Maths
What is the equation of a circle, and how do you find it?
Quick answer
A circle with centre (a, b) and radius r has the equation (x − a)² + (y − b)² = r². If the centre is the origin, it is simply x² + y² = r².
In one sentence
The equation of a circle is a rule that every point (x, y) on the circle obeys: its distance from the centre is always the radius, which gives (x − a)² + (y − b)² = r².
Explained
A circle is the set of all points that are the same distance, the radius r, from one fixed point, the centre. On a coordinate grid we can turn that sentence into an equation.
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Equation of a circle with centre (a, b) and radius r (standard form):
(x − a)² + (y − b)² = r²
Where does it come from? Take any point P(x, y) on the circle and the centre C(a, b). Draw a right-angled triangle with CP as the hypotenuse: the horizontal side is x − a and the vertical side is y − b. By Pythagoras' theorem, (x − a)² + (y − b)² = CP² = r². It is the same idea as the distance formula, squared.
The most important special case is a circle centred on the origin: a = 0 and b = 0, so
x² + y² = r²
For example, x² + y² = 25 is the circle with centre (0, 0) and radius 5. It passes through (5, 0), (0, 5), (−5, 0), (0, −5) and also (3, 4), because 3² + 4² = 25.
Reading the equation. The signs inside the brackets are the opposite of the centre's coordinates: (x − 2)² means a = 2, but (x + 2)² = (x − (−2))² means a = −2. And the number on the right is r², not r: take the square root to get the radius. In the US the centre is often written (h, k), giving (x − h)² + (y − k)² = r²; it is the same formula.
Inside, on or outside? Put a point's coordinates into the left-hand side. If you get exactly r², the point is on the circle; less than r² means inside; more means outside.
General form. If you multiply out the brackets you get the general form x² + y² + Dx + Ey + F = 0. To find the centre and radius from it, complete the square for x and for y. The centre is (−D/2, −E/2) and r² = (D/2)² + (E/2)² − F. If that number is negative, the equation is not a circle at all.
Examples
| Centre | Radius | Equation |
|---|---|---|
| (0, 0) | 3 | x² + y² = 9 |
| (2, 5) | 4 | (x − 2)² + (y − 5)² = 16 |
| (−3, 1) | 6 | (x + 3)² + (y − 1)² = 36 |
| (4, −2) | √7 | (x − 4)² + (y + 2)² = 7 |
| (0, −5) | 1 | x² + (y + 5)² = 1 |
Worked example 1 (centre and a point): find the circle with centre (−1, 2) that passes through (5, 10). The radius is the distance between them: r² = (5 − (−1))² + (10 − 2)² = 36 + 64 = 100. So the equation is (x + 1)² + (y − 2)² = 100, and r = 10.
Worked example 2 (general form): find the centre and radius of x² + y² − 4x + 6y − 3 = 0. Group and complete the square: (x² − 4x) + (y² + 6y) = 3, so (x − 2)² − 4 + (y + 3)² − 9 = 3, which gives (x − 2)² + (y + 3)² = 16. Centre (2, −3), radius 4.
Common mistakes
- Getting the signs of the centre wrong. (x + 3)² + (y − 1)² = 36 has centre (−3, 1), not (3, −1). Fix: write it as (x − (−3))².
- Forgetting to square the radius. A circle with radius 6 is … = 36, not … = 6. Fix: the right-hand side is r².
- Giving r² as the radius. From x² + y² = 49 the radius is √49 = 7, not 49.
- Completing the square on one side only. When you add 4 and 9 inside the brackets, take them away again (or add them to the other side) so the equation stays balanced.
Step by step
Step by step
- Find the centreRead it from the question, or use the midpoint if you know the two ends of a diameter.
- Find r²Use the given radius, or the squared distance from the centre to a point on the circle.
- SubstitutePut a, b and r² into (x − a)² + (y − b)² = r².
- CheckPut a known point of the circle into your equation: the two sides must be equal.
- From general formGroup the x terms and the y terms, complete the square for each, then read off the centre and r.
A tangent to a circle is perpendicular to the radius at the point where it touches. That fact lets you find the equation of a tangent line, as in question 16 below.
Practice questions
Equation of a circle
Try each one, then check your answer.
- Write the equation of the circle with centre (0, 0) and radius 5.
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x² + y² = 25 - Write the equation of the circle with centre (2, 3) and radius 4.
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(x − 2)² + (y − 3)² = 16 - Write the equation of the circle with centre (−1, 4) and radius 3.
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(x + 1)² + (y − 4)² = 9 - Find the centre and radius of (x − 5)² + (y + 2)² = 49.
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Centre (5, −2), radius 7 - Find the centre and radius of x² + (y − 1)² = 10.
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Centre (0, 1), radius √10 ≈ 3.16 - Is the point (3, 4) on the circle x² + y² = 25?
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3² + 4² = 9 + 16 = 25: yes - Is (1, 2) inside, on or outside (x − 4)² + (y − 6)² = 16?
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(1 − 4)² + (2 − 6)² = 9 + 16 = 25 > 16: outside - A circle has centre (1, −2) and passes through (4, 2). Find its equation.
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r² = 3² + 4² = 25: (x − 1)² + (y + 2)² = 25 - The ends of a diameter are (−2, 1) and (6, 7). Find the equation of the circle.
