Parts of a Circle

Name the lines and regions of a circle: radius, diameter, chord, tangent, arc, sector and segment, and draw or shade each one.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 28 real past-paper questions below

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The Lines of a Circle

Every circle has a centre, usually called O. The edge is the circumference. Four lines have names you must know.

Line	What it does
Radius	centre to the edge
Diameter	edge to edge, through the centre
Chord	edge to edge, missing the centre
Tangent	touches the outside at one point

Rule

A diameter is two radii end to end: $d = 2 r$ . A radius of 7 cm gives a diameter of 14 cm.

Name the Line

The 1-mark question: "Write down the mathematical name of the line AB." Ask three questions in order.

Steps

1
Does it touch the circle at one point only? Tangent.
2
Does it go through the centre? Diameter if it goes edge to edge, radius if it stops at the centre.
3
Both ends on the edge, missing the centre? Chord.

Common mistake

Writing "chord" for a line through the centre. It is a chord, but the mark needs the exact name: diameter.

Radius and Diameter

Exams often give the diameter and ask for the radius, or the other way round. One formula links them:

Formula

d = 2 r

$d$ is the diameter and $r$ is the radius. A diameter is two radii end to end. To go back the other way, halve it: $r = d \div 2$ .

Worked example

1
$d = 15.4 cm$ — The question gives the diameter. We want the radius.
2
$r = d \div 2$ — The radius is half the diameter, because the diameter is two radii.
3
$r = 15.4 \div 2$ — Put 15.4 in place of d.
4
$r = 7.7 cm$ — 15.4 ÷ 2 = 7.7. The radius keeps the same units, cm.

Circles in a row: three equal circles, radius 6 cm, sit in a row inside a rectangle. Each touches its neighbours and the sides. Find the rectangle's length and width.

Worked example

1
$d = 2 \times 6 = 12 cm$ — Lengths across circles count in diameters. Each circle is 12 cm wide.
2
$length = 3 \times 12$ — Three circles touch side by side with no gaps, so the length is three diameters.
3
$length = 36 cm$ — 3 × 12 = 36.
4
$width = 12 cm$ — There is only one row, and each circle touches the top and bottom, so the width is one diameter.

Common mistake

Doubling when you were given the diameter. Check which one the question gives: diameter → halve it, radius → double it.

Arcs, Sectors and Segments

The other 1-mark question: "Shade a sector" or "Shade a segment". Both are regions, so you shade an area.

Part	What it is
Arc	a piece of the edge
Sector	between two radii and an arc: a pizza slice
Segment	between a chord and an arc
Semicircle	half a circle, cut off by a diameter

Common mistake

Mixing up sector and segment. A sector has a corner at the centre. A segment has no corner at the centre: it is cut off by a chord.

Memory trick

Sector: a slice from the centre. Segment: sliced off by a chord.

Key idea

Minor means the smaller part, major the larger part. The minor arc goes the short way round; the major arc goes the long way. Sectors and segments are named the same way.

Drawing on a Circle

Draw questions are worth 1 mark each. The mark is for where your line starts and stops, so use a ruler and be exact.

Rule

Radius: from the centre O to the edge. Diameter: through O, edge to edge. Chord: edge to edge, missing O. Every end must sit exactly on the circle.

To draw a tangent at a point P on the circle:

Steps

1
Lightly join O to P with a ruler. This is the radius OP.
2
Put the right-angle corner of a set square (or a protractor) at P, lined up along OP.
3
Draw a long straight line through P at 90° to OP. It touches the circle at P only.

Common mistake

A “tangent” that cuts into the circle. If your line crosses the edge twice, it is not a tangent. It must touch at P only.

Key idea

“Mark a point on the circumference” means a dot exactly on the edge, not inside or outside it. Label it with the letter the question asks for.

Which Method? Read the Question

Read the command word first: write, draw or shade.

You see	You do
Name the line	tangent? diameter? radius? chord?
Draw a radius, diameter or chord	ruler, ends exactly on the edge; a diameter goes through O
Draw a tangent at P	a straight line through P at 90° to OP
Given the diameter, find the radius	halve it
Shade a sector	two radii and the arc between
Shade a segment	a chord and the arc beside it
Length round the edge	Circumference and Area of a Circle
Angles with tangents	Tangent Properties of a Circle

Key idea

These questions need no working. The mark is for the exact word or an exact drawing, so “line” or “edge” scores nothing.

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