CircularMEASURES

STANDARD 9

r

Perimeter of a circle is 2пr

П ≈3.141

AB

AB is part of the circle ,it is known as Arc

AB

O

The angle made by joining the end points of an arc to

the center of the circle is called the central angle of

an arc.

Eg: angle AOB is the central angle of the arc AB

For an arc of central angle x°,arc length =2пr × x/360

r = radius of the circle

If we draw regular polygons with more and more sides within the circle, their areas would get closer and closer to the area of the circle.

By joining the vertices of the polygon to the center of the circle, we can divide the polygon into triangles

The area of the polygon is the sum of the areas of these triangles; and all these triangles are congruent. So we need only multiply the area of one triangle by the number of triangle.

To find the area of the triangleDraw the perpendicular from the center of the circle to a side of the polygon

h

s

Area of the triangle is½ × s × h = 1/2sh

If the polygon has n sides, then its area is

n × 1/2 × sh = ½ n s h

n s is the sum of all sides of the polygon ie. Perimeter so we can denote p= ns

Therefore ,Area of the Polygon =

½ ph

Thus as the number of sides of the polygon increases, in the area ½ ph of the polygon, the perimeter gets closer and closer to the perimeter of the circle and the perpendicular distance h gets closer and closer to the radius of the circle.Thus area of the circle = ½ × perimeter of

the circle× radius

Area of the circle =1/2 × 2п× radius (r)× radius

= п𝑟2