Circle Segment Area Calculator
Calculate the area of a circular segment.
About Circle Segment Area Calculator
A circular segment is the region between a chord and its arc. Area = sector area - triangle area.
$$A = \frac{r^2}{2}(\theta - \sin\theta)$$
How to use this calculator
- Enter the circle radius.
- Enter the central angle in radians.
- Click calculate to get the segment area.
- Use the formula and example to check your setup.
The formula explained
$$ \text{Area} = \frac{r^2}{2}\left(\theta - \sin\theta\right) $$
- \(\text{Area}\) = area of the circular segment
- \(r\) = radius of the circle
- \(\text{theta}\) = central angle in radians
- \(\text{sin}\) = sine of the central angle
Step by step method
- Find the radius of the circle that contains the segment.
- Measure the central angle that subtends the segment, making sure it is in radians.
- Substitute the values into the segment area formula.
- Compute the result and confirm the units are square units.
Worked example
Suppose a circle has radius 10 and the segment is formed by a central angle of 1.2 radians.
- Use \(\text{Area} = \frac{r^2}{2}(\theta - \sin\theta)\).
- Substitute the values: \(\text{Area} = \frac{10^2}{2}(1.2 - \sin(1.2))\).
- Compute \(10^2 = 100\), so \(\text{Area} = 50(1.2 - 0.9320)\).
- Then \(\text{Area} = 50(0.2680) = 13.4\).
Answer. The segment area is 13.4 square units.
Tips and common mistakes
- Make sure the angle is in radians, not degrees.
- Use the radius, not the diameter.
- A larger angle gives a larger segment area, up to the full circle.
- If your angle is from a drawing, double-check that it is the central angle.
Frequently asked questions
What is a circular segment?+
A circular segment is the region between a chord and the arc above it. It is part of a circle, not a full sector. This calculator finds the area of that region.
Do I need radians or degrees?+
Use radians for the formula this calculator uses. If you only have degrees, convert them before calculating. That avoids the most common input mistake.
What is the difference between a segment and a sector?+
A sector is the wedge-shaped slice made by two radii and an arc. A segment is the smaller region cut off by a chord and the arc. They are related, but they are not the same shape.
Can this calculator handle a very small segment?+
Yes, as long as you enter the radius and angle correctly. Very small angles produce very small areas, so careful input matters. The result will still be in square units.
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