StudyMathOnline: Mensuration

Circle

Radius of a circle

radius of the circle is the distance between any point on the circumference of the circle to the center of the circle.

Diameter of a circle

Diameter of a circle is a line segment joining any two points on the circle and passing through the center of the circle.

Note: Diameter of any circle is twice the radius

Area of a circle

Area of a circle is total area enclosed by the circumference of the circle and it is defined as
area of circle =
πr²

Circumference of a circle

Circumference of a circle is the length of the outside perimeter of the circle.
Circumference of a circle = 2πr

Arc of a circle

Arc of a circle is a part of the circumference of the circle. In the figure given below there are two points ( X and Y) on the circumference of the circle. The part of the circumference between X and Y makes an arc of the circle.

Now, We can go from X to Y on a circumference in anticlockwise and clockwise direction. In the figure below, if we go from X to Y in clockwise direction then its one arc and if we go from X to Y in anti clockwise direction then its another arc. Thus any two points on the circle makes two arcs.

These are called as minor arc and major arc as one of them will be greater than the other except in the case when both point lie on the diameter of a circle.

In the figure above,
the minor arc XY subtends an angle of θ at the center
the minor arc XY subtends an angle of (360 - θ) at the center

Calculation for the length of the arc of the circle:

Length of the minor arc = (θ/360)x (2πr)

Length of the major arc ={ (360 - θ)/360} x ( 2πr)

Sector of a circle

Sector of a circle is a part of the area of the circle. In the figure given below there are two points ( X and Y) on the circumference of the circle and from these points two radius have been drawn to the centre of the circle. The part of the area between radius at X and radius at Y makes a sector of the circle.

Now, We can go from X to Y on a circumference in anticlockwise and clockwise direction. In the figure below, if we go from X to Y in clockwise direction then its one sector and if we go from X to Y in anti clockwise direction then its another sector. Thus any two points on the circle makes two sectors on the circle.

These are called as minor sector and major sector as one of them will be greater than the other except in the case when both point lie on the diameter of a circle.

In the figure above,
the minor arc XY subtends an angle of θ at the center
the minor arc XY subtends an angle of (360 - θ) at the center

Calculation for the area of the sector of the circle:

Length of the minor arc = (θ/360)x ( πr²)

Length of the major arc ={ (360 - θ)/360} x ( πr²)

Few examples

Some solved problems based on the circle:

Problem 1 : An arc subtends an angle of 60 degrees at the center of the circle. Find the length of the arc if the radius of the circle is 10cm.

Solution:

Using the formula for calculating the length of the arc discussed above:

Length of the arc = (θ/360)(2 πr)
Here,
   r = 10 cm
   θ = 60 degrees
   π = 3.14

Put the values in the equation, we have
   Length of the arc (XY) = (60/360)(2 x 3.14 x 10)
                                                 = (1/6)(62.8)
                                                 = 10.47 cm

Problem 2: Find the area of the sector XY, if the points XY subtends an angle of 120 at the center of the circle with radius 6cm.

Solution:

Using the formula discussed before to calculate the area of the sector,

Area of the sector = (θ/360)x ( πr²)

Given that,

radius = 6cm

θ = 120

π = 3.14

Put these values in the formula above to get the answer,

Area of the sector = {(120)/360} x ( 3.14 x 6 x 6)
= (1/3) x ( 113.04)
= 37.68 square cm.

Learn about triangle:Mensuration Triangle

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Pages

Thursday, May 29, 2014

Mensuration - Circle