Mensuration circle

Mensuration of Class 8

CIRCLE:

Circle

d = 2r	diameter
	radius
C = 2πr	circumference
>	radius

C = π× d	circumference
	diameter

	circumference
	area

K = πr2	area
	radius

	area
	diameter

	area
	circumference
	radius

SEGMENT OF A CIRCLE:

Segment of a Circle

	chord length
	circle radius
	circle center to chord midpoint distance

	segment area
	circle radius

	central angle
s = θ× r	arc length
	circle radius

h = r -t	segment height
r = h + t	circle radius
t = r – h	circle center to chord midpoint distance

SECTOR OF A CIRCLE:

A sector of a circle is a pie-shaped region bounded by an arc and an angle.
On this page we'll show you how to find the arc length a and the area A of the sector.

Sector of a Circle

	sector area
	circle radius
	central angle

ARC OF A CIRCLE:

Arc of a Circle

S = r ×θ	arc length
	circle radius
	central angle

We'll use a circle with radius 15 cm and central angle 60°.

circle

The first step is to find out what fraction of the circleis represented by the central angle 60°:

The sector represents 1/6 of the circle,or 0.1666... of the circle.

To find the arc length a, we now need to find the circumference (the distance around the full circle), and then find 1/6 of it (multiply it by 0.1666...)

This is the length of the arc a.

C = 2πr

= 2π⋅ 15

= 94.248

a = ⋅ 94.248 cm

= 15.7 cm

This method will work for any sector in any circle. You just need to find the fraction of the circle that the sector angle represents, and then find that fraction of the circumference.

Here's a quicker formula to use, that summarizes what we did in one calculation.

SOLVED EXAMPLES

question 1.Find the circumference and area of a circle of radius 6.5 cm.

Solution: Here r = 6.5 cm

Circumference of the circle

question 2.A wire is bent into the form of a square of side 30 cm. The wire is straightened and bent into the form of a circle. What will be the radius of the circle so formed? (Take π = 22/7)

Solution:Length of wire = Perimeter of the square = (30 × 4) cm = 120 cm.

When this wire is bent into the form of a circle, then clearly the circumference of the circle would be 120 cm.

Let r be the radius of this circle.

Then, circumference