Find the centre

The idea

A circle is the set of all points on a plane that are equidistant from a given point on that plane, its centre; the distance from the centre to any point on the circle is the radius.

Raindrop circles

Raindrops falling on water make circles. So do the full moon and the sun in the sky. Big or small, every circle has a centre, and the book says: *all points on the circle are at equal distance from the centre*.

Take any point on the edge of a circle. How far is it from the centre?

• The same as every other point on the edge
• It depends on which point you take
• Less than the others, if it is near the top

Find the centre

This round cut-out has a hidden centre. Four strings run from the pin to its edge.

Equal distance

A circle is the set of all points on a plane that are equidistant from a given point, its centre. The distance from the centre to any point on the circle is the radius.

The book turns the idea around: every point of a circle is at the same distance from the centre, so we define a circle by that.

Notes

A circle is the set of all points on a plane equidistant from a given point, its centre. The distance from the centre to any point on the circle is the radius.

Check yourself

A circle has centre O and radius 7 cm. P is a point on the circle. How far is P from O?

A, B and C are points on a circle with centre O, and OA = 4 cm. What is OB + OC, in cm?

Answer: 8 cm

OB = 4 and OC = 4, because B and C are on the same circle as A. So OB + OC = 4 + 4 = 8 cm.

The radius of a circle is 6 cm. The point Q is 5 cm from the centre. Where is Q?

Which of these describes the centre of a circle?

• 7 cm — correct. Yes. Every point on the circle is a radius away from the centre.
• Less than 7 cm. Points nearer than 7 cm are inside the circle. A point on the circle is exactly 7 cm away.
• It depends where P is on the circle. No. Every point of the circle is at the same distance from the centre. That is what makes it a circle.
• On the circle. A point on the circle is 6 cm from the centre. Q is only 5 cm away.
• Inside the circle — correct. Yes. The points of the circle are 6 cm from the centre, so a point only 5 cm away is not on it. It lies inside.
• Outside the circle. Outside points are farther than 6 cm from the centre. Q is nearer.
• A point on the edge of the circle. Points on the edge are on the circle. The centre is inside, equally far from all of them.
• Any point inside the circle. Only one point inside is equally far from every point on the circle.
• The point that is the same distance from every point on the circle — correct. Yes. That is why the strings were all equal when the pin was on it.