TMUA Geometry and Circle Theorems

Circle theorems

• The angle at the centre is twice the angle at the circumference on the same arc.
• The angle in a semicircle is $90^\circ$.
• Opposite angles of a cyclic quadrilateral sum to $180^\circ$.
• A tangent is perpendicular to the radius, and the alternate segment theorem links the tangent-chord angle to an angle in the segment.

Polygons

Exterior angles of any polygon sum to $360^\circ$, so a regular $n$-gon has exterior angle $\dfrac{360^\circ}{n}$.

Pythagoras in 3D

The space diagonal of a cuboid with edges $a,b,c$ has length $\sqrt{a^2+b^2+c^2}$.

Drawing the figure

Sketch before you calculate and mark every given angle and length. Two radii form an isosceles triangle, so label them equal and use the equal base angles. Check that the angles in each triangle total $180^\circ$.

Worked examples

Common mistakes

• Applying the angle at the centre theorem to an angle on the minor arc when the point is on the major arc.
• Using interior angles instead of exterior angles for polygon sums.
• Treating a 3D diagonal as a 2D one.
• Assuming a figure is symmetric when the question does not say so.

Test yourself on this topic with our free practice questions.