Circle theorems are essential rules that help you solve angle...
Maths136 views·Updated 28 Jun 2026·2 pagesUnderstanding GCSE Maths: Circle Theorems Made Simple
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chloe m@chloem.980
1
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Core Circle Theorems
Angle at the centre is always twice the angle at the circumference when both angles are formed by the same two points on the circle. This is one of the most useful theorems you'll encounter – if you spot an angle at the centre, you immediately know the circumference angle is half that size.
Cyclic quadrilaterals are four-sided shapes where all vertices touch the circle's edge. The key rule here is simple: opposite angles always add up to 180°. So if one angle is 70°, the opposite one must be 110°.
Angles in a semicircle are always 90°. Whenever you see an angle formed by drawing lines from both ends of a diameter to any point on the circumference, you've got yourself a right angle – no measuring needed!
Angles subtended by the same arc are equal. If two angles are created using the same two points on the circumference, they'll always be identical in size.
Quick tip: Look for the centre of the circle first – it often holds the key to solving the problem!
2
of 2
Tangent Properties and Alternate Segments
Tangent lines create right angles with the radius at the point where they touch the circle. Remember, a tangent only touches the circle at one point, never crosses through it.
Equal tangents from an external point mean that if you draw two tangent lines from the same outside point to a circle, both tangent segments will be exactly the same length. This property is brilliant for solving problems involving tangent lengths.
Alternate segment theorem might sound complex, but it's actually straightforward. When you have a tangent and a chord meeting at the same point, the angle between them equals the angle in the alternate segment (the bit of the circle on the opposite side of the chord).
These tangent properties often combine with other circle theorems in exam questions, so practise spotting them in different contexts.
Exam hint: Always mark right angles when you spot tangent-radius relationships – it makes the rest of the problem much clearer!
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Maths136 views·Updated 28 Jun 2026·2 pagesUnderstanding GCSE Maths: Circle Theorems Made Simple
C
chloe m@chloem.980
Circle theorems are essential rules that help you solve angle problems involving circles. These theorems appear frequently in GCSE Maths exams, so mastering them will boost your confidence and marks significantly.
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of 2
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Core Circle Theorems
Angle at the centre is always twice the angle at the circumference when both angles are formed by the same two points on the circle. This is one of the most useful theorems you'll encounter – if you spot an angle at the centre, you immediately know the circumference angle is half that size.
Cyclic quadrilaterals are four-sided shapes where all vertices touch the circle's edge. The key rule here is simple: opposite angles always add up to 180°. So if one angle is 70°, the opposite one must be 110°.
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Angles subtended by the same arc are equal. If two angles are created using the same two points on the circumference, they'll always be identical in size.
Quick tip: Look for the centre of the circle first – it often holds the key to solving the problem!
2
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Tangent Properties and Alternate Segments
Tangent lines create right angles with the radius at the point where they touch the circle. Remember, a tangent only touches the circle at one point, never crosses through it.
Equal tangents from an external point mean that if you draw two tangent lines from the same outside point to a circle, both tangent segments will be exactly the same length. This property is brilliant for solving problems involving tangent lengths.
Alternate segment theorem might sound complex, but it's actually straightforward. When you have a tangent and a chord meeting at the same point, the angle between them equals the angle in the alternate segment (the bit of the circle on the opposite side of the chord).
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