Equation of a Circle
Practise using x² + y² = r² for circles centred at the origin in this GCSE Higher worksheet, aimed at approximately Grade 7. Find radii, write equations from measurements and graphs, test points and find intersections with horizontal or vertical lines. The final two questions offer a clearly labelled extension to circles with other centres. Jump to the questions
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Worksheet preview and key skills
Worksheet preview
Eight questions build from finding a radius to solving coordinate problems. The first six focus on circles centred at (0, 0); the final two are stretch questions about other centres.
For example, a circle with diameter 12 has radius 6, so its equation is x² + y² = 36. To test a point, substitute both coordinates and compare their squared sum with 36.
What you’ll practise
- Finding radii and writing equations from radii, diameters, areas and a coordinate grid.
- Testing points and finding both intersections with a horizontal or vertical line.
- Stretch: interpreting centre form and using a centre and a point to write an equation.
Use the interactive worksheet below, or read the Topic guide for the method and worked example.
Equation of a Circle
Complete each answer, then select Check answer. Give exact values unless a rounding accuracy is stated; each numbered question is worth one mark.
Topic guide
A circle centred at the origin
The main GCSE focus here is x² + y² = r², where the centre is (0, 0) and r is the radius. Every point (x, y) on the circle is distance r from the origin. Its horizontal and vertical distances form a right-angled triangle, so Pythagoras gives x² + y² = r². Squaring also makes this work for negative coordinates.
Finding the radius
The number on the right is the radius squared. For x² + y² = 64, r = √64 = 8, not 64. A radius is positive. For x² + y² = 30, r = √30, or 5.48 to two decimal places. Only round when asked.
If the equation is scaled, first divide every term by the common coefficient: 3x² + 3y² = 72 becomes x² + y² = 24. The radius is then √24.
Writing an equation
Square the radius and put the result on the right. A radius of 3.5 gives x² + y² = 12.25. A radius of 3√2 gives r² = 9 × 2 = 18, so x² + y² = 18.
If the diameter is 18, halve it to get r = 9, then square: x² + y² = 81. If the area is 27π, use A = πr² to get r² = 27 directly, giving x² + y² = 27. On a coordinate grid, read the radius from the origin to an axis crossing, checking the scale.
Testing a point
For the circle x² + y² = 85, test (−6, 7): (−6)² + 7² = 36 + 49 = 85, so the point lies on the circle. The point (−6, 6) gives 72, so it does not. Use both coordinates and compare with the right-hand side.
Finding two intersections
For x² + y² = 100 and y = −8, substitute to get x² + 64 = 100. Thus x² = 36 and x = −6 or 6. The intersections are (−6, −8) and (6, −8). For a vertical line, substitute its fixed x-coordinate and solve for y in the same way.
Common mistakes
- Confusing r with r², or using the diameter as the radius.
- Forgetting to square the coefficient in a surd radius.
- Treating a negative coordinate squared as negative: (−4)² = 16.
- Giving just one intersection when two square roots are needed.
- Rounding an exact answer when no approximation was requested.
Stretch / extension: other centres
This is extension material for the final two questions; the main GCSE focus above remains circles centred at the origin. In (x − h)² + (y − k)² = r², the centre is (h, k). For example, (x + 1)² + (y − 4)² = 9 has centre (−1, 4) and radius 3. Notice the opposite visible signs. Given a centre (h, k) and a point (a, b), calculate r² = (a − h)² + (b − k)², then insert h, k and r² into centre form.