Circle theorems: Where do they come from?

What are Circle Theorems?

Circle theorems are a set of mathematical rules related to the properties and relationships of angles, lines, and segments in and around a circle. Key theorems include the angle at the center being twice the angle at the circumference when subtended by the same arc, angles in the same segment being equal, the angle in a semicircle being a right angle, and the opposite angles of a cyclic quadrilateral summing to 180 degrees. These theorems help in solving geometric problems involving circles.

Today, we'll be outlining each of these theorems in detail, along with their practical applications.

Circle Theorems Pro Tip: Watch Out for Isosceles

Isosceles triangles frequently trip people up with circle theorems, especially during the preparation for exams, so make sure to take your time and beat exam anxiety. We offer live interactive lessons to help you avoid this. Ensure you're keeping an eye out for equal angles and sides while correctly identifying triangles.

Circle theorems:

The Angle at the Centre is 2 Times the Angle at the Circumference

The centre angle is larger because the basic shape of the triangle creates the widening that doubles the angles.

To use this theorem practically:

Locate the key parts of the circle you'll use for the theorem. In this case, the angles at the centre and the circumference.
Using other facts to double-check, determine the angle at the centre (or at the circumference)
If you've discovered the angle at the centre, divide your result by two to discover the angle at the circumference. You can do the opposite of this to find the other angle.

The Angle in a Semicircle is a Right Angle

A circle can be simplified down and divided to learn more about it. One way of doing this is to divide the circle into a semicircle. This can help identify a right angle within the semicircle.

To see this theorem in action:

Locate the key areas of the circle. For this, you'll need the diameter of the circle, which will act as the bottom of your semicircle. You'll also need chords, the straight lines between two points on the circumference.
Use other facts to determine other angles in the triangle.
Use this theorem to calculate and narrow down the other angles within the triangle. One of the angles is at 90°, so the other two angles in the triangle will add up to 90, for a total of 180° in the triangle.

Angles in the Same Segment are Equal

When a circle is divided into two segments (major and minor), angles in the same segment can be counted as equal. The example above shows that they'll be equal when two triangles are created with the same chord that creates the segments.

To use this theorem:

Locate the key parts of the circle. In this case, you'll look for the chord that divides the circle into two segments. The larger is the major segment, and the smaller is the minor segment. You'll also need to identify the points on the circumference that make up the triangle.
Determine an angle at the circumference within the same segment using other angle facts.
This theorem states that one angle within the same segment equals the other angle. This means the unknown angle will be the same as the known one.

Opposite Angles That Are In a Cyclic Quadrilateral Add Up To 180°

A cyclic quadrilateral is a four-sided shape that can be inscribed into a circle. Think of it like a square box that’s been squished while being shipped. It’s an irregular shape that has four vertices on the circumference of the circle and is connected by chords.

Opposite angles within a cyclic quadrilateral will add to a total of 180°. This theorem will help you narrow down your options when determining the value of angles within a cyclic quadrilateral.

To use this theorem practically:

Pinpoint the key parts of the circle. Here, it's the four vertices positioned along the circumference of the circle and the chords that connect them.
Determine an angle within the quadrilateral using other angle facts.
Locate the angle opposite the angle you already know the value of. Subtract this value from 180 to find the value of the unknown angle.

The Angle Between the Chord and the Tangent is the Same as the Angle that's in the Alternate Segment

This circle theorem uses a tangent. Tangents are straight lines that touch the outside circumference of the circle with a single point of contact. It's at a 90° angle to the radius. By finding the angle between the chord and the tangent of the radius, you also find the angle in the alternate segment.

To use this theorem:

Find the key parts of the circle. In this case, you're looking for the vertices on the circumference, drawing the tangent outside the circle, and identifying the major and minor segments.
Using other angle facts, including other circle theorems, determine the angle between the tangent and chord.
This value is the same as the angle in the alternate segment.