Two angles on the same arc
Mark two points A and B on a circle with center O. Join A and B to the center: the angle AOB is called the angle at the center.
Now choose any other point P on the circle, on the long way round from A to B, and join P to A and to B. The angle APB is called an angle at the circumference, because its corner P is on the circle.
Both angles open toward the same part of the circle, the arc AB on the far side from P, so both are said to stand on the arc AB. The theorem is about these two angles: the angle at the center is twice the angle at the circumference.
The angle at the center, AOB, and the angle at the circumference, APB, both stand on the arc AB at the bottom.
Move P, and nothing changes
Keep A and B where they are, and slide P along the circle. The lines PA and PB swing round, but the angle between them stays the same, wherever P goes on that arc. The angle at the center does not move either, since A, B and O are fixed.
Drag P along the circle. The angle at P reads 70° wherever it goes, and the angle at the center reads 140°.
Twice as big
Here the angle at the center is 140° and the angle at P is 70°, and 140 = 2 × 70: the angle at the center is exactly twice the angle at the circumference.
This holds for every pair of points A and B, and every position of P on the long arc. If the angle at the center is 100°, the angle at the circumference on the same arc is 100° ÷ 2 = 50°. If the angle at the circumference is 38°, the angle at the center is 2 × 38° = 76°.
Why: the line from P through the center
Draw the straight line from P through the center O, and continue it until it meets the circle again at a point D. It splits the figure into two triangles, AOP and BOP.
OA and OP are both radii, so they are equal, and triangle AOP is isosceles. The angles opposite its equal sides are equal: call the angle OAP and the angle OPA each a. In the same way OB = OP, so triangle BOP is isosceles; call the angle OBP and the angle OPB each b. The angle at P is made of the two parts, so APB = a + b.
The angle AOD is an exterior angle of triangle AOP: it lies between the side OA and the side PO continued past O. An exterior angle of a triangle equals the sum of the two interior angles opposite it, so AOD = a + a = 2a. In triangle BOP, the exterior angle DOB = b + b = 2b.
The angle at the center is the two parts together: AOB = 2a + 2b = 2(a + b). And a + b is the angle at P. So the angle at the center is twice the angle at the circumference.
The line from P through O meets the circle again at D. The two angles marked a are equal, and so are the two marked b. The exterior angle AOD is 2a, and in the same way DOB is 2b. Here a = 45° and b = 25°, so AOD = 90°, DOB = 50°, and the angle at the center is 140°, twice the 70° at P.
Other positions of P
The figure above has the center O inside the angle APB, so the line PD runs between PA and PB. Move P close to B and the center falls outside the angle at P. The same two isosceles triangles still appear, but PA and PB now lie on the same side of PO, and b is the larger part. The angle at P is the difference of the two parts, b − a, and the angle at the center is the difference of the two exterior angles, 2b − 2a = 2(b − a). It is still twice the angle at P.
If P moves across the chord AB onto the short arc, the angle APB stands on the long arc instead, and the angle at the center on that arc is the reflex angle, 360° − 140° = 220°. The theorem still holds: the angle at P is 220° ÷ 2 = 110°.
P on the short arc. The angle at P stands on the long arc, and the angle at the center on that arc is the reflex angle on the upper side, 220°: half of 220° is 110°.
The usual mistakes
Copying the angle. The angle at the center and the angle at the circumference are not equal: if the angle at the circumference is 35°, the angle at the center is 70°, not 35°.
Halving or doubling the wrong way. The angle at the center is the larger one. From the center to the circumference, halve; from the circumference to the center, double.
Using angles on different arcs. The theorem pairs an angle at the center with an angle at the circumference that stand on the same arc, between the same two points A and B.