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Equation of a Circle

HigherHigher tier onlyAQAEdexcelOCR

Practise equation of a circle with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through using the equation of a circle, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. x² + y² = r² is a circle centred on the origin with radius r.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The equation of a circle centred on the origin is \(x^2 + y^2 = r^2\), where \(r\) is the radius. It comes directly from Pythagoras' theorem applied to any point on the circle.

The crucial detail is that the number on the right is the radius squared, not the radius. So \(x^2 + y^2 = 25\) describes a circle of radius \(5\), and reading the radius as \(25\) is the standard error.

Deciding whether a point lies on the circle simply means substituting its coordinates. If the left side equals \(r^2\) the point is on the circle; if it is smaller the point is inside, and if larger it is outside.

Revision notes

The equation

For a circle centred at the origin with radius \(r\), every point satisfies \(x^2 + y^2 = r^2\).

So a circle of radius \(6\) has equation \(x^2 + y^2 = 36\). To find the radius from an equation, take the square root of the constant.

Testing a point

Substitute the coordinates into the left-hand side and compare with \(r^2\).

For \(x^2 + y^2 = 25\) and the point \((3, 4)\): \(9 + 16 = 25\), so the point lies on the circle.

Inside or outside

A smaller result means the point is inside the circle; a larger result means outside.

For the same circle, \((1, 2)\) gives \(1 + 4 = 5\), which is less than \(25\), so the point lies inside.

Key points

  • A circle at the origin has equation \(x^2 + y^2 = r^2\).
  • The constant is the radius squared.
  • Take the square root to find the radius.
  • Substitute a point to test whether it lies on the circle.
  • A smaller result means inside.
  • A larger result means outside.

Worked examples

Example 1

Find the radius of the circle \(x^2 + y^2 = 49\).

Working

\[r^2 = 49\]the constant is the radius squared
\[r = 7\]take the square root

Example 2

Write the equation of a circle centred at the origin with radius \(9\).

Working

\[r^2 = 81\]square the radius
\[x^2 + y^2 = 81\]write the equation

Example 3

Does the point \((6, 8)\) lie on the circle \(x^2 + y^2 = 100\)?

Working

\[6^2 + 8^2 = 36 + 64\]substitute the coordinates
\[= 100\]the result equals the constant
\[\text{Yes, it lies on the circle}\]state the conclusion

Common mistakes

  • Reading the constant as the radius.

    In x² + y² = 25 the radius is 5, not 25. The constant is r squared.

  • Forgetting to square the radius when writing an equation.

    A radius of 9 gives 81 on the right-hand side.

  • Not squaring both coordinates when testing a point.

    Both x and y must be squared before adding.

  • Assuming a point is on the circle without checking.

    Substitute and compare with r² before concluding.

Exam tips

  • Write down r² before doing anything else.
  • Take the square root to find the radius from an equation.
  • Substitute both coordinates when testing a point.
  • Compare the result with r² to decide inside, on, or outside.

Key terms

Radius
The distance from the centre to any point on the circle.
Origin
The point \((0,0)\) where the axes cross.
Circle equation
\(x^2 + y^2 = r^2\) for a circle centred at the origin.
Substitute
To replace variables with given values.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.