M l aggarwal understanding icse mathematics class 9 Solutions for Chapter 16 mensuration

Find the area of a triangle whose base is 6 cm and corresponding height is 4 cm.

Find the area of a triangle whose sides are

(i) 3 cm, 4 cm and 5 cm

(ii) 29 cm, 20 cm and 21 cm

(iii) 12 cm, 9.6 cm and 7.2 cm

Find the area of a triangle whose sides are 34 cm, 20 cm, and 42 cm. hence, find the length of the altitude

corresponding to the shortest side.

The sides of a triangular field are 975 m, 1050 m, and 1125 m. If this field is sold at the rate of Rs 1000

per hectare, find its selling price. (1 hectare = 10000 \left.\mathbf{m}^{2}\right)

The base of a right-angled triangle is 12 cm and its hypotenuse is 13 cm long. Find its area and the

perimeter.

Find the area of an equilateral triangle whose side is 8 m. Give your answer correct to two decimal

places.

If the area of an equilateral triangle is 81 \sqrt{3} \mathrm{cm}^{2} \text { find its perimeter. }

(i) If the length of the sides of a triangle are in the ratio 3: 4: 5 and its perimeter is 48 cm, find its area.

(ii) The sides of a triangular plot are in the ratio 3: 5: 7 and its perimeter is 300 m. Find its area.

ABC is a triangle in which AB = AC = 4 cm and ∠A = 90^{\circ}

Calculate the area of △ABC. Also find the length of perpendicular from A to BC.

Find the area of an isosceles triangle whose base is 6 cm and perimeter is 16 cm.

. The sides of a right-angled triangle containing the right angle are 5x cm and (3x – 1) cm. Calculate the

length of the hypotenuse of the triangle if its area is 60 \mathrm{cm}^{2}

In △ABC, ∠B = 90^{\circ}, \mathrm{AB}=(2 \mathrm{A}+1) \mathrm{cm} \text { and } \mathrm{BC}=(\mathrm{A}+1) \mathrm{cm} . \text { If the area of the } \Delta \mathrm{ABC} \text { is } 60 \mathrm{cm}^{2}, \text { find its } perimeter.

If the perimeter of a right angled triangle is 60 cm and its hypotenuse is 25 cm, find its area.

The perimeter of an isosceles triangle is 40 cm. The base is two third of the sum of equal sides. Find the

length of each side.

If the area of an isosceles triangle is 60 \mathrm{cm}^{2} and the length of each of its equal sides is 13 cm, find its

base.

. Find the area of the quadrilateral ABCD in which ∠BCA =90^{\circ}, AB = 13 cm and ACD is an equilateral

triangle of side 12 cm.

The length of a rectangular hall is 5 cm more than its breadth if the area of the hall is 594 \mathrm{m}^{2}, \text { find its }

(a) The diagram (i) given below shows two paths drawn inside a rectangular field 50 m long and 35 m

wide. The width of each path is 5 meters. Find the area of the shaded portion.

(b) In the diagram (ii) given below, calculate the area of the shaded portion. All measurements are in

centimeters.

The shaded region of the given diagram represents the lawn in front of a house. On three sides of the

lawn there are flower beds of width 2 m.

(i) Find the length and the breadth of the lawn.

(ii) Hence, or otherwise, find the area of the flower – beds.

The cost of enclosing a rectangular garden with a fence all-around at the rate of Rs 15 per metre is Rs

5400. If the length of the garden is 100 m, find the area of the garden.

A rectangular plot of land measures 41 metres in length and 22.5 metres in width. A boundary wall 2

metres high is built all around the plot at a distance of 1.5 m from the plot. Find the inner surface area of

the boundary wall.

A wire when bent in the form of an equilateral triangle encloses an area of 36 \sqrt{3} \mathrm{cm}^{2}

Find the area

enclosed by the same wire when bent to form:

(i) a square, and

(ii) a rectangle whose length is 2 cm more than its width.

ABCD is a parallelogram with sides AB = 12 cm, BC = 10 cm and diagonal AC = 16 cm. Find the area

of the parallelogram. Also find the distance between its shorter sides.

Diagonals AC and BD of a parallelogram ABCD intersect at O. Given that AB = 12 cm and

perpendicular distance between AB and DC is 6 cm. Calculate the area of the triangle AOD.

The area of a rhombus is 98 \mathrm{cm}^{2} . If one of its diagonal is 14 cm, what is the length of the other diagonal?

The perimeter of a rhombus is 45 cm. If its height is 8 cm, calculate its area.

The distance between parallel sides of a trapezium is 12 cm and the distance between mid-points of

other sides is 18 cm. Find the area of the trapezium.

The area of a trapezium is 540 \mathrm{cm}^{2} If the ratio of parallel sides is 7: 5 and the distance between them is

18 cm, find the length of parallel sides.

The area of a parallelogram is 98 \mathrm{cm}^{2} If one altitude is half the corresponding base, determine the base

and the altitude of the parallelogram.

A rectangle has twice the area of a square. The length of the rectangle is 12 cm greater and the width is

8 cm greater than 2 side of a square. Find the perimeter of the square.

. (a) The figure (i) given below shows the cross-section of the concrete structure with the measurements

as given. Calculate the area of cross-section.

(b) The figure (ii) given below shows a field with the measurements given in meters. Find the area of the

field.

(c) Calculate the area of the pentagon ABCDE shown in fig (iii) below, given that AX = BX = 6 cm, EY =

CY = 4 cm, DE = DC = 5 cm, DX = 9 cm and DX is perpendicular to EC and AB.

