[Q] Mensuration Tricks - 2

This is a famous question. Just remember whenever you are forming a circle and then a square,

the side of that square is given by, a = 1.6*r (approx.), where r = radius of the circle

I have written approx. because the actual formula is 1.57*r, but it will make the calculations a bit lengthy. So, just find 1.6*r and the answer will be little less than that. Like here

Side = 1.6*84 = 134.4, so the answer is 132 cm

Answer: (A)

When the wire is bent in the form of a circle of radius 84cm, that means the circumference (or the length of the wire) of the circle is 2*π*84 = 44*12 cm

Now this wire forms a square of (let’s say) side ‘a’

Then, 4a (perimeter of the square) = 44*12

Hence a = 132 cm

If the given rectangular sheet of paper (length =l, breadth = b) is rolled across its length to form a cylinder, having a height b, then volume of cylinder = (l*l*b)/4π

If rolled across its breadth, then = (b*b*l)/4π

In this question the sheet is rolled along its length, so volume = (l*l*b)/4π = 12*12*5/(4* π)

Volume = 180/ π cm³

Answer: (C)

In this question, the ratio of surface areas is given and they are asking the ratio of volumes. The word “sphere” is useless here. In such questions, just imagine area as A² and volume as A³. Now A³ is given and you have to find A³. How will you do it? Simple, first take the square-root of A² to convert it into A, and then take the cube of A to find A³.

So for solving this question, we just have to take the square-root of 4:9. The ratio will become 2:3. Then take the cube of 2:3. Hence the answer is 8:27

Answer: (C)

Here again ratio of areas is given, that means A² is given, and we have to find A. So 4:9 will become 2:3

Answer: (A)

Diameter and perimeter are directly proportional, P = D*π, where P is the perimeter and D is the diameter.

Hence a 75% increase in diameter means a 75% increase in perimeter

 Answer: (D)

The area of base and the volume of a cone are directly proportional V = A * h/3, where V = volume and h = height of the cone

Hence a 100% increase in the area of the base would mean a 100% increase in the volume

Answer: (B)

A is increased by 50% hence A² (or surface area) will increase by (1.5*1.5 – 1)*100 % = 125%

Note: Similarly A³ (or volume) will increase by (1.5*1.5*1.5 – 1)*100 % = 237.5%

Answer: (A)

Where ever the word “melting” is used in mensuration, it means only one thing – equate the volume

The volume of the rectangular block = l*b*h = 21*77*24 cm³

Now this volume will be equal to the volume of the sphere formed after melting the block

Volume of sphere = (4/3) * π * r³ = 21*77*24

Hence, r = 21 cm

Answer: (A)

The water rises by 5.6 cm. Take this 5.6 cm as the height of the cylindrical beaker and find its volume.

Volume of a cylinder = π*r*r*h = π * (7/2) * (7/2) * 5.6

Volume of the marbles (spherical in shape) = (4/3) * π * r³ = (4/3) * π * 0.7 * 0.7 * 0.7

No. of marbles dropped = Volume of beaker/Volume of a marble = 150

Answer: (B)

Let the radius of the big sphere be R.

Volume of a cone = (1/3) * π * R³ (since radius and volume are same as the radius of the sphere)

Let the radius of the smaller sphere = r

Then volume of cone = volume of smaller sphere

(1/3) * π * R³ = (4/3) * π * r³

r : R = 1 : 2^2/3

Surface area of smaller sphere(s) = 4 * π * r²

Surface area of larger sphere(S) = 4 * π * R²

S/s = (r/R)² = 1 : 2^4/3

Answer: (D)

When a cone is hollowed out from a cylinder, we get the above figure

The whole surface area of the remaining solid = Area of A + Area of B + Area of C

A = curved surface area of the cone

B = curved surface area of the cylinder

C = area of the cylindrical base

A = π * r * l, where l = slant height of the cone, which is  or

Hence A = π*3*5 = 15π

B = 2πrh = 2π*3*4 = 24π

C = πr² = π*3² = 9π

The whole surface area of the remaining solid = 15π + 24π + 9π = 48π

Answer: (C)

Given, AB = 3 cm, BC = 6 cm and OF = 1 cm

Height of the cone (AC) = √(6^2 - 3^2 ) = 3√3 cm

Triangles ABC and CFO are similar (RHS similarity)

So, OC/BC = OF/AB

OC = 2 cm, therefore CG = 3 cm (OG = 1 cm)

Now, ABC and CEG are similar

GE/AB = CG/AC

So, GE =

Required volume = Volume of cone (CDE) - Volume of Sphere

                                = 3π – (4/3)π

                                = (5/3)π

Answer: (C)

If you have any doubt in this article, please drop a comment.

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