NCERT Class 10 Chapter 12 Areas Related to Circles CBSE Board Sample Problems Long Answer

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Question

All the vertices of a rhombus lie on a circle. Find the area of rhombus, if the area of circle is

Solution

Rhombus lie on a circle

Rhombus Lie on a Circle

Diagonal of a rhombus are perpendicular bisector of each other. D

Each diagonal is diameter of the circle.

Now. Area of circle

:. Diameter of the circle = 40 cm = each diagonal of the rhombus

Area of the circle

Question

Find the area of the minor segment of a circle of radius 14 cm, when its central angle is . Also find the area of the corresponding major segment.

Solution

Circle of radius 14 cm

Circle of Radius 14 Cm

In

Also AO=BO

is an equilateral triangle.

Area of equilateral

Area of sector

Area of minor segment

Area of circle

Area of major segment

Question

In figure, APB and AQO are semicircle, and AC = OB. If the perimeter of the figure is 40 cm, find the area of the shaded region.

APB and AQO are semicircle

APB and AQO Are Semicircle

Solution

Let r be the radius of the semicircle APB, i.e. OB = OA = r, then is the radius of the semicircle AQO.

Given Penmeter of the figure is 40cm.

... Length of are APB + length of are

Now, area of shaded portion - area of semicircle APLI + area of semicircle AQO

By putting r=7cm

Required area of shaded

Portion is

Question

Solution

In Figure, is Shown a Sector OAP of a Circle with Centre O, Containing is Perpendicular to the Radius OA and Meets OP Produced at B. Prove That the Perimeter of Shaded Region is Ution

Length are

Perimeter of shaded region

Question

An elastic belt is placed around the rim of a pulley of radius 5 cm. From one point C on the belt, the elastic belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from the point O. Find the length of the belt that is still in contact with the pulley. Also find the shaded area, (use and )

An elastic belt

An Elastic Belt

Solution

An elastic belt

An Elastic Belt

Given: AO = 5 cm and OP = 10 cm

In right,

Length 0f

Hence, length of belt in contact

Now, in right OAT we have

Area of sector OACB

Shaded Area = Area of — Area of OACB

Question

A wire is in the form of a circle of radius 28cm. It is rebent into a square form. Find the length of the side of the square

Solution

Length of the wire= circumference of the circle

Let each side of the square be x cm

Perimeter = length of the wire

Question

The sum of radii of two circles is 140cm and the difference of their circumference is 88cm. Find the diameters of the circles.

Solution

Let radius of first circle be r cm.

Radius of second circle

Circumference of first circle

Circumference of second circle

Difference of two circumference

Radius of second circle

Diameter of first circle

Diameter of second circle

Question

PQRS is a diameter of a circle of radius 6 cm. The length PQ, QR and RS are equal. Semi-circles are drawn on PQ and QS as diameters as shown in the figure. Find the area and perimeter of the shaded region.

Diameter of a circle of radius 6 cm

Diameter of a Circle of Radius 6 Cm

Solution

Perimeter=ir6 + ir4 + ir2 = l2ir

= 37.68 cm

Area of the shaded region

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