R.D. Sharma solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० chapter 13 - Areas Related to Circles [Latest edition]

Find the circumference and area of circle of radius 4.2 cm

Find the circumference of a circle whose area is 301.84 cm².

Find the area of circle whose circumference is 44 cm.

A road which is 7 m wide surrounds a circular park whose circumference  is 352 m . Find the area of the road .

The circumference of two circles are in ratio 2:3. Find the ratio of their areas

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

The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

The outer circumference of a circular race-track is 528 m . The track is everywhere 14 m wide. Calculate the cost of levelling the track at the rate of 50 paise per square metre.

`(use pi=22/7). `

A circular park is surrounded by a road 21 m wide. If the radius of the park is 105 m, find the area of the road.

A circle of radius 7.5 cm is inscribed in a square. Find the area outside the circle and inside the square.

The area of a circle inscribed in an equilateral triangle is 154 cm². Find the perimeter of the triangle. [Take `sqrt(3) = 1.73`.]

A car travels 1 kilometre distance in which each wheel makes 450 complete revolutions. Find the radius of the its wheels.

A square of diagonal 8 cm is inscribed in a circle. Find the area of the region lying outside the circle and inside the square. 

A path of 4 m width runs round a circular grassy plot whose circumference is `163 3/7` m Find:

1. the area of the path
2. the cost of gravelling the path at the rate of ₹ 1.50 per square metre
3. the cost of turfing the plot at the rate of 45 paise per m².

 A path of width 3.5 m  runs around a semi-circular grassy plot whose perimeter is 72 m . Find the area of the path .

`("Use"  pi= 22/7) ` 

Find the area enclosed between two concentric circles of radii 3.5 cm and 7 cm. A third concentric circle is drawn outside the 7 cm circle , such that the area enclosed between it and the 7 cm circle is same as that between the two inner circles . Find the radius of the third circle correct to one decimal place.

An archery target has three regions formed by three concentric circles as shown in figure. If the diameters of the concentric circles are in the ratios 1 : 2 : 3, then find the ratio of the areas of three regions.

Find in terms of π, the length of the arc that subtends an angle of 30° at the centre of a circle of radius 4 cm.

Find the angle subtended at the centre of a circle of radius 5 cm by an arc of length `((5π)/3)`cm.

An arc of length 20𝜋 cm subtends an angle of 144° at centre of circle. Find the radius of the circle.

An arc of length 15 cm subtends an angle of 45° at the centre of a circle. Find in terms of 𝜋, radius of the circle.

A sector of circle of radius 4cm contains an angle of 30°. Find the area of sector

The area of sector is one-twelfth that of the complete circle. Find the angle of the sector .

In a circle of radius 35 cm, an arc subtends an angle of 72° at the centre. Find the length of arc and area of sector

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

A sector is cut-off from a circle of radius 21 cm the angle of sector is 120°. Find the length of its arc and its area.

In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find the area of the sector formed by the arc. (Use π = `22/7`)

The length of the minute hand of a clock is 5 cm. Find the area swept by the minute hand during the time period 6 : 05 am and 6 : 40 am.

The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector. 

AB is a chord of a circle with centre O and radius 4 cm. AB is of length 4 cm. Find the area of the sector of the circle formed by chord AB.

In a circle of radius 6 cm, a chord of length 10 cm makes an angle of 110° at the centre of the circle. Find the circumference of the circle.

In a circle of radius 6 cm, a chord of length 10 cm makes an angle of 110° at the centre of the circle. Find the area of the circle.

In a circle of radius 6 cm, a chord of length 10 cm makes an angle of 110° at the centre of the circle. Find the length of the arc AB.

In a circle of radius 6 cm, a chord of length 10 cm makes an angle of 110° at the centre of the circle. Find the area of the sector OAВ.

A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope. Find the area of that part of the field in which the horse can graze. Also, find the increase in grazing area if length of rope is increased to 10 m. (Use π = 3.14)

Figure, shows a sector of a circle, centre O, containing an angle θ°. Prove that perimeter of the shaded region is `r(tan θ + sec θ + (πtheta)/180^circ - 1)`.

Figure, shows a sector of a circle, centre O, containing an angle 𝜃°. Prove that area of the shaded region is `r^2/2(tanθ - (πθ)/180^circ)`.

The diagram shows a sector of circle of radius ‘r’ can containing an angle 𝜃. The area of sector is A cm2 and perimeter of sector is 50 cm. Prove that

(i) 𝜃 =`360/pi(25/r− 1)`

(ii) A = 25r – r²

A chord PQ of length 12 cm subtends an angle of 120° at the centre of a circle. Find the area of the minor segment cut off by the chord PQ. 

