Circular Segment
A circular segment is a slice of a circle defined by an arc and the chord that connects its endpoints.
Put simply, it's the area within a circle that lies between a chord and the arc above it.
This is also known as a single-based circular segment.
A chord is essentially a line segment linking two distinct points on the circumference of the circle. The length of the chord and the arc it spans determine the circular segment's size and shape. Meanwhile, the arc is the portion of the circle's edge that stretches between the chord's endpoints.
Circular Segment Formulas
The area Asc of a circular segment can be found by subtracting the area of the triangle formed by the chord and the circle's radii from the area of the circular sector As.
$$ A_{sc} = \frac{r^2}{2} \cdot ( \frac{\pi \alpha}{180°} - \sin \alpha ) $$
Where α is the central angle in degrees, and r is the radius of the circle.
Note: The central angle is formed by two radii that connect the circle’s center to the chord's endpoints. This angle's measurement dictates the arc's length and, consequently, the circular segment's area.
Expressed in radians, the area formula becomes
$$ A_{sc} = \frac{1}{2} r^2 \cdot ( \alpha - \sin \alpha ) $$
Demonstration: The area of a circular sector with central angle α in degrees is $$ \frac{ \alpha}{360} \cdot \pi r^2 $$. The area of a triangle with vertices at the ends of the chord and the circle's center can be calculated using the trigonometric triangle area formula based on two sides and the included angle. $$ \frac{1}{2} \cdot a \cdot b \cdot \sin \alpha $$ Where a and b are the sides, and α is the included angle. In the case of the triangle with vertices at the chord's ends and the circle's center, both a and b are equal to the circle's radius r. Thus, the triangle's area is $$ \frac{1}{2} \cdot r^2 \cdot \sin \alpha $$. The area of the circular segment is then the difference between the circular sector's area and the triangle's area $$ A_{sc} = \frac{\alpha}{360} \cdot \pi r^2 - \frac{1}{2} r^2 \sin(\alpha) $$. Factoring out r2/2 gives us the formula to calculate the circular segment's area with an angle in degrees $$ A_{sc} = \frac{r^2}{2} \cdot ( \frac{\pi \alpha}{180°} - \sin \alpha ) $$. To convert to radians, measure the flat angle with π radians instead of 180° and simplify. $$ A_{sc} = \frac{r^2}{2} \cdot ( \frac{\pi \alpha}{\pi} - \sin \alpha ) $$ $$ A_{sc} = \frac{r^2}{2} \cdot ( \alpha - \sin \alpha ) $$
If the circular segment represents less than half of the circle, its area can be calculated by subtracting the triangle OAB's area from that of the corresponding circular sector.
Conversely, if the circular segment exceeds half the circle (semicircle), its area is the sum of the circular sector's area and triangle OAB's area.
The arc length L spanned by the chord can be calculated as:
$$ L = \frac{ \alpha }{180} \cdot \pi r $$
Double-Based Circular Segment
A double-based circular segment is the part of a circle enclosed by two parallel chords.
The area of a circular segment with two bases is calculated by subtracting the areas of the two segments formed by the individual chords AB and CD from the total area A of the circle.
And so on.