GeometryFree

Circle Segment Area Calculator

Circle segment area calculator computes the area between a chord and its arc from radius and central angle. A circle with radius 7 and 60-degree central angle has a segment area of approximately 4.44 square units.

Enter Values

Radius (r)

Central Angle (degrees)

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

A = \frac{r^2}{2}(\theta - \sin\theta)

How it works: The segment area equals the sector area minus the triangle area formed by the two radii and the chord.

Worked Example

Find the segment area for radius 7 and central angle 60 degrees.

1Step 1: Convert to radians: 60 x pi/180 = 1.0472 rad.

2Step 2: Sector area = (r^2/2) x theta = (49/2) x 1.0472 = 25.66.

3Step 3: Triangle area = (r^2/2) x sin(theta) = (49/2) x sin(60) = 24.5 x 0.866 = 21.22.

4Step 4: Segment area = sector - triangle = 25.66 - 21.22 = 4.44 square units. For a semicircle (180 degrees): segment area = sector area = pi x r^2 / 2 = 76.97.

Circle Segment Area Explained

A circle segment is the region between a chord (straight line connecting two points on the circle) and the arc it cuts off. It equals the sector area minus the triangle area.

Segment area = sector area - triangle area = (r^2/2)(theta - sin(theta)), where theta is in radians
Sector area = (r^2/2) x theta (the "pie slice" from center to arc)
Triangle area = (r^2/2) x sin(theta) (the triangle formed by the two radii and the chord)
Chord length = 2r x sin(theta/2). Sagitta (segment height) = r(1 - cos(theta/2))

Enter the angle in degrees; this calculator converts to radians automatically. For a full semicircle (180 degrees), the segment area equals the semicircle area.

You can also calculate changes using our Circle Geometry Theorem Solver, Semicircle Calculator, Circumference Calculator or Ellipse Area Calculator.

Frequently Asked Questions

What is the segment area for radius 7 and 60-degree angle?

Segment area = (49/2)(1.0472 - sin(1.0472)) = 24.5 x (1.0472 - 0.8660) = 24.5 x 0.1812 = 4.44 square units. This is a small fraction (2.88%) of the full circle area (153.94 sq units).

What is the difference between a sector and a segment?

A sector is the "pie slice" from the center to the arc (bounded by two radii and an arc). A segment is the region between a chord and its arc (no radii involved). Segment area = sector area - triangle area. For a 60-degree angle: sector = 25.66, triangle = 21.22, segment = 4.44.

Do I enter degrees or radians?

Enter degrees. The calculator converts to radians internally using the formula: radians = degrees x pi/180. For example, 60 degrees = 1.0472 radians, 90 degrees = 1.5708 radians, 180 degrees = pi radians.

How do I calculate chord length from the angle?

Chord length = 2 x r x sin(theta/2). For radius 7 and 60 degrees: chord = 2 x 7 x sin(30) = 14 x 0.5 = 7. The chord equals the radius when the central angle is 60 degrees, forming an equilateral triangle.

What is a practical use of circle segment area?

Engineers use segment area to calculate the cross-sectional area of a partially filled circular pipe or tank. If a pipe with radius 7 inches is filled to a certain depth, the area of the water cross-section is a circle segment. This determines flow capacity in hydraulic engineering.

Can I use this Circle Segment Area Calculator on my own web page?

You can. Look for the "Embed" button near the top of this calculator. It lets you pick a size, border style, and color palette, then gives you an iframe tag to paste into any webpage. The widget is responsive, loads fast, and costs nothing. More details at calculory.com/services/embed-calculators.

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I can help you understand the circle segment area calculator formula, interpret your results, and answer follow-up questions.

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