Library / Foundations: Maths / FMA-52
Foundations: Maths
Radius of a circle from its centre and a point on it
Equation of a circle with centre (a, b) and radius r, solved here for the radius from a point (x, y) on the circle.
- Ref
- FMA-52
- Rev
- v1
- Field
- Foundations: Maths
- Result
- r
A circle with centre (a, b) and radius r is the set of points (x, y) with (x − a)² + (y − b)² = r². Solving for r from a point on the circle gives the distance from the centre to that point. For a circle centred at the origin, a = b = 0. A tangent at a point is perpendicular to the radius to that point. A point is inside the circle if its distance from the centre is less than r.
Equation of a circle with centre (a, b) and radius r, solved here for the radius from a point (x, y) on the circle. Every point on the circle is the same distance r from the centre.
Calculator
- Circumference, 2πr31.42
- Area, πr²78.54
- Slope of the tangent at the point (if the radius is not vertical)−0.75
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A circle centred at (0, 0) through (3, 4) has radius 5: the 3–4–5 triangle when the centre is the origin. Its circumference is 31.42 and its area 78.54.
- A point at distance 5.5 from the centre would lie outside the circle and one at 4.5 inside, which is how a point is tested against a circular boundary.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| b y coordinate of the centre | −50 to 50 | 3.073 | 54.08 | 51.01 | +2 | −27.9 % |
| a x coordinate of the centre | −50 to 50 | 4.014 | 53.15 | 49.14 | +2 | −17.5 % |
| y y coordinate of a point on the circle | −50 to 50 | 3 | 50.09 | 47.09 | +2 | +34.2 % |
| x x coordinate of a point on the circle | −50 to 50 | 4 | 50.16 | 46.16 | +2 | +28.1 % |
Sweep of x coordinate of the centre
| x coordinate of the centre | Radius |
|---|---|
| −50 | 53.15 |
| −37.5 | 40.7 |
| −25 | 28.28 |
| −12.5 | 16.01 |
| 0 | 5 |
| 12.5 | 10.31 |
| 25 | 22.36 |
| 37.5 | 34.73 |
| 50 | 47.17 |
Sweep one variable
Variables and how each one contributes
Radius
What the equation gives you. Everything else feeds it.
x coordinate of a point on the circle
The x coordinate of a point known to lie on the circle. Raises the radius when the point moves further from the centre in x, and lowers it when the point moves towards it. Raising it by 1 (to 4) moves the result by +13.1 %, from 5 to 5.657.
y coordinate of a point on the circle
The y coordinate of a point known to lie on the circle. Raises the radius when the point moves further from the centre in y, and lowers it when the point moves towards it. Raising it by 1 (to 5) moves the result by +16.6 %, from 5 to 5.831.
x coordinate of the centre
The x coordinate of the centre of the circle. Lowers the radius when the centre moves towards the point in x, and raises it when it moves away. Raising it by 1 (to 1) moves the result by −10.6 %, from 5 to 4.472.
y coordinate of the centre
The y coordinate of the centre of the circle. Lowers the radius when the centre moves towards the point in y, and raises it when it moves away. Raising it by 1 (to 1) moves the result by −15.1 %, from 5 to 4.243.
Worked example with the values above
- Offsets from the centre: x − a = 3 and y − b = 4.
- Square and add: 9 + 16 = 25.
- r = √25 = 5.
Assumptions and limits
- Cartesian coordinates in a plane with the same length unit on both axes.
- The point really lies on the circle; otherwise the formula returns the distance from the centre to the point.
Common mistakes
- Using x² + y² = r² when the centre is not at the origin.
- Forgetting to square the offsets, or forgetting the square root when solving for r.
Statistics and uncertainty
Errors in the point and centre coordinates combine in quadrature in the radius: an error of ±u in each of two perpendicular offsets gives a radius error of about u at most. Measuring the circle through three well-spread points is more reliable than relying on one.
Where it is used
- Setting up a circular path, hole or arc on a drawing.
- Testing whether a point is inside, on or outside a circular boundary.
Maths you need
- Pythagoras: the square of the hypotenuse is the sum of squares
- Coordinates: positions along perpendicular axes