Understanding the Equation of a Circle Revision Notes | GCSE AQA Higher Further Maths

1. Where Does the Equation Come From?

Picture a circle drawn on a coordinate grid, centred exactly at the origin. Every single point on that circle is the same distance from $(0, 0)$ — that distance is the radius $r$. That one fact is all you need to build the equation.

You already know Pythagoras: for a right-angled triangle with legs $x$ and $y$ and hypotenuse $r$, the relationship is $x^2 + y^2 = r^2$. A circle centred at the origin is nothing more than the set of all points $(x, y)$ where that Pythagorean relationship holds exactly.

The equation $x^2 + y^2 = r^2$ captures the spec requirement directly: centre $(0, 0)$, radius $r$. Recognising this form — and being able to write it down from given information — is the core skill this lesson builds.

2. Reading the Equation — Finding the Radius

When an equation is already in the form $x^2 + y^2 = r^2$, the right-hand side tells you $r^2$, not $r$. You need to square-root it to find the actual radius. This is where many students drop a mark.

Worked example: Finding the Radius from the Equation

A circle has equation $x^2 + y^2 = 49$. Write down the radius of the circle.

1. $x^2 + y^2 = r^2$, so $r^2 = 49$ — Match the given equation to the standard form. The right-hand side gives $r^2$. [M1]
2. $r = \sqrt{49} = 7$ — Take the positive square root. $7 \times 7 = 49$, confirming $r = 7$. [A1]

Answer: Radius $= 7$

3. Writing the Equation from Given Information

The second key skill is the reverse: you are given the centre $(0, 0)$ and the radius, and you must write down the equation. The method is to square the radius and substitute into $x^2 + y^2 = r^2$.

Worked example: Writing the Equation Given the Radius

Write down the equation of a circle with centre $(0, 0)$ and radius $5$.

1. $x^2 + y^2 = r^2$ — Always start from the standard form for a circle centred at the origin. [M1]
2. $r = 5$, so $r^2 = 5 \times 5 = 25$ — Square the radius explicitly — write out the multiplication to avoid errors.
3. $x^2 + y^2 = 25$ — Replace $r^2$ with 25. This is the complete equation. [A1]

Answer: $x^2 + y^2 = 25$

Notice that the equation never contains $r$ itself — only $r^2$ appears on the right-hand side. Writing $x^2 + y^2 = 5$ when the radius is 5 is one of the most common errors in this topic.

4. Worked Examples — Both Directions

Exam questions test both directions: reading the radius from an equation, and writing the equation from a radius. The worked examples below cover both, including a case with a non-integer radius.

Worked example: Writing the Equation — Surd Radius

Write down the equation of a circle with centre $(0, 0)$ and radius $\sqrt{7}$.

1. $x^2 + y^2 = r^2$ — Start from the standard form.
2. $r = \sqrt{7}$, so $r^2 = (\sqrt{7})^2 = 7$ — Squaring a square root removes the root sign: $(\sqrt{7})^2 = 7$. This is a neat result — no surds remain in the equation.
3. $x^2 + y^2 = 7$ — The equation is clean and exact. [A1]

Answer: $x^2 + y^2 = 7$

Worked example: Reading the Radius — Surd Answer

A circle has equation $x^2 + y^2 = 11$. Write down the radius, giving your answer in surd form.

1. $r^2 = 11$ — The right-hand side of $x^2 + y^2 = r^2$ is 11.
2. $r = \sqrt{11}$ — 11 is not a perfect square, so the radius is left in surd form. This is the exact answer. [A1]

Answer: Radius $= \sqrt{11}$

5. Using the Equation to Check Points

The equation $x^2 + y^2 = r^2$ does more than describe the circle — it lets you test whether any point lies on, inside, or outside it. Substitute the coordinates into the left-hand side and compare with $r^2$.

• If $x^2 + y^2 = r^2$: the point lies exactly on the circle.
• If $x^2 + y^2 < r^2$: the point lies inside the circle.
• If $x^2 + y^2 > r^2$: the point lies outside the circle.

Worked example: Checking Whether a Point Lies on the Circle

A circle has equation $x^2 + y^2 = 25$. Determine whether the point $(3, 4)$ lies on, inside, or outside the circle.

1. $x^2 + y^2 = 3^2 + 4^2$ — Replace $x$ with 3 and $y$ with 4 in the left-hand side.
2. $3^2 = 3 \times 3 = 9$ and $4^2 = 4 \times 4 = 16$ — Square each coordinate separately and explicitly.
3. $9 + 16 = 25 = r^2$ — The result equals $r^2 = 25$, so the point lies exactly on the circle. [A1]

Answer: $(3, 4)$ lies on the circle $x^2 + y^2 = 25$.

6. Deliberate Practice

These problems cover every variation the exam can throw at you: writing equations, reading radii, and checking points. Work through each one before checking the answer.

7. Exam Technique and Common Mistakes

This topic is short but precise — marks are lost through small, predictable errors rather than a lack of understanding. The guidance below targets the mistakes that appear most often in mark schemes.

Both specification requirements are now covered: you can state that $x^2 + y^2 = r^2$ is the equation of a circle with centre $(0, 0)$ and radius $r$, and you can write down the equation from given information — the two skills the exam assesses directly.