A chord AB of a circle, of radius 14 cm makes an angle of 60° at the centre of the circle. Find the area of the minor segment of the circle. Also, find the area of the major segment of the circle. `("Use"  π = 22/7)`

 A chord of a circle of radius 20 cm sub tends an angle of 90⁰ at the centre . Find the area of the corresponding major segment of the circle
( Use \[\pi = 3 . 14\]) 

A chord 10 cm long is drawn in a circle whose radius is 5√2 cm. Find the area of both segments

The radius of a circle with centre O is 5 cm (Figure). Two radii OA and OB are drawn at right angles to each other. Find the areas of the segments made by the chord AB (Take π = 3.14).

In the follwing figure AB is the diameter of a circle, centre O. C is a point on the circumference such that ∠COB = 𝜃. The area of the minor segment cut off by AC is equal to twice the area of sector BOC. Prove that `sin  θ/2 cos  θ/2 = π(1/2 - θ/120^circ)`.

A chord of a circle subtends an angle of θ at the centre of circle. The area of the minor segment cut off by the chord is one eighth of the area of circle. Prove that `8 sin  θ/2 cos  θ/2 + π = (πθ)/45`.

A plot is in the form of rectangle ABCD having semi-circle on BC. If AB = 60m and BC = 28m, find the area of plot.

In the following figure, PQRS is a square of side 4 cm. Find the area of the shaded square. 

Four cows are tethered at four corners of a square plot of side 50m, so that’ they just cant reach one another. What area will be left ungrazed.

A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20 m × 16 m. Find the area of the field in which the cow can graze. 

A calf is tied with a rope of length 6 m at the corner of a square grassy lawn of side 20 m . If the length of the rope is increased by 5.5 m , find the increase in area of the grassy lawn in which the calf can graze .

In the following figure, ABCD is a square with side `2sqrt(2)` cm and inscribed in a circle. Find the area of the shaded region. (Use π = 3.14).

In the following, a square OABC is inscribed in a quadrant OPBQ. If OA = 15 cm, find the area of shaded region (use π = 3.14).

In the following figure, from a rectangular region ABCD with AB = 20 cm, a right triangle AED with AE = 9 cm and DE = 12 cm, is cut off. On the other end, taking BC as diameter, a semicircle is added on outside' the region. Find the area of the shaded region. [Use π =]`22/7` [CBSE 2014]

From each of the two opposite corners of a square of side 8 cm, a quadrant of a circle of radius 1.4 cm is cut. Another circle of radius 4.2 cm is also cut from the centre as shown in the following figure. Find the area of the remaining (Shaded) portion of the square. (Use π = 22/7)

In the following figure, ABCD is a rectangle with AB = 14 cm and BC = 7 cm. Taking DC, BC and AD as diameters, three semi-circles are drawn as shown in the figure. Find the area of the shaded region.

In the following figure, ABCD is a rectangle, having AB = 20 cm and BC = 14 cm. Two sectors of 180° have been cut off. Calculate:

the area of the shaded region.

In the following figure, ABCD is a rectangle, having AB = 20 cm and BC = 14 cm. Two sectors of 180° have been cut off. Calculate:

 the  length of the boundary of the shaded region.

In the following figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the (i) quadrant OACB (ii) shaded region.

In the following figure a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 21 cm, find the area of the shaded region.

In the following figure, OABC is a square of side 7 cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region. (Use π = 22/7)

Find the area of the shaded region in the following figure, if AC = 24 cm, BC = 10 cm and O is the centre of the circle. (Use π = 3.14)

ABCDEF is a regular hexagon with centre O (in the following figure). If the area of triangle OAB is 9 cm², find the area of : (i) the hexagon and (ii) the circle in which the haxagon is incribed.

A child makes a poster on a chart paper drawing a square ABCD of side 14 cm. She draws four circles with centre A, B, C and D in which she suggests different ways to save energy. The circles are drawn in such a way that each circle touches externally two of the three remaining circles (in the following figure). In the shaded region she write a message 'Save Energy'. Find the perimeter and area of the shaded region.
(Use π = 22/7)

In the following figure, AB and CD are two diameters of a circle perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region. 

In the the following figure, PSR, RTQ and PAQ are three semicircles of diameter 10 cm, 3 cm and 7 cm respectively. Find the perimeter of shaded region.

