shape & space : circles


subtended angles




The parts of a circle


circle parts


centre - the point within the circle where the distance to points on the circumference is the same.

radius - the distance from the centre to any point on the circle. The diameter is twice the radius.

circumference(perimeter) - the distance around a circle.

chord is a straight line joining two points on the circumference.

diameter - a chord(of max. length) passing through the centre

sector - a region enclosed by two radii and an arc.

segment - the region enclosed by a chord and an arc of the circle.

tangent - a straight line making contact at one point on the circumference, such that the radius from the centre is at right angles to the line.


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Subtended angles

When a chord subtends an angle on the circumference of a circle, the angle subtended at the centre of the circle is twice the angle.


circle - subtended angles #1


A diameter subtends a right-angle at the circumference.


90 deg. subtended angle

angle XPZ = 90 deg.


Angles subtended by a chord onto the circumference of a circle are equal.


subtended angles #2

angle ADB = angle ACB


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chord #1


The line joining the centre of a circle and the mid-point of a chord is perpendicular to the chord. The chord is bisected into two equal halves.



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circles - tangents


The tangents to a circle from a point are equal in length.



the tangents subtend equal angles at the centre of the circle

angle POA = angle POB


the angles between the tangents and the line joining the centre of the circle and the point are equal.

angle APO = angle BPO

note : Triangle APO and triangle BPO are congruent.


The angle between a tangent and a chord

The angle between a tangent and a chord is equal to the angle subtended by the chord in the opposite segment.


circle - tangent #2

angle ZPY = angle PXY


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Cyclic quadrilaterals

Opposite angles in a cyclic quadrilateral add up to 180 deg.


circle -  quads


As with all quadrilaterals, the sum of the interior angles = 360 deg.

Any exterior angle of a cyclic quadrilateral equals the interior opposite angle.


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