Circle Theorems Calculator

Apply the angle at the centre circle theorem: enter the angle you know — at the centre or at the circumference — and get the matching angle, the cyclic quadrilateral's opposite angle, and the centre angle's classification.

Quick Facts

Angle at the Centre Theorem
Central angle = 2 × inscribed angle
True whenever both angles stand on the same arc, for any inscribed angle strictly between 0° and 180°.
Cyclic Quadrilateral Theorem
Opposite angles sum to 180°
Any quadrilateral with all four vertices on the circle has opposite angles that are supplementary.

Your Results

Calculated
Angle at the centre
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2 × angle at the circumference (same arc)
Angle at the circumference
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Inscribed angle standing on the same arc
Cyclic quadrilateral opposite angle
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180° − angle at the circumference
Centre angle type
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Acute, right, obtuse, straight, or reflex

Ready

Choose which angle you know, enter its value, then press Calculate.

Formula and method for the Angle at the Centre Theorem

The angle at the centre theorem is one of the core circle theorems in Euclidean geometry: for two angles that stand on the same arc, the angle formed at the centre of the circle is always exactly twice the angle formed at any point on the remaining circumference. Written as a formula, if θ is the angle at the circumference, the angle at the centre is 2θ.

How the calculation works

This calculator works in either direction. Choose which angle you already know — at the centre or at the circumference — and enter its value in degrees. If you supply the circumference angle, the centre angle is found by doubling it. If you supply the centre angle, the circumference angle is found by halving it. From that pair, the calculator also reports the opposite angle in a cyclic quadrilateral (180° minus the circumference angle) and classifies the centre angle as acute, right, obtuse, straight, or reflex.

Common mistakes

  • Different arcs: the doubling relationship only holds when both angles are subtended by the same arc. Mixing up which arc an angle sits on gives the wrong pairing.
  • Reflex angles: if the circumference angle is obtuse (over 90°), doubling it produces a centre angle over 180° — a reflex angle. That is not an error; it is the theorem describing the angle measured the long way round the centre.
  • Degree limits: a centre angle must stay below 360°, and a circumference (inscribed) angle must stay below 180°, since it sits inside a triangle formed with a chord.

Where these theorems come up

  • GCSE and A-level geometry exams routinely test the angle at the centre, cyclic quadrilateral, and alternate segment theorems together
  • Technical drawing and CAD work uses circle theorems to fix unknown angles when only partial dimensions are given
  • Structural and mechanical design (gears, arches, curved trusses) relies on the same inscribed-angle relationships to position points on a circular arc
  • Navigation and surveying use circle theorems to calculate bearings and angles from arc and chord measurements

Frequently Asked Questions

What is the angle at the centre circle theorem?
It states that the angle subtended at the centre of a circle by an arc is twice the angle subtended at any point on the circumference by the same arc. In short, central angle = 2 × inscribed angle, whenever both angles stand on the same arc.
Why does my central angle come out greater than 180°?
When the angle at the circumference is obtuse (greater than 90°), doubling it passes 180°, producing a reflex angle at the centre. This is correct: the theorem still holds, it is just describing the angle measured the long way round the centre.
How does this relate to angles in a semicircle?
If the angle at the circumference is exactly 90°, the central angle works out to 180° — a straight line, meaning the chord passes through the centre and is a diameter. This is Thales' theorem: any angle inscribed in a semicircle is a right angle.
What is the cyclic quadrilateral connection?
The angle at the circumference and its cyclic-quadrilateral opposite angle always sum to 180°, because both are inscribed angles standing on the two arcs that together make up the whole circle. This calculator reports that opposite angle automatically.

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