Circles • Topic 4 of 4

Inscribed & Central Angles

A central angle has its vertex at the centre, and it equals the measure of the arc it cuts off. An inscribed angle has its vertex on the circle, and it equals half the arc it intercepts — so an inscribed angle is always half the central angle that stands on the same arc. The most-tested consequence is Thales’ theorem: an angle inscribed in a semicircle (one whose sides pass through the ends of a diameter) is exactly 90°. That turns "a triangle inscribed in a circle with one side as the diameter" into an instant right triangle, ready for the Pythagorean theorem. Because the figure is not drawn to scale, always confirm from the wording whether an angle is central or inscribed before applying the halving rule.

A central angle at the centre and an inscribed angle on the same arc, the inscribed angle being halfInscribed and central anglesO80°40°Inscribed angle = half the central

✅ Solved examples

1. A central angle measures 80°. Find the inscribed angle that intercepts the same arc.
An inscribed angle is half the central angle on the same arc: 80° / 2 = 40°.
2. An inscribed angle measures 35°. Find the central angle that intercepts the same arc.
The central angle is twice the inscribed angle: 2 × 35° = 70°.
3. A triangle is inscribed in a circle with one side being a diameter. What is the measure of the angle opposite that diameter?
By Thales’ theorem, an angle inscribed in a semicircle is 90°.
4. An inscribed angle intercepts an arc of 100°. Find the inscribed angle.
Inscribed angle = ½ × intercepted arc = ½ × 100° = 50°.

✏️ Practice — try these, take hints as needed

1. A central angle measures 120°. Find the inscribed angle on the same arc.
Inscribed = ½ central.
120 / 2.
Halve it.
60°
2. An inscribed angle measures 45°. Find the central angle on the same arc.
Central = 2 × inscribed.
2 × 45.
Double it.
90°
3. A triangle is inscribed in a circle so that one side is a diameter. Find the angle opposite the diameter.
Use Thales’ theorem.
An angle in a semicircle.
It is a right angle.
90°
4. An inscribed angle intercepts an arc of 140°. Find the inscribed angle.
Inscribed = ½ arc.
½ × 140.
Halve the arc.
70°
5. An inscribed angle measures 25°. Find the arc it intercepts.
Intercepted arc = 2 × inscribed angle.
2 × 25.
Double it.
50°

📝 Topic test — 8 questions

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