Updated: Sep 4, 2021
Students of geometry often find circles particularly difficult to deal with. Unlike other shapes, circles lack the straight sides joined at definite angles which much of geometry equips the student to understand. Nevertheless, getting to grips with circles is of central importance to geometry and its applications. Architects and engineers often have to make precise calculations about circles when designing towers or wheels. Astronomers need a good understand of circles when studying the orbits of the planets. As such, mathematicians since the ancient Greek thinker Euclid have developed a set of simple equations which describe the properties of circles. This post will outline these ‘circle theorems’ by using diagrams and easy-to-understand language. It will then go through some hints and tips for making the best of circle theorems in your work.
As simple as possible
This post will go through the circle theorems without all of the unnecessary jargon that you find in some explanations. Nevertheless, there are a few technical terms and ways of talking which should help us along.
One way that of talking about angles that makes understanding circle theorems easier is to name angles after the three points which join the lines that make them up. Take a look at the triangle in figure 1. This triangle is made up of three points: q, r, and s. If we wanted to talk about this triangle, we would need an easy way to identify which of the three angles we’re talking about. The way we do this is to list the three points which join the lines which make up the angle, with the point at the angle in the middle.
So, for example, the blue angle in figure 1 is made up of two lines from points q, r, and s (with point q at the angle itself). Therefore, we can label the blue angle rqs. In a similar way, we can call the red angle qsr and the green angle qrs.
Some more definitions can help us understand concepts in this article better:
A ‘chord’ is a straight line that extends between two points on the circumference on the circle.
A ‘tangent’ is a straight line which only touches the circumference of a circle at a single point.
A tringle is said to be ‘inscribed’ within a circle if each of its corners lies on the circumference of the circle.
The ‘vertex angle’ of a triangle is the angle opposite its base. So, on figure 1, if we take the line between r and s as the base of the triangle, then the angle in blue is the vertex angle.
Theorem 1: The centre angle is always twice the size of the circumference angle.
In figure 2 we have three points, a, b, and c, around the circumference of a circle and a point at the centre of the circle, which we can call point m. The angle amc is always double the angle abc, for any set of three points along the circumference.
Theorem 2: The vertex angles of inscribed triangles with the same base chord will always be the same.
In figure 3 we have four points on the circumference of a circle, a, b, c, and d. There are two triangles, both with the chord between a and b as their base. One triangle has c as its third point, while the other has d. The angles acb and adb (the two ‘vertex angles’ of the triangles) will always be the same.
Theorem 3: The length of two tangents from a point outside the circle to the circumference of the circle will always be the same.
In figure 4 we have two tangents which touch different points on the circumference of a circle, a and b. They then cross each other at another point, c. The length between a and c will always be equal to the length between b and c.
Theorem 4: Opposite angles of a quadrilateral inscribed within a circle will always add up to 180°
In figure 5 we have a quadrilateral whose four corners are points on the circumference of a circle. The opposite angles of this quadrilateral will add up to 180°. For example, cab added to cdb will add to 180°.
Theorem 5: The angle between a chord, which is one side of a triangle inscribed within a circle, and a tangent is always equal to the angle of the triangle opposite that chord.
In figure 6, we have an inscribed triangle, whose corners are at points a, b, and c on the circumference of a circle. We also have a tangent between points d and e, which meets the circle at point c. The angle bce is equal to the angle of the triangle opposite that chord, cab. The same is true of angles dca and cba.
Theorem 6: The vertex angle of an inscribed triangle will be a right angle if it has the diameter as its base.
In figure 7, we have a triangle inscribed within a circle. The base of the triangle is a chord that goes through the middle of the circle. As such, it is the length of the diameter of the circle. This means that the angle opposite the base, which is angle bac on figure 7, will always be a right angle (90°), no matter where on the circle it lies.
Theorem 7: The angle between the radius and a tangent is always a right angle.
In figure 8, we have the radius of the circle, represented by the line between the midpoint m and a point of the circumference b. There is also a tangent, between lines a and c, which meets the circle at point b. As the radius line meets the edge of the circle at the point where the tangent meets the circle, the angle between them must be a right angle.
Theorem 8: If a line that goes through the centre of a circle is perpendicular to a chord, then it will always bisect that chord.
In figure 9 we have a line which goes through the centre of the circle, point m. We also have a chord, between points a and c. The two lines meet at point b. The angles mbc and mba are right angles. From this we know that the line through the centre ‘bisects’ the chord; it cuts the chord in half, so to speak. What this means is that the part of the line between a and b must be equal in length to the part of the line between b and c.
How do I deal with so many circle theorems?
When studying circle theorems, it is easy to get overwhelmed by questions containing complex language and multiple parts. Here are some hints and tips for making the best use of circle theorems in your work.
Firstly, practice makes perfect. While it may seem dull, nothing beats actually using the circle theorems themselves to answer questions and solve problems. Fundamentally, circle theorems are important as they provide a way of understanding and manipulating the world around us. As such, the practical skill of using circle theorems is best developed through just using them.
Secondly, these theorems are best understood intuitively. It is important not only to know the right equations and technical terms, but also to understand why these theorems describe circles the way that they do. Just sitting with a list of the theorems and trying to learn them word for word will not help you very much. You have to look at the circles themselves and just 'see' why, for example, the vertex angle of an inscribed triangle must always be a right angle if it has the diameter as its base. You must imagine all the different ways of constructing whichever shape is relevant to the theorem, whether it be a triangle or quadrilateral, inside of a circle. In doing so, you get an intuitive feel for the properties of circles and of inscribed shapes. This will mean that when you are faced with questions in exams or homework, you don't have to sift through a mental list of theorems that you've remembered word for word. Instead, you simply know what to do just by looking.
Fingers crossed 🤞
Hopefully now you’ll be in a better position to tackle questions involving circles head on. To help you to revise, why not try to construct your own circles on paper. Draw some tangents, or perhaps an inscribed triangle. You can practice applying the theorems by working out what you can discover about the circle's you've constructed using them. Alternatively, why not head over to studysquare.co.uk to test yourself on what you’ve learnt:
Get hold of our Exam Revision Guide and let’s turn your exam experience into a success story 😀 → https://www.studysquare.co.uk/pdf
Logic Enthusiast is an independent writer and is studying for an MA in Philosophy at the University of Edinburgh. He is particularly interested in Logic and the Philosophy of Science.