Areas Related to Circles
This chapter covers the area and circumference of a circle, and extends this to finding the area of sectors and segments of a circle, along with areas of combinations of plane figures.
Circumference of a circle = 2 x pi x r. Area of a circle = pi x r^2, where r is the radius and pi is approximately 22/7 or 3.14.
Area of a sector with angle theta (in degrees) = (theta/360) x pi x r^2. Area of the corresponding minor segment = Area of sector - Area of the triangle formed by the two radii and the chord.
Q. Find the area of a circle whose circumference is 44 cm.
Circumference = 2 x pi x r = 44, so r = 44 / (2 x 22/7) = 44 x 7 / 44 = 7 cm. Area = pi x r^2 = (22/7) x 7 x 7 = 154 cm^2.
A sector is the region enclosed by two radii and the arc between them. A segment is the region enclosed by a chord and the arc it cuts off. The minor segment corresponds to the minor sector, and the major segment to the major sector.
Q. Find the area of a sector of a circle with radius 21 cm and a central angle of 60 degrees.
Area of sector = (theta/360) x pi x r^2 = (60/360) x (22/7) x 21 x 21 = (1/6) x 22 x 3 x 21 = (1/6) x 1386 = 231 cm^2.
If the radius of a circle is doubled, its area becomes:
Correct answer: (C) 4 times the original area
The region enclosed by a chord and the corresponding arc of a circle is called a _______.
Segment
The area of a semicircle is half the area of the full circle with the same radius.
True — a semicircle is a sector with a central angle of 180 degrees, which is exactly half of 360 degrees, so its area is exactly half that of the full circle.