Understanding the Equation of a Circle Revision Notes | GCSE AQA Higher Further Maths
The equation of a circle with centre (0, 0) and radius r is x² + y² = r², so x² + y² = 49 has radius 7. Notes, worked examples and 11 questions answered.
Updated 14 Mar 2026 · Revise Right Team
Understanding the Equation of a Circle is part of Equation of a circle with centre (0, 0) in GCSE AQA Higher Further Maths.
3 min
Reading time
11
Lessons in this topic
GCSE AQA Higher
Pathway
Reviewed by Revise Right
Content review
What this lesson covers
The equation of a circle with centre (0, 0) and radius r is x² + y² = r², so x² + y² = 49 has radius 7. Notes, worked examples and 11 questions answered.
State the equation of a circle with centre (0, 0) and radius r as $x^2 + y^2 = r^2$
Identify the radius of a circle from its equation in the form $x^2 + y^2 = r^2$
Write down the equation of a circle given its centre at the origin and its radius
Verify whether a given point lies on, inside, or outside a circle using its equation
Key ideas to keep in view
Use this lesson to stay inside Equation of a circle with centre (0, 0) while connecting the detail back to the wider Coordinate Geometry (2 dimensions only) topic.
Equation of a circle (centre at origin): Algebraic relationship of the form $x^2 + y^2 = r^2$, where every point $(x, y)$ on the circle is exactly distance $r$ from the origin. Used when the centre is at $(0, 0)$.
Radius: The fixed distance from the centre of a circle to any point on its circumference. In $x^2 + y^2 = r^2$, the radius equals $\sqrt{r^2}$, so you must square-root the right-hand side to find it.
Revision notes
Read the core explanations from Understanding the Equation of a Circle before testing yourself from memory.
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.
$x^2 + y^2 = r^2$, so $r^2 = 49$ — Match the given equation to the standard form. The right-hand side gives $r^2$. [M1]
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 $5$.
$x^2 + y^2 = r^2$ — Always start from the standard form for a circle centred at the origin. [M1]
$r = 5$, so $r^2 = 5 \times 5 = 25$ — Square the radius explicitly — write out the multiplication to avoid errors.
$x^2 + y^2 = 25$ — 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 $\sqrt{7}$.
$x^2 + y^2 = r^2$ — Start from the standard form.
$r = \sqrt{7}$, 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.
$x^2 + y^2 = 7$ — 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.
$r^2 = 11$ — The right-hand side of $x^2 + y^2 = r^2$ is 11.
$r = \sqrt{11}$ — 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.
$x^2 + y^2 = 3^2 + 4^2$ — Replace $x$ with 3 and $y$ with 4 in the left-hand side.
$3^2 = 3 \times 3 = 9$ and $4^2 = 4 \times 4 = 16$ — Square each coordinate separately and explicitly.
$9 + 16 = 25 = r^2$ — 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.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.
What is the equation of a circle with centre (0, 0)?
The equation of a circle with centre (0, 0) and radius r is x² + y² = r². Every point (x, y) on the circle is the same distance r from the origin, so the Pythagorean relationship x² + y² = r² holds exactly for every point on it.
How do you find the radius from the equation of a circle?
The right-hand side of x² + y² = r² gives r², not r, so square-root it to find the radius. For x² + y² = 49, r = √49 = 7. For x² + y² = 11, 11 is not a perfect square, so the radius is √11.
How do you write the equation of a circle given its radius?
Square the radius and substitute into x² + y² = r². A circle with centre (0, 0) and radius 5 has equation x² + y² = 25, not x² + y² = 5. With radius √7, r² = 7, so the equation is x² + y² = 7.
How do you check if a point lies on a circle?
Substitute the coordinates into x² + y² and compare with r²: if they are equal, the point lies exactly on the circle. For x² + y² = 25 and the point (3, 4), 3² + 4² = 9 + 16 = 25, so (3, 4) lies on the circle.
What is the radius of x² + y² = 49?
The radius of the circle x² + y² = 49 is 7. Matching to x² + y² = r² gives r² = 49, and taking the positive square root gives r = √49 = 7.
How do you tell if a point is inside or outside a circle?
A point is tested by substituting its coordinates into x² + y² and comparing with r². If x² + y² = r² the point lies on the circle, if x² + y² < r² it lies inside, and if x² + y² > r² it lies outside.
What is the most common mistake with the circle equation?
The most common mistake with the circle equation is writing r instead of r² on the right-hand side. A circle with radius 5 and centre (0, 0) is x² + y² = 25, not x² + y² = 5.
Why is the equation of a circle x² + y² = r²?
The equation of a circle with centre (0, 0) is x² + y² = r² because every point on the circle is the same distance r from the origin. By Pythagoras, a right-angled triangle with legs x and y and hypotenuse r gives x² + y² = r².
What is the radius of x² + y² = 11?
The radius of x² + y² = 11 is √11. Since r² = 11 and 11 is not a perfect square, the radius is left in surd form as the exact answer.
Does (3, 4) lie on the circle x² + y² = 25?
The point (3, 4) lies exactly on the circle x² + y² = 25, because 3² + 4² = 9 + 16 = 25, which equals r².
What is the equation of a circle with radius √7?
The equation of a circle with centre (0, 0) and radius √7 is x² + y² = 7. Squaring a square root removes the root sign, (√7)² = 7, so no surds remain in the equation.