Circle Vocabulary and Properties

Grade 3

Parts of a circle

You need precise vocabulary before tackling circle theorems. A radius runs from the centre to any point on the circumference. A diameter passes through the centre — it is twice the radius. A chord connects two points on the circumference without passing through the centre. An arc is a piece of the circumference. A sector is a pie-slice region bounded by two radii and an arc. A segment is the region between a chord and an arc.

A tangent touches the circle at exactly one point and is perpendicular to the radius at that point — that right angle is a key property in many questions.

Key property: tangent and radius

A tangent to a circle is always perpendicular to the radius drawn to the point of contact. This gives you a right angle to work with — use Pythagoras or trigonometry from there.

Worked example

EXAMPLE

A tangent from external point T touches a circle of radius 5 cm at point P. The distance from T to the centre O is 13 cm. Find the length TP.

Angle OPT = 90° (tangent perpendicular to radius). By Pythagoras: TP² = OT² − OP² = 13² − 5² = 169 − 25 = 144. TP = √144 = 12 cm.

Try this one

A chord \( PQ \) is drawn in a circle of centre \( O \) and radius \( 9 \) cm. The perpendicular distance from \( O \) to the chord \( PQ \) is \( 4 \) cm.

Work out the length of the chord \( PQ \). Give your answer to 1 decimal place. [4]

Nova's hint:

The perpendicular from the centre to a chord bisects the chord, creating a right-angled triangle.

Ask yourself: 'If I drop a perpendicular from the centre, what right-angled triangle appears, and which side does Pythagoras give me?' The radius is the hypotenuse.

Try to Solve it with Nova?