Sizing up a slice of the circle
Imagine a pizza with a 16 inch diameter cut into eight equal slices. One slice is on your plate and a friend asks whether that slice is really one-eighth of the whole pizza. It should be, but how many square inches of pizza is it, exactly? That is a sector problem: a sector is the wedge of a circle cut out by two radii and the arc between them, and its area depends on just two numbers, the radius and the central angle.
The sector area formula
Formula: Sector area = (theta / 360) x pi x r^2
In this formula theta is the central angle of the sector measured in degrees, r is the radius of the circle, and pi is the familiar constant 3.14159 and beyond. The logic is simple: pi x r^2 is the area of the whole circle, and theta / 360 is the fraction of the full 360 degree circle that your sector occupies. A 90 degree sector is one quarter of the circle, so its area is one quarter of pi x r^2.
How to use this calculator
- Measure or enter the radius of the circle, not the diameter.
- Enter the central angle in degrees.
- Run the calculation and read the sector area in square units.
- Sanity check the answer: it must be smaller than the full circle area pi x r^2.
Worked example
Take the pizza slice: diameter 16 inches, so the radius r = 8 inches, and each of the eight slices has a central angle theta = 360 / 8 = 45 degrees. Step 1: find the fraction of the circle, 45 / 360 = 0.125. Step 2: square the radius, 8^2 = 64. Step 3: multiply by pi, 3.14159 x 64 = 201.06 square inches for the whole pizza. Step 4: take the slice fraction, 0.125 x 201.06 = 25.13. Each slice covers about 25.13 square inches.
Where this comes in handy
Sectors show up in more places than pizza night. Gardeners lay out curved beds and sprinkler zones as sectors, bakers scale pie and tart recipes by slice area, engineers size fan blades and gear teeth, and designers build pie charts where each slice angle represents a share of the data. Any time a round shape is divided from its center, the sector formula gives the area of each piece.
Mistakes people often make
- Using the diameter instead of the radius. Always halve the diameter first; the formula needs r, not 2r.
- Mixing degrees and radians. This version of the formula needs degrees. If your angle is in radians, convert first by multiplying by 180 / pi.
- Forgetting to square the radius. It is r^2, not r, a slip that makes answers far too small.
What this number does not tell you
The result assumes a perfect circle and an exact angle. In the real world a hand-cut pizza slice or a roughly marked garden bed will be a little off, so treat the answer as a precise calculation of an approximate measurement. Rounding the radius to the nearest unit keeps the result honest about how carefully the inputs were measured.
Working with other shapes too? Try the rectangle area calculator, the trapezoid area calculator, or the area under the curve calculator for trickier regions.