Free Maths Calculator

Equation of a Circle Calculator

Three circle jobs in one place. Build the equation of a circle from its centre and radius, read the centre and radius back out of an equation by completing the square, or find the equation of the tangent at a point on x² + y² = r², the GCSE Higher question. A drawing shows the circle each time.

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    Worked Example

    Find the equation of the tangent to x² + y² = 25 at the point (3, 4).

    1. Gradient of the radius to (3, 4): 4 ÷ 3
    2. Tangent gradient, the negative reciprocal:−34
    3. y − 4 = −34(x − 3), so y = −34x + 254

    Answer: y = −34x + 254, or 3x + 4y = 25

    Try These Yourself

    Write the equation of the circle with centre (0, 0) and radius 6.Show answerHide answer

    Answer: x² + y² = 36

    Square the radius: 6² = 36.

    Find the centre and radius of (x − 1)² + (y + 2)² = 9.Show answerHide answer

    Answer: Centre (1, −2), radius 3

    Change the signs inside the brackets, and square root the 9.

    Find the tangent to x² + y² = 10 at (1, 3).Show answerHide answer

    Answer: x + 3y = 10

    Radius gradient 3, so tangent gradient −13 through (1, 3).