Higher Mathematics

Higher Maths Equation of a circle

Practise equation of a circle for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Find a circle's centre and radius from standard or expanded form and build its equation.

Topic practiceFormula supportMixed revision

Before you start

  • Complete the square.
  • Recognise squared distance.
  • Read coordinates with signs carefully.
Higher Mathematics lesson

Explanation

The standard circle equation displays the centre and radius. Expanded equations are converted by completing the square separately in x and y.

The signs inside the brackets are opposite to the centre coordinates.

Visual support

Circle centre and radiusThe centre and one radius are marked on a coordinate circle, matching the standard equation of a circle.xyCPr

(x − a)2 + (y − b)2 = r2 has centre (a, b) and radius r

Method and rules

  • (x − a)2 + (y − b)2 = r2 has centre (a, b) and radius r
  • Complete the square in x and y separately.
  • A point is on the circle when it satisfies the equation.

Worked examples

Worked example 1

Convert expanded form

Find the centre and radius of x2 + y2 − 6x + 4y − 12 = 0.

  1. Group x-terms and y-terms.
  2. Use x2 − 6x = (x − 3)2 − 9 and y2 + 4y = (y + 2)2 − 4.
  3. Move constants to obtain standard form.

Answer: (x − 3)2 + (y + 2)2 = 25; centre (3, −2), radius 5

Worked example 2

Method check

State the centre and radius of (x + 4)2 + (y − 1)2 = 9.

  1. Reverse the bracket signs and square-root the right side.

So: Centre (-4, 1), radius 3.

Watch out

  • Reading the centre with the bracket signs unchanged.
  • Calling r2 the radius
  • Completing only one of the two squares.

Exam reminder

For equation of a circle, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.

Related Higher Maths topics

Continue with the methods that connect most closely to this topic.

Tangents to circlesLine and circle intersectionsDistance and midpointAll Higher Maths topicsHigher Maths worksheets