Circle Theorem Problems

Topic Overview

In order to solve various circle theorem problems, you need to know the meanings of several mathematical terms regarding circles:

The 'radius' of a circle is the term used to describe the distance from the centre of a circle to its edge.

The 'diameter' of a circle is the term used to describe the distance between one edge of a circle through its centre, to the edge on the other side of the circle. The diameter of a circle is twice the value of its radius.

The 'circumference' is the term used to describe the distance around the edge of a circle. The circumference of a circle can be calculated by the formula:

\[2*\pi *Radius\]

The 'tangent' to a circle is the term used to describe a line which touches a circle at one point; without cutting across the circle. The angle between a tangent and the radius of a circle is 90°, and tangents which are positioned from a point outside the circle are equal in length.

An 'arc' is the term used to describe part of the circumference of a circle.

A 'chord' is a term used to describe a line which goes from one point to another on the circle's circumference.

A 'sector' is the term used to describe a portion of a circle. For example: a semicircle is a sector because it is a half portion of the whole circle. Similarly, a quadrant is a sector because it is a quarter portion of the whole circle.

A 'segment' is the term used to describe the portion of a circle between a chord of a circle and its associated arc. 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. These are demonstrated by the diagram below:

Image not set

Similarly, two radiuses or 'radii' can divide a circle into a minor sector and major sector, as demonstrated in the diagram below:

Image not set

Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various circle theorem problems. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Understand the terminology used to describe circles
  • Calculate the circumference and area of a circle
  • Understand and construct geometrical proofs using circle theorems

Listed below are a series of summaries and worked examples to help you solidify your knowledge about circle theorems.

1-Calculating the lengths of arcs and areas of sectors

In order to calculate the length of an arc or the area of a sector, you must calculate the value of the angle which is made by the arc or sector at the centre of a circle.

For example, if the angle is a right angle (90°), then the arc in question is a quarter of the circumference of a circle and that sector area is a quarter of the area of the circle.

Therefore, you can use the formulae for the circumference of circle and the area of a circle in order to calculate the length of the arc and the area of the sector respectively:

\[Circumference = 2*\pi *Radius\]
\[Area = \pi *Radiu{s^2}\]

Example
(a) - Calculate the length of the arc and the area of the sector of a circle which had a radius of 4cm.

Image not set

Solution
(a) - The angle which is made by the arc or sector at the centre of the circle is a right angle (90°).

Therefore, you can calculate the length of the arc and area of the sector using the formula mentioned above:

Arc length = 1/4 x circumference of the circle

Sector area = 1/4 x area of the circle

Therefore the arc length is equal to:

\[\frac{1}{4}(2*\pi *4) = \frac{{8\pi }}{4} = 2\pi = 6.2831...\]

and the sector area is equal to:

\[\frac{1}{4}(\pi *{4^2}) = \frac{{16\pi }}{4} = 4\pi = 12.5663...\]

Therefore, the length of the arc = 6.28 cm (2dp) and the area of the sector = 12.57 cm2 (2dp)

As a rule, you can calculate the length of an arc and the area of a sector by finding the angle which is made by the arc or sector at the centre of the circle and calculating what proportion it is of a whole turn (360°).

Once you know this proportion, you can multiply the circumference of the circle and the area of the circle by this proportion in order to calculate the arc length and sector area respectively:

Example
(a) - Calculate the length of the arc and the area of the sector

Image not set

Solution
(a) - First you need to calculate the proportion of this sector in relation to a whole rotation. You can do so by dividing the angle which is made by the arc or sector at the centre of the circle by 360°:

\[360/40 = 9\]

Therefore, the sector is 1/9 of the entire circle. Using this information, you can calculate the arc length and sector area:

\[\frac{1}{9}(2*\pi *10) = \frac{{20\pi }}{9} = 6.981...\]
\[\frac{1}{9}(\pi *{10^2}) = \frac{{100\pi }}{9} = 4\pi = 34.906...\]

Therefore the length of the arc = 6.98cm (2dp) and the area of the sector = 34.91cm2 (2dp)

2-The various properties of angles within a circle

Within the topic of circle theorems there are a series of rules relating to the angles within a circle. It is important that you memorise these rules as you will require them in order to solve various circle theorem problems during your GCSE maths exam. These angle rules are listed below:

1. The angle subtended at the centre of a circle is double the size of the angle subtended at the edge from the same two points

The term 'subtended' means 'made' or 'hanging down'.

The diagrams below demonstrate this rule:

Image not set

From these diagrams, you can see that the angle made at the centre of the circle is exactly double the angle made at the edge of the circle from the same points.

2. Angles which are in the same segment are equal, i.e. angles subtended (made) by the same arc at the circumference are equal.

The diagram below demonstrates this rule:

Image not set

3. The angles which are within a semicircle add up to 90°

The diagram below demonstrates this rule:

Image not set

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.

4. 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 180°.

In the cyclic quadrilateral below, angles A + C = 180°, and angles B + D = 180°.

Image not set

5. The angle between the tangent and radius is 90° and tangents from a point outside the circle are equal in length

The diagram below demonstrates this rule:

Image not set

3-Alternate segment theorem
The 'alternate segment theorem' states that:

The angle between a tangent and its chord is equal to the angle in the 'alternate segment'.

During your GCSE maths exam, you may be required to use this alternate segment theorem to solve various mathematical problems:

Example
(a) - From the diagram below, calculate the size of angle 'x' and angle 'y'

Image not set

Solution
(a) - Using the alternate segment theorem, you know that the angle between a tangent and its chord is equal to the angle in the 'alternate segment'.

Therefore, you know that angle 'x' is equal to 60°.

Once you know the value of 'x', you can calculate the value of 'y'. Given that the angles in a triangle add up to 180°;

\[180 = x + y + 40\]
\[180 = 60 + 40 + y\]
\[y = 180 - 100 = 80\]

Therefore, using the alternate segment theorem and the properties of triangles, you can calculate that angle x = 60° and angle y = 80°.

Exam Tips

  1. Write down ALL of your working out when solving circle theorem problems.
  2. Remember that the angle subtended at the centre of a circle is double the size of the angle subtended at the edge from the same two points.
  3. Bear in mind that angles which are in the same segment are equal, i.e. angles subtended (made) by the same arc at the circumference are equal.
  4. Memorise the rule that the angles which are within a semicircle add up to 90° and that the opposite angles in a cyclic quadrilateral add up to 180°.
  5. Remember that the angle between the tangent and radius is 90° and that tangents from a point outside the circle are equal in length.
  6. Memorise the alternate segment theorem which states that the angle between a tangent and its chord is equal to the angle in the alternate segment.

Topic Summary

Scoring the maximum amount of marks for circle theorem problems is largely dependent upon your ability to recognise and recall various angle rules and circle formulae.

Consequently, it is important that you memorise the angle rules mentioned throughout this guide, as well as refreshing your memory on the basic properties of circles and triangles by studying the revision guides on 'Perimeter, Area, Volume’, 'Congruent Triangles', 'Polygons' and 'Proof'.

Another great way to memorise the different angle rules and circle theorems is by attempting as many past paper questions as possible. By attempting these practice questions under examination conditions, you can familiarise yourself with the various angles rules and learn how to use them to solve a wide range of mathematical problems. With practice, you will soon be able to tackle any circle theorem questions with ease and score the highest possible marks for this topic when you sit your GCSE maths exam.

Related Topics

  • Polygons
  • Congruent Triangles
  • Trigonometry and Pythagoras
  • Construction and Loci Problems
  • Perimeter, Area, Volume
  • Transformations
  • Proof