Maths- Circles

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Segments and arcs

When a chord divides a circle into segments, it produces a minor segment and major segment. It also divides the circumference into a minor arc and major arc.

image: circle: divided by a line near the top half, smaller half: minor segment, larger half: major segment, arrows shows out-side edge of major segment: major arcm, out-side edge of minor segment: minor arc. (

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In a similar way, two radiuses (or radii) divide the circle into a minor sector andmajor sector.

image: circle, triangular segment: minor sector, rest of circle: major sector. (

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The angle at the centre of a cirle

The angle at the centre of a circle

One of the rules of geometry is that the angle at the centre of a circle is double the size of the angle at the edge from the same two points.

Have a look at the diagrams below showing this rule:


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Angles in the same segment

Angles in the same segment are equal

Angles subtended (made) by the same arc at the circumference are equal.

image: circle with triangular shapes inside: top corner of each triangular shape: A. (

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Angles in a semicircle

Angles in a semicircle are 90°

The angle at the centre (AOB) is twice the angle at the circumference (APB). As AOB is 180°, it follows that APB is 90°. AOB is the diameter, so it follows that the angle in a semicircle is always a right angle:

image: circle with triangular shape inside: top corner: P, bottom right: B, straight through: O to A on the bottom left. ( 

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Angles in a cyclic quadrilateral

Opposite angles in a cyclic quadrilateral add up to 180°

A cyclic quadrilateral is a quadrilateral whose vertices all touch the circumference of a circle. The opposite angles add up to 180o. The angles A + C= 180o, B + D = 180o.image: circle with a quadrilateral shape inside: top corner: A, right: B, bottom: C, left: D. (

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Angles between the tangent and radius

The angle between the tangent and radius is 90°

image: a tangent ( circle with solid circle inside placed central point, a line represents radius placed at 90 degrees , straight-line off the left edge of circle. (

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Continuing of angles between tangents and radius'

tangent to a circle is a line which just touches the circle.


A tangent is always at right angles to the radius where it touches the circle.

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Alternate segment theorem

Alternate segment theorem

The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.The angle between a tangent and chord is equal to the angle made by that chord in the alternate segment.In this diagram we can use therule to see that the yellow angles are equal, and the blue angles ar equal

image: a circle with a triangle inside, 4 angles are marked, one of the triangle's angles is missing (

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Tangents from a point outside the cirlce

Tangents from a point outside the circle are equal in length

 The two tangents fit together:

image: stage 2 (

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