Angle at the centre is twice the angle at the circumference

Same arc: centre angle = 2 × circumference angle.

Key formulas

Both angles must stand on the identical arc. Mark the arc before writing 2×.

Arc AB subtends 40° at the circumference → 80° at the centre.

If the arc is a semicircle, the centre angle is 180°, so the circumference angle is 90° — angle in a semicircle.

Problem. An arc subtends 42° at the circumference. Find the angle at the centre on the same arc.

Solution. Angle at centre = 2 × 42° = 84°.

Using the reflex arc doubles the wrong quantity.

Circumference angle on an arc is 28°. Centre angle on that arc is:

Anywhere on the remaining part of the circle, on the same side of chord AB.