In the following figure, two circles with centres A and B touch each other at the point C. If AC = 8 cm and AB = 3 cm, find the area of the shaded region.

In the following figure, O is the centre of a circular arc and AOB is a straight line. Find the perimeter and the area of the shaded region correct to one decimal place. (Take π = 3.142)

In the following figure, shows a kite in which BCD is the shape of a quadrant of a circle of radius 42 cm. ABCD is a square and Δ CEF is an isosceles right angled triangle whose equal sides are 6 cm long. Find the area of the shaded region.

In the following figure, ABCD is a trapezium of area 24.5 cm² , If AD || BC, ∠DAB = 90°, AD = 10 cm, BC = 4 cm and ABE is quadrant of a circle, then find the area of the shaded region. [CBSE 2014]

From a thin metallic piece, in the shape of a trapezium ABCD, in which AB || CD and ∠BCD = 90°, a quarter circle BEFC is removed (in the following figure). Given AB = BC = 3.5 cm and DE = 2 cm, calculate the area of the remaining piece of the metal sheet.

In the following figure, ABC is an equilateral triangle of side 8 cm. A, B and C are the centres of circular arcs of radius 4 cm. Find the area of the shaded region correct upto 2 decimal places. (Take π =3.142 and`sqrt3` = 1.732).

The inside perimeter of a running track (shown in the following figure) is 400 m. The length of each of the straight portion is 90 m and the ends are semi-circles. If the track is everywhere 14 m wide. find the area of the track. Also find the length of the outer running track.

In the following figure, the square ABCD is divided into five equal parts, all having same area. The central part is circular and the ines AE, GC, BF and HD lie along the diagonals AC and BD of the square. If AB = 22 cm, find:

 the circumference of the central part.

In the following figure, the square ABCD is divided into five equal parts, all having same area. The central part is circular and the lines AE, GC, BF and HD lie along the diagonals AC and BD of the square. If AB = 22 cm, find:

the perimeter of the part ABEF.

In the following figure find the area of the shaded region. (Use π = 3.14) 

A circle is inscribed in an equilateral triangle ABC is side 12 cm, touching its sides (the following figure). Find the radius of the inscribed circle and the area of the shaded part.

In the following figure, an equilateral triangle ABC of side 6 cm has been inscribed in a circle. Find the area of the shaded region. (Take π = 3.14). 

Find the area of a shaded region in the the following figure,where a circular arc of radius 7 cm has been drawn with vertex A of an equilateral triangle ABC of side 14 cm as centre.   (Use π = 22/7 and \[\sqrt{3}\] = 1.73)

In the following figure, ABCD is a square of side 2a, Find the ratio between

(i) the circumferences

(ii) the areas of the in circle and the circum-circle of the square. 

In the following figure, there are three semicircles, A, B and C having diameter 3 cm each, and another semicircle E having a circle D with diameter 4.5 cm are shown. Calculate:

(i) the area of the shaded region

(ii)  the cost of painting the shaded region at the rate of 25 paise per cm² , to the nearest rupee.

In the following figure, ABC is a right-angled triangle, ∠B = 90°, AB = 28 cm and BC = 21 cm. With AC as diameter a semicircle is drawn and with BC as radius a quarter circle is drawn. Find the area of the shaded region correct to two decimal places.

In the following figure, the boundary of the shaded region consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm, find

1. the length of the boundary.
2. the area of the shaded region.

In the following figure, AB = 36 cm and M is mid-point of AB. Semi-circles are drawn on AB, AM and MB as diameters. A circle with centre C touches all the three circles. Find the area of the shaded region.

In the following figure, ABC is a right angled triangle in which ∠A = 90°, AB = 21 cm and AC = 28 cm. Semi-circles are described on AB, BC and AC as diameters. Find the area of the shaded region.

In the following figure, shows the cross-section of railway tunnel. The radius OA of the circular part is 2 m. If ∠AOB = 90°, calculate:
 the height of the tunnel

In the following figure, shows the cross-section of railway tunnel. The radius OA of the circular part is 2 m. If ∠AOB = 90°, calculate: 

the perimeter of the cross-section 

In the following figure, shows the cross-section of railway tunnel. The radius OA of the circular part is 2 m. If ∠AOB = 90°, calculate:

the area of the cross-section.

In the given figure, ABCD is a trapezium with AB || DC, AB = 18 cm DC = 32 cm and the distance between AB and DC is 14 cm. Circles of equal radii 7 cm with centres A, B, C and D have been drawn. Then find the area of the shaded region.
(Use \[\pi = \frac{22}{7}\] 

Sides of a triangular field are 15 m, 16 m and 17 m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 7 m each to graze in the field. Find the area of the field which cannot be grazed by the three animals.

