Circle Theorem: Angle in a Semicircle Is 90°
(KS3, Year 8)

The Lesson

A triangle drawn from two ends of a diameter of a circle makes an angle of 90° at the circumference. It is a right triangle because the angle at the circumference is a right angle.

Why Is the Angle in a Semicircle 90°?

The circle theorem where the angle in a semicircle is 90° is a special case of the circle theorem where the angle at the center is twice the angle at the circumference. The semicircle is bounded at the diameter. The angle made by the diameter at the center is 180°.

Let the angle at the circumference be θ.

The angle at the center is twice the angle at the circumference.
180° = 2 × θ θ = 180° ÷ 2 θ = 90°
This is why the angle in a semicircle is 90°.

Lesson Slides

The slider below shows a real example of the circle theorem that the angle in a semicircle is 90°. Open the slider in a new tab

What Is a Semicircle?

A semicircle is half a circle. The curved edge is half a circumference, and the straight edge is the diameter.

All Angles in a Semicircle Are 90°

No matter on which point of the circumference the triangle is drawn to, the angle will be 90°.

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See Also

What is a triangle? What is an angle? What is a right triangle? What is a right angle? What is a circle? What are the circle theorems?