Segment Area Calculator

How to Calculate Segment Area and Height

A circular segment is the region of a circle enclosed by a chord and an arc. Understanding how to calculate its area and height is important in geometry and has practical applications in various fields.

Segment Formulas

The key formulas for calculating segment properties are:

• Area: \(A = \frac{r^2}{2}(\theta - \sin\theta)\)
• Height: \(h = r(1 - \cos\frac{\theta}{2})\)
• Chord length: \(c = 2r \sin(\frac{\theta}{2})\)
• Arc length: \(s = r\theta\)

Where:

• \(A\) is the area of the segment
• \(h\) is the height of the segment
• \(r\) is the radius of the circle
• \(\theta\) is the central angle in radians
• \(c\) is the length of the chord
• \(s\) is the length of the arc

Calculation Steps

1. Convert the central angle from degrees to radians: \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180°}\)
2. Calculate the segment area using the formula: \(A = \frac{r^2}{2}(\theta - \sin\theta)\)
3. Calculate the segment height using the formula: \(h = r(1 - \cos\frac{\theta}{2})\)
4. If needed, calculate the chord length and arc length using their respective formulas

Example Calculation

Let's calculate the area and height of a segment with a central angle of 60° and a radius of 10 units:

1. Convert 60° to radians: \(\theta = 60° \times \frac{\pi}{180°} = \frac{\pi}{3} \approx 1.0472\) radians
2. Calculate the area: \(A = \frac{10^2}{2}(1.0472 - \sin(1.0472)) \approx 27.3137\) square units
3. Calculate the height: \(h = 10(1 - \cos(\frac{1.0472}{2})) \approx 1.3397\) units
4. Chord length: \(c = 2 \times 10 \times \sin(\frac{1.0472}{2}) \approx 10\) units
5. Arc length: \(s = 10 \times 1.0472 \approx 10.472\) units

Visual Representation

This diagram illustrates a circular segment. The segment is shown in blue, bounded by the chord (green) and the arc. The central angle θ is formed by the two radii (red), and the segment height h (purple) is the perpendicular distance from the chord to the arc at its midpoint.