## The Equation of a Circle

The equation of a circle is an important mathematical problem that you may encounter in your Edexcel or AQA maths exams. Learn how to recognise the equation of a circle, form an equation of a circle given its radius and centre, and use it to solve problems.

The general equation of a circle is (x – h)2 + (y – k)2 = r2, where (h, k) represents the centre and r represents the radius. You can convert a circle’s general equation into standard form by using the technique known as “completing the square.”

## Circumference

The circumference of a circle is the linear distance around the edge of the circular shape. It is proportional to the diameter, d, and relates to the famous mathematical constant, pi (p).

The standard equation of a circle is (x – a)2 + (y – b)2 = r2, where (a,b) is the centre of the circle and r is its radius. The radius of a circle is the fixed point where all points on the circle are equidistant from the centre.

To find the equation of a circle, start by marking points r units up, down, left, and right from the centre. Then draw a line from these four points. Then substitute x, y, c, and d in the equation to find the circle’s centre and radius.

## Radius

The radius of a circle is the line that connects the centre to any point on its edge. It is usually denoted by ‘r’ or ‘R’ and is always half the length of a circle’s diameter.

To find the radius, first, remember that all points on a circle’s circumference are equidistant from the centre O. This definition is important because it explains why the area, circumference and radius all change with a step-wise increase in the circle’s radius.

The standard equation for a circle is (x-h)2 + (y-k)2 = r2, where the centre of the circle is (h, k) and r is the radius. The polar form of the equation is also often used to represent a circle whose centre passes through the origin.

## Center

The centre of a circle is the point inside the circle at an equal distance from all points on its circumference. It is also a constant distance from all points on the radius.

The equation of a circle is (x – h)2 + (y – k)2 = r2, where h and k are the x and y coordinates of the centre. The radius of a circle is the length of the line that connects two points on the circle and passes through the centre.

A circle’s centre can be found using a geometric compass and a straightedge. Accurately finding the circle’s centre is important because you will need it for many calculations. You can also use a ruler to ensure your measurements are accurate. You can also draw a circle on graph paper to understand where the centre is. You can also find the circle’s centre with a digital calculator, although you’ll need to round the answer.

## Angles

Several different types of angles are formed in a circle. These include central, inscribed, interior, and exterior angles.

Two radii whose vertex lies at the circle’s centre form a central angle. This is the most important angle, as it determines the number of degrees in a circle.

The measure of a central angle equals the sum of the measures of the arcs it intercepts. This means that a central angle can subtend any angle on the circumference of a circle and an angle at its centre.

An inscribed angle is an angle whose vertex lies on a circle and whose sides contain chords of the circle. This angle has a measure of half the measure of the arcs it intercepts.