---
title: "Circle Theorems: Angles at the Centre and in a Semicircle"
description: "Use the fact that the angle at the centre is twice the angle at the circumference on the same arc, and that the angle in a semicircle is a right angle."
canonical: https://lightmysky.com/learn/mathematics/circle-theorems-angles-at-the-centre-and-in-a-semicircle-mt_kOwb61sfUR
source: https://lightmysky.com/learn/mathematics/circle-theorems-angles-at-the-centre-and-in-a-semicircle-mt_kOwb61sfUR.md
retrieved: 2026-09-12
---

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# Circle Theorems: Angles at the Centre and in a Semicircle

Use the fact that the angle at the centre is twice the angle at the circumference on the same arc, and that the angle in a semicircle is a right angle.

Subject: Mathematics · Area: Geometry · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/circle-theorems-angles-at-the-centre-and-in-a-semicircle-mt_kOwb61sfUR

## Ready when they can

- Find a missing angle from the centre and circumference relationship
- Recognise the semicircle result as the case where the centre angle is 180 degrees
- Name the theorem as the reason for each step of the working

## Lesson: Centre angles and semicircles

The angle at the centre is twice the angle at the edge on the same chord. With a centre angle of 80 degrees, the edge angle is 40. Always check both angles stand on the same chord, then halve or double.

**Example.** A diameter makes a centre angle of 180 degrees, so the edge angle is half of that: 90 degrees. Every triangle in a semicircle holds a right angle. With one acute angle of 35 degrees, the third angle is 55, since angles total 180.

**Tip.** Name the theorem beside each step, since reasons earn marks. The diameter case has its own name: angle in a semicircle. When the reflex angle is 260 degrees, the minor angle is 100, so the edge angle is 50. Say which angle you halved.

**Recap.** Halve the centre angle for the edge, spot the right angle in a semicircle, and name the theorem each time.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Arc Length, Sector Area and Segments](https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ)

## Opens up

- [Circle Theorems: Cyclic Quadrilaterals and the Same Segment](https://lightmysky.com/learn/mathematics/circle-theorems-cyclic-quadrilaterals-and-the-same-segment-mt_11GlmkiqB1)