Circle segment - perimeter and area
P = 2πr. α / 360°
A =(πr2*360/α) - triangle area
A = area
P = perimeter
α = angle
r = radius
t = chord
WHAT IS IT ?
A circle segment is a region of a circle enclosed by a chord and the corresponding arc. The chord is a straight line connecting two points on the circle, and the arc is the part of the circle's perimeter that lies between those two points. Circle segments can be classified into two types:
1. **Minor Segment**: The smaller area enclosed by the chord and the arc.
2. **Major Segment**: The larger area enclosed by the chord and the arc.
The properties of a circle segment include the chord, which is the straight boundary, and the arc, which is the curved boundary. Circle segments are used in various applications in mathematics, engineering, and design, especially in calculating areas and lengths related to circular shapes.
CALCULATION:
EXAMPLE:
The radius of the circle is 5 cm, the angle α is 60 degrees
For the calculation, we first need to calculate the circular sector using the formula A =(πr²*α)/360, thus
A = (3.14*5²*60)/360 = (3.14*25*60)/360 = 13.08 cm²
So, the area of the circular sector is 13.08 cm².
To further calculate the area of the circular segment, we need to subtract the area of the triangle from the circular sector.
The area of the triangle is calculated using the sine of the angle, where the sine of the angle is the ratio of the opposite side to the hypotenuse. The hypotenuse in our case is the radius, and the angle needed for the sine is half of α, i.e., 30 degrees.
So,
sin 30 degrees = (1/2 of the chord) / hypotenuse (radius)
0.5 = (1/2 of the chord) / 5
0.5 * 5 = (1/2 of the chord)
2.5 cm = (1/2 of the chord), so the entire chord is 5 cm
We calculate the height h using the Pythagorean theorem
c² = a² + b²
5 squared (radius) = 2.5 squared + b²
25 = 6.25 + b²
18.75 = b²
4.33 = b, so the height
We calculate the triangle using the formula
A = (a * h) / 2
A = (5 * 4.33) / 2
A = 10.82 cm²
The area of the circular segment is then equal to the area of the circular sector minus the area of the triangle, thus 13.08 - 10.82 equals 2.21 cm²
We calculate the perimeter of the circular segment using the formula
O = 2πr · α / 360°
O = (2*3.14*60)/360
O = 5.24 cm
