should be 2 × the radius.

What if you aren't given the radius?

Your compass is always set to the radius, so first find it.

Given the diameter: r = d ÷ 2.
Given the circumference: first d = C ÷ π, then r = d ÷ 2.

Three circles drawn to the same scale (smaller than real size). Left: radius 3 cm. Middle: diameter 4 cm, so the compass is set to 2 cm. Right: circumference 12.56 cm, so d = 12.56 ÷ 3.14 = 4 cm and the radius is 2 cm: the same size as the middle circle.

Worked example

Example 1: Given the diameter

Construct a circle with a diameter of 9 cm.

Find the radius: r = 9 ÷ 2 = 4.5 cm.
Open the compass to 4.5 cm (the pencil halfway between 4 and 5 on the ruler).
Mark the centre, place the point, and turn.
Check: a line through the centre, across the circle, measures 9 cm.

Set the compass to 4.5 cm.

Worked example

Example 2: Given the circumference

Construct a circle with a circumference of 25.12 cm.

d = C ÷ π ≈ 25.12 ÷ 3.14 = 8 cm.
r = 8 ÷ 2 = 4 cm.
Construct a circle with a radius of 4 cm.
Check: wrap a string around your circle; it should be about 25 cm.

Set the compass to 4 cm.

Worked example

Example 3: Too big for a compass

A sign needs a circle with a diameter of 30 cm. Your compass only opens to 10 cm. How can you draw it?

The radius is 30 ÷ 2 = 15 cm.
Tie a loop in one end of a string and slip a pencil through it. Measure 15 cm along the string from the pencil tip and hold that spot of the string firmly on the centre with your finger.
Keep the string tight and move the pencil all the way around.
Another way: poke two holes 15 cm apart in a strip of cardboard. Pin one hole at the centre with a thumbtack (ask a grown-up; tacks are sharp) and put a pencil through the other.

Use a 15 cm string or cardboard strip as a giant compass.

Worked example

Example 4: The beanbag target

Construct three concentric circles with radii 2 cm, 4 cm and 6 cm.

Mark one centre point.
Set the compass to 2 cm and draw the first circle.
Without moving the centre, open to 4 cm and draw, then 6 cm and draw.
Check: the gap between each pair of circles is 2 cm all the way around.

Three circles with the same centre, each 2 cm apart. (For the real beanbag target from the start of the lesson, with an outside circumference of about 1.9 m = 190 cm: d ≈ 190 ÷ 3.14 ≈ 60.5 cm, so the radius is about 30 cm. That's a job for the string method!)

The beanbag target from Example 4: three concentric circles with radii 2 cm, 4 cm and 6 cm, all drawn from one centre point.

Construction tips

Your compass may slip open while you draw. Re-check the radius with a ruler before each circle.
Turn the paper instead of your wrist if that's easier.
When working from a circumference, round the radius sensibly (to the nearest millimetre, 0.1 cm) because a ruler can't measure more exactly than that.

Hands-on activity

Circle flower design

You need: a compass (or the string-and-pencil method), a pencil, paper on top of a piece of cardboard, coloured pencils. Ask a grown-up to help with the compass point.

Construct a circle with a radius of 3 cm.
Keep the same compass width. Put the point anywhere on the circle and draw another circle.
Put the point on one of the places where the two circles cross and draw again. Keep going around until you have drawn 6 circles around the first one.
Colour the flower-petal pattern in the middle.
Then go outside (with a grown-up's OK): with chalk and a 1.5 m piece of rope, have one person hold the end at the centre and draw a circle with a diameter of 3 m on a driveway or a paved yard, away from traffic.

What to look for: the petals meet neatly at the centre when the compass width stays the same.

Just for fun

Why did the compass win the drawing contest?

It ran circles around the competition.

Your turn

You need a circle with a diameter of 12 cm. What width do you open the compass to?

A circle has a circumference of 18.84 cm. What are its diameter and radius?

What radius do you use to construct a circle with a circumference of 31.4 cm?

Where do you put the point of the compass?

Write the steps to construct a circle with a radius of 4 cm.

How long should a rope be, from the centre to the chalk, to draw a circle with a diameter of 3 m?

More practice: warm-up

d = 8 cm. What is r?

r = 2.5 cm. What is d?

You need a circle with a radius of 7 cm. How wide do you open the compass?

C = 62.8 cm. What is d?

Are all the radii of one circle the same length?

More practice: on your own

Find the radius you need to construct a circle with a circumference of 28.26 cm.

You want a circle with a circumference of about 50 cm. What radius should you use, to the nearest tenth of a centimetre?

A circle has a radius of 3 cm. You draw a second circle with the same centre and double the circumference. What radius do you use?

Two circles each have a radius of 3 cm. Their centres are 4 cm apart. Do the circles cross? Explain.

Lea had to construct a circle with a circumference of 15.7 cm. She calculated 15.7 ÷ 3.14 = 5 and set her compass to 5 cm. What went wrong?

Challenge

A round garden bed needs edging that is 9.42 m long. How long should the rope be between the centre stake and the marker you use to outline the bed?

A circle fits exactly inside a square with 10 cm sides, touching all four sides. What is its radius and circumference?

Construct a circle with a radius of 4 cm. Keep the same compass width and step around the circle, marking points where the pencil lands each time. How many steps fit? What shape do the points make?

Check your understanding

Quiz

You need a circle with a diameter of 10 cm. What width do you set the compass to?

A circle has a circumference of 12.56 cm. What radius do you use to construct it?

Where does the compass point go?

How can you draw a circle that is too big for your compass?

Why does a compass draw a circle?

A circle has a circumference of 47.1 cm. What is its diameter?

For parents and teachers

Big idea: E2.4 asks students to construct circles when given the radius, diameter or circumference. Every construction comes down to finding the radius.

How to help at home: A basic compass is inexpensive. Help your child hold it by the top and turn smoothly. The string-and-pencil method and a chalk circle on the driveway are fun ways to make big circles.

Common difficulties:

Setting the compass to the diameter. Ask, "What does the compass measure?" (the radius)
Forgetting to halve after dividing a circumference by π.
Compass slipping. Re-check the width with a ruler.

Safety: compass points and thumbtacks are sharp. Work on cardboard, keep points down, and supervise younger siblings. Chalk circles outside only where a grown-up says it's safe, away from traffic.