The perimeter of a rectangular plot is 180 m and its area is 1800 \mathrm{m}^{2}. Take the length of the plot as x m. Use

the perimeter 180 m to write the value of the breadth in terms of x. Use the value of the length, breadth and

the area to write an equation in x. Solve the equation to calculate the length and breadth of the plot.

Find the length of the diameter of a circle whose circumference is 44 cm.

Find the radius and area of a circle if its circumference is 18π cm.

Find the perimeter of a semicircular plate of radius 3.85 cm.

(a) In the figure (i) given below, AC and BD are two perpendicular diameters of a circle ABCD. Given

that the area of the shaded portion is 308 \mathrm{cm}^{2}

calculate

(i) the length of AC and

(ii) the circumference of the circle

(b) In the figure (ii) given below, AC and BD are two perpendicular diameters of a circle with center O. If

AC = 16 cm, calculate the area and perimeter of the shaded part. (Take π = 3.14)

A bucket is raised from a well by means of a rope that is wound around a wheel of diameter 77 cm.

Given that the bucket ascends in 1 minute 28 seconds with a uniform speed of 1.1 m/sec, calculate the

number of complete revolutions the wheel makes in raising the bucket.

The wheel of a cart is making 5 revolutions per second. If the diameter of the wheel is 84 cm, find its

speed in km/hr. Give your answer correct to the nearest km.

The circumference of a circle is 123.2 cm. Calculate:

(i) the radius of the circle in cm.

(ii) the area of the circle in \mathrm{cm}^{2}, \text { correct to the nearest } \mathrm{cm}^{2} \text { . }

(iii) the effect on the area of the circle if the radius is doubled.

A copper wire when bent in the form of a square encloses an area of 121 \mathrm{cm}^{2} . If the same wire is bent

into the form of a circle, find the area of the circle.

A copper wire when bent into an equilateral triangle has area 121 \sqrt{3} \mathrm{cm}^{2}

If the same wire is bent into

the form of a circle, find the area enclosed by the wire.

(a) In the figure (i) given below, AOBC is a quadrant of a circle of radius 10m. Calculate the area of the

shaded portion. Take π = 3.14 and give your answer correct to two significant figures.

(b) In the figure (ii) given below, OAB is a quadrant of a circle. The radius OA = 3.5 cm and OD = 2 cm.

Calculate the area of the shaded portion.

A circular field has a perimeter of 660 m. A plot in the shape of a square having its vertices on the

the circumference is marked in the field. Calculate the area of the square field.

(a) In the figure (i) given below, the boundary of the shaded region in the given diagram consists of four

semicircular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is

3.5 cm, calculate

(i) the length of the boundary.

(ii) the area of the shaded region.

(b) In the figure (ii) given below, a piece of cardboard, in the shape of a trapezium ABCD, AB || CD and

∠BOD = 90^{\circ} , quarter circle BFEC is removed. Given AB = BC = 3.5 cm and DE = 2 cm. Calculate the area

of the remaining piece of the cardboard.

In the figure (i) given below, a chord AB of a circle of radius 10 cm subtends a right angle at the centre

O. Find the area of the sector OACB and of the major segment. Take π = 3.14.

Find the surface area and volume of a cube whose one edge is 7 cm.

A classroom is 10 m long, 6 m broad and 4 m high. How many students can it accommodate if one

student needs 1.5 \mathrm{m}^{2} of floor area? How many cubic metres of air will each student have?

The volume of a cuboid is 3600 \mathrm{cm}^{3} and its height is 12 cm. The cross-section is a rectangle whose length

and breadth is in the ratio 4: 3. Find the perimeter of the cross-section.

The length of the longest rod which can be kept inside a rectangular box is 17 cm. If the inner length and

the breadth of the box are 12 cm and 8 cm respectively, find its inner height.

A closed rectangular box has inner dimensions 90 cm by 80 cm by 70 cm. Calculate its capacity and the

area of tin-foil needed to line its inner surface.

A cube of 11 cm edge is immersed completely in a rectangular vessel containing water. If the

dimensions of the base of the vessel are 15 cm × 12 cm, find the rise in the water level in centimeters correct

to 2 decimal places, assuming that no water overflows.

A rectangular container, whose base is a square of side 6 cm, stands on a horizontal table and holds

water up to 1 cm from the top. When a cube is placed in the water and is completely submerged, the water

rises to the top and 2 \mathrm{cm}^{3} of water overflows, calculate the volume of the cube.

(a) Two cubes, each with 12 cm edge, are joined end to end. Find the surface area of the resulting

cuboid.

(b) A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube?

Also, find the ratio between the surface area of the original cube and the sum of the surface areas of the

new cubes.

A cube of a metal of 6 cm edge is melted and cast into a cuboid whose base is 9 cm × 8 cm. Find the

height of the cuboid.

A rectangular plot is 24 m long and 20 m wide. A cubical pit of edge 4 m is dug at each of the four

corners of the field and the soil removed is evenly spread over the remaining part of the plot. By what

height does the remaining plot get raised?

The inner dimensions of a closed wooden box are 2 m, 1.2 m, and 0.75 m. The thickness of the wood is

2.5 cm. Find the cost of wood required to make the box if 1 \mathrm{m}^{3} \text { of wood costs } \mathrm{Rs} 5400 .

A square brass plate of side x cm is 1 mm thick and weighs 4725 g. If one cc of brass weights 8.4 gm,

find the value of x.

A swimming pool is 50 metres long and 15 metres wide. Its shallow and deep ends are 1 ½ metres and

14 ½ metres deep respectively. If the bottom of the pool slopes uniformly, find the amount of water

required to fill the pool.