Library / Geometry, coordinates, surveying and trigonometry / GEO-09
Geometry, coordinates, surveying and trigonometry
Area of a circular ring
Area of an annulus between two circles with the same centre.
- Ref
- GEO-09
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- A [cm²]
The area between two circles with the same centre is the area of the outer disc minus the inner one, πR² − πr². Factorising the difference of squares gives π(R − r)(R + r). That form is the ring width times the mean circumference, which makes it a good check: width × 2π × mean radius.
Area of an annulus between two circles with the same centre. The factorised form is the area of the outer disc less the area of the inner disc.
Calculator
- Ring width, R − r2 cm
- Mean radius, (R + r)/24 cm
- Width times mean circumference, (R − r) × 2π × mean radius50.27 cm²
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 washer with outer radius 5 cm and inner radius 3 cm has an area of 50.27 cm², a 2 cm wide ring.
- With r = 0 it becomes the full disc πR² = 78.54 cm².
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Outer radius | 3.5 to 30 cm | 10.21 | 2,799 | 2,789 | +10% | +32.8 % |
| r Inner radius | 0 to 4.9 cm | 3.11 | 78.54 | 75.43 | +10% | −11.8 % |
Sweep of Inner radius
| Inner radius (cm) | Ring area (cm²) |
|---|---|
| 0 | 78.54 |
| 0.6125 | 77.36 |
| 1.225 | 73.83 |
| 1.838 | 67.93 |
| 2.45 | 59.68 |
| 3.063 | 49.08 |
| 3.675 | 36.11 |
| 4.288 | 20.79 |
| 4.9 | 3.11 |
Sweep one variable
Variables and how each one contributes
Ring area
What the equation gives you. Everything else feeds it.
Outer radius
The outer radius, in cm. A larger outer radius raises the area, and quickly, since it enters squared. Raising it by 10 % (to 5.5) moves the result by +32.8 %, from 50.27 to 66.76 cm².
Inner radius
The inner radius, in cm, from 0 up to R. A larger hole removes more area, so the ring area falls. Raising it by 0.1 (to 3.1) moves the result by −3.81 %, from 50.27 to 48.35 cm².
Worked example with the values above
- Width R − r = 5 − 3 = 2 cm; sum R + r = 8 cm.
- Product: 2 × 8 = 16.
- A = π × 16 = 50.265 cm².
Assumptions and limits
- A true circle, with angles in radians where the formula uses them.
- Lengths are measured from the same centre, in one consistent unit.
Common mistakes
- Entering a central angle in degrees where radians are needed.
- Using the full chord in place of the half chord when working with the right triangle.
- Using the major arc when the minor arc is meant.
Statistics and uncertainty
Circle results depend on the radius squared or on trigonometric functions of the angle, so errors in the radius are doubled in areas. The segment and rise formulas can lose digits for very small angles or very shallow arcs.
Where it is used
- Cutting, rolling and setting out curved plates, pipes and arches.
- Partly filled pipes and tanks, washers and flanges, and tangent lines past a circular object.
Maths you need
- Radians: angle measured as arc length over radius
- Right triangle: Pythagoras and sine over hypotenuse
- Circle area: π times radius squared