In the given figure, the side of square is 28 cm and radius of each circle is half of the length of the side of the square, where O and O' are centres of the circles. Find the area of shaded region.

In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park.

What is the ratio of the areas of a circle and an equilateral triangle whose diameter and a side are respectively equal?

If the circumference of two circles are in the ratio  2 : 3, what is the ratio of their areas?

Write the area of the sector of a circle whose radius is r and length of the arc is l. 

What is the length (in terms of π)  of the arc that subtends an angle of 36° at the centre of a circle of radius 5 cm?

What is the angle subtended at the centre of a circle of radius 6 cm by an arc of length 3 π cm?

What is the area of a sector of a circle of radius 5 cm formed by an arc of length 3.5 cm? 

In a circle of radius 10 cm, an arc subtends an angle of 108° at the centre. what is the area of the sector in terms of π? 

If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square?

Write the formula for the area of a sector of angle \[\theta\] (in degrees) of a circle of radius r. 

Write the formula for the area of a segment in a circle of radius r given that the sector angle is \[\theta\] (in degrees). 

If the adjoining figure is a sector of a circle of radius 10.5 cm, what is the perimeter of the sector? (Take \[\pi = 22/7\]) 

If the diameter of a semi-circular protractor is 14 cm, then find its perimeter.

A piece of wire 22 cm long is bent into the form an arc of a circle subtending an angle of 60° at its centre. Find the radius of the circle. `["Use"  π = 22/7]`

Find the area of the largest triangle that can be inscribed in a semi-circle of radius runits.

An arc subtends an angle of 90° at the centre of the circle of the radius 14 cm. Write the area of minor sector thus formed in terms of π.

Find the area of sector of circle  of radius 21 cm and central angle 120⁰.

What is the area of a square inscribed in a circle  of diameter p cm ?

Is it true to say that area of a segment of a circle is less than the area of its corresponding sector? Why?

If the numerical value of the area of a circle is equal to the numerical value of its circumference , find its radius.

Find the ratio of the area of the circle circumscribing a square to the area of the circle inscribed in the square .

Find the length of the arc of a circle which subtends an angle of 60° at the centre of the circle of radius 42 cm.

In the following, ABCD is a square of side 10 cm. A sector of radius 5 cm is cut out from one of the corners. Find the area of the shaded region. (Take π = 3.14)

In the given figure, the shape of the top of a table is that of a sector of a circle with centre O and ∠AOB = 90°. If AO = OB = 42 cm, then find the perimeter of the top of the table.

In the following figure, three sectors of a circle of radius 5 cm, making angles 35°, 50° and 95° at the centre are shaded. Find the area of the shaded region. `("Use"  π = 22/7)`

If the area of a circle is 154 cm², then is perimeter is ______.

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is ______.

The area of the circle that can be inscribed in a square of side 6 cm is ______.

The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm is ______.

The area of the square that can be inscribed in a circle of radius 8 cm is ______.

It is proposed to build a single park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park would be ______.

The radius of a circle whose circumference is equal to the sum of the circumferences of the two circles of diameters 36 cm and 20 cm, is ______.

If the difference between outer and inner radii of a circular ring is 14 cm, then the difference between outer and inner circumferences, is ______.

The area of a sector whose perimeter is four times its radius r, is ______.

Two circles touch each other externally. The sum of their areas is 490 π cm². Their centres are separated by 28 cm. The difference of their perimeters is ______.

If `l` is the arc length of a sector of a circle of radius r and A is the area of the sector, then A : `l` = ______.

If the circumferences of two circles are in the ratio c₁ : c₂, then the ratio of their areas is ______.

The area of a sector of angle θ° of a circle of radius r is ______.

The length of the arc of a sector of angle θ° of a circle of radius r is ______.

The area of the minor segment of angle θ° of a circle of radius r is ______.

The difference in the areas of the regular hexagon circumscribing a circle of radius 10 cm and the regular hexagon inscribed in the circle is ______.

In figure, ABCD is a square of side 10 cm and a circle is inscribed in it. The area of the shaded part is ______.

In figure, two circles of radii 7 cm each are shown. ABCD is a rectangle and AD and BC are the radii. The area of the shaded region is ______.

The area of the largest circle that can be drawn inside a rectangle of length a cm and breadth b cm (a > b) is ______.

The number of revolutions made by a circle of radius r to cover a distance s is ______.