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Centre = midpoint (2, 4); r² = (6 − 2)² + (7 − 4)² = 16 + 9 = 25: (x − 2)² + (y − 4)² = 25 - Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.
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(x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25: centre (3, −2), radius 5 - Find the centre and radius of x² + y² + 8x − 2y + 8 = 0.
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(x + 4)² − 16 + (y − 1)² − 1 + 8 = 0, so (x + 4)² + (y − 1)² = 9: centre (−4, 1), radius 3 - Write (x − 2)² + (y + 5)² = 9 in general form.
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x² − 4x + 4 + y² + 10y + 25 − 9 = 0: x² + y² − 4x + 10y + 20 = 0 - Where does (x − 3)² + (y − 1)² = 25 cross the y-axis?
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Put x = 0: 9 + (y − 1)² = 25, so (y − 1)² = 16 and y − 1 = ±4: (0, 5) and (0, −3) - Does x² + y² + 2x + 4y + 10 = 0 describe a circle?
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(x + 1)² + (y + 2)² = 1 + 4 − 10 = −5. A square sum cannot be negative: no, it is not a circle. - Find the circle through (0, 0), (6, 0) and (0, 8).
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In x² + y² + Dx + Ey + F = 0: (0, 0) gives F = 0; (6, 0) gives 36 + 6D = 0, D = −6; (0, 8) gives 64 + 8E = 0, E = −8. So x² + y² − 6x − 8y = 0: (x − 3)² + (y − 4)² = 25 - Find the tangent to x² + y² = 25 at the point (3, 4).
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The radius to (3, 4) has slope 4/3, so the tangent has slope −3/4: y − 4 = −¾(x − 3), which gives 3x + 4y = 25
Word problems
Circles on a map
Try each one, then check your answer.
- A phone mast at (0, 0) reaches everything within 8 km. Write the equation of the edge of its range. Is a house at (5, 6) (in km) covered?
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x² + y² = 64; 5² + 6² = 61 < 64, so the house is inside: yes, covered. - A garden sprinkler at (3, 2) waters everything up to 4 m away. Write the equation of the edge of the wet area. Does it reach a rose bush at (6, 6)?
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(x − 3)² + (y − 2)² = 16; (6 − 3)² + (6 − 2)² = 9 + 16 = 25 > 16: no (the bush is 5 m away). - On a plan in metres, the edge of a round pond is x² + y² − 10x − 8y + 32 = 0. Find the centre and the radius of the pond.
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(x − 5)² + (y − 4)² = 25 + 16 − 32 = 9: centre (5, 4), radius 3 m - A Ferris wheel has its centre 20 m above the ground and a radius of 18 m. With the ground as the x-axis and the centre at (0, 20), write its equation and find the height of the lowest seat.
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x² + (y − 20)² = 324; lowest point 20 − 18 = 2 m above the ground. - A goat is tied to a post at (2, −1) with a 5 m rope. Can it reach a carrot at (5, 3)?
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(5 − 2)² + (3 + 1)² = 9 + 16 = 25 = 5², so the carrot is exactly on the edge: yes, just.
Questions people ask
What is the standard form of the equation of a circle?
(x − a)² + (y − b)² = r², where (a, b) is the centre and r the radius. It is also called centre-radius form; in the US it is usually written with (h, k).
What is the general form of the equation of a circle?
x² + y² + Dx + Ey + F = 0 (in the UK often written x² + y² + 2gx + 2fy + c = 0). Complete the square to turn it back into standard form and read the centre and radius.
How do you find the equation of a circle from two points?
If the two points are the ends of a diameter, the centre is their midpoint and the radius is half the distance between them. Two points on the circle alone are not enough: many circles pass through them, so you need one more fact.
How do you find the equation of a circle through 3 points?
Put each point into x² + y² + Dx + Ey + F = 0. That gives three equations in D, E and F; solve them, then complete the square. Another way is to draw the perpendicular bisectors of two chords: they cross at the centre.
Can you find the area from the equation of a circle?
Yes: read off r and use area = πr² and circumference = 2πr. For x² + y² = 25, r = 5, so the area is 25π ≈ 78.5 square units and the circumference 10π ≈ 31.4 units.
What is the parametric equation of a circle?
x = a + r cos t and y = b + r sin t, where t runs from 0 to 360° (or 0 to 2π). As t changes, the point goes once round the circle.
Quick quiz
Equation of a circle
Check what you know with 4 quick questions.
Question 1 of 4
What are the centre and radius of (x − 3)² + (y + 4)² = 36?
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Related topics
Parts of a circle
A circle has a centre, a radius, a diameter and a circumference. The radius runs from the centre to the edge, and the diameter is twice the radius.
Distance formula
The distance between (x₁, y₁) and (x₂, y₂) is d = √((x₂ − x₁)² + (y₂ − y₁)²). It is Pythagoras' theorem used on a coordinate grid.
Pythagoras' theorem
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c², where c is the longest side (the hypotenuse). For sides 3 and 4, the hypotenuse is 5.
Completing the square
Completing the square rewrites x² + bx + c as (x + b/2)² − (b/2)² + c. It solves quadratics, finds the vertex (turning point) of a parabola and proves the quadratic formula.
Angles in parallel lines
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.