How To Solve Perimeter Area and Volume Questions Quickly

How to Solve Perimeter, Area, and Volume questions:-

• Geometry is concerned in calculating the length, perimeter, area and volume of various geometric figures and shapes

Type 1: Find the area, perimeter, length, breadth and some other sides of the shapes

Question 1: The sides of a triangle are in the ratio $\frac{1}{2}$:$\frac{1}{3}$: $\frac{1}{4}$. Find the smallest side of the triangle, if the perimeter of the triangle is 78 cm. 

Options: 

A. 20 cm

B. 24 cm

C. 36 cm

D. 18 cm

Solution:    According to the question, ratio of the side of the triangle are $\frac{1}{2}$:$\frac{1}{3}$: $\frac{1}{4}$ = 6: 3 : 4

Perimeter of the triangle = 78 cm

Therefore 6x + 4x + 3x = 78

                     13x = 78

x = 6

The Sides are : 6 x 6 = 36 cm

6 x 4 = 24 cm

6 x 3 = 18 cm

Correct option: D

Question 2 A rope makes 120 rounds of cylinder with base radius 10 cm. How many times it can go round a cylinder with base radius 20 cm?

Options: 

A. 70

B. 60 cm

C. 45 cm

D. 50 cm

Solution:    Le the round be a

If radius is more, then rounds will be less as the length of the ropes remains the same

x = 2*π*10*120…(1)

Similarly,

x = 2 * π * 20 * a…(2)

From (1) and (2)

10 * 120 = 20 * a

=> a = 60

Correct option: B

Question 3 Find the area of a parallelogram with base 16 cm and height 7 cm.

Options:

A. 112 cm²

B. 128 cm²

C. 102 cm²

D. 212 cm²

Solution:     Area of parallelogram = b * h

Area of parallelogram = 16 * 7 = 112 cm²

Correct option: A

Type 2: How To Solve Perimeter, Area and Volume Questions Quickly.By finding the volume and total surface area

Question 1: A cube of 5 cm was cut into as many 1 cm cubes as possible. Find out the ratio of the surface area of the larger cube to that of the surface areas of the smaller cubes?

Options: 

A. 1:2

B. 2:3

C. 1:5

D. 1:3

Solution:    Volume of the original cube = 5³ = 125 cm³.

Volume of each smaller cubes = 1 cm³. It means there are 125 smaller cubes.

Surface area of the cube = 6a²

Surface area of the larger cube = 6a² = 6 * 5² = 6 * 25 = 150

Surface area of one smaller cubes = 6 (1²) = 6

Now, surface area of all 125 cubes = 125 * 6 = 750

Therefore,

Required ratio = Surface area of the larger cube: Surface area of smaller cubes

= 150: 750

= 1: 5

Correct option: C

Question 2: The curved surface area of a cylindrical pillar is 264 m² and its volume is 924m³. Find the ratio of its diameter to its height.

Options: 

A. 7:4

B. 3:4

C. 6:5

D. 7:3

 Solution:    Volume of cylinder = πr2h

Curved Surface area of cylinder = 2 πrh

$\frac{\text{ Volume of cylinder }}{\text{ Curved Surface area of cylinder }}$

= $\frac{ π r^2 h}{2πrh}$ = $\frac{924}{264}$

r = $\frac{924}{264}\times 2$

r = 7

Curved Surface area of cylinder = 2πrh = 264

$2\times \frac{22}{7}\times 7\times h = 264$

$h = 264\times \frac{7}{22}\times \frac{1}{2}\times \frac{1}{7}$

h = 6

Now, required ratio = 2r/h = 2 *7/6 = 14/6 = 7/3 = 7:3

Correct option: D

Question 3: The volumes of two cones are in the ratio 1:10. The radius of the cones are in the ratio of 1: 2. What is the ratio of the height of the cone?

Options: 

A. 3:4

B. 3:5

C. 2:5

D. 1:3

Solution:    Volume of cone = $\frac{1}{3}$ πr²h

$\frac{V_{1}}{V_{2}}$= $\frac{1}{10}$

Now the ratio of the radius of both the cone = 1: 2

$\frac{V_{1}}{V_{2}}$=$\frac{(\frac{1}{3} π(r_{1})^2h_{1} )}{(\frac{1}{3} π(r_{2})^2h_{2}}$=

$\frac{(1)^2 h_{1}}{(2)^2 h_{2}}$

On solving we get$\frac{h_{1}}{h_{2}}$ = $\frac{2}{5}$

Therefore, the ratio of the height of the cone = 2:5

Correct option: C