Area Quiz

Question 1

The length of a rectangle is increased by 20% and its width is decreased by 10%. What is the percentage change in its area?

Accepted Answer

Let the original length of the rectangle be L and the original width be W.
Original Area (A) = L × W.

New length (L') = L × (1 + 20/100) = L × (1 + 0.20) = 1.2L.
New width (W') = W × (1 - 10/100) = W × (1 - 0.10) = 0.9W.

New Area (A') = L' × W' = (1.2L) × (0.9W) = 1.08LW.

Percentage change in area = ((New Area - Original Area) / Original Area) × 100
= ((1.08LW - LW) / LW) × 100
= (0.08LW / LW) × 100
= 0.08 × 100 = 8%.

Since the value is positive, it represents an 8% increase in the area.

Question 2

The sides of a triangular field are 13 meters, 14 meters, and 15 meters. Find the area of the field.

Accepted Answer

Let the sides of the triangle be a = 13 m, b = 14 m, and c = 15 m.

First, calculate the semi-perimeter (s) of the triangle:
s = (a + b + c) / 2
s = (13 + 14 + 15) / 2
s = 42 / 2
s = 21 m.

Now, use Heron's formula to find the area of the triangle:
Area = √[s × (s - a) × (s - b) × (s - c)]
Area = √[21 × (21 - 13) × (21 - 14) × (21 - 15)]
Area = √[21 × 8 × 7 × 6]
Area = √[(3 × 7) × (2 × 2 × 2) × 7 × (2 × 3)]
Area = √[(2^4) × (3^2) × (7^2)]
Area = (2^2) × 3 × 7
Area = 4 × 3 × 7
Area = 12 × 7
Area = 84 sq m.

Question 3

The circumference of a circular park is 176 meters. What is the area of the park? (Take π = 22/7)

Accepted Answer

Let the radius of the circular park be 'r' meters.

The circumference (C) of a circle is given by the formula C = 2πr.
Given C = 176 m.

So, 2 × (22/7) × r = 176
(44/7) × r = 176
r = (176 × 7) / 44
r = 4 × 7
r = 28 meters.

The area (A) of a circle is given by the formula A = πr².
A = (22/7) × (28)²
A = (22/7) × 28 × 28
A = 22 × 4 × 28
A = 88 × 28
A = 2464 sq m.

Question 4

The lengths of the two diagonals of a rhombus are 24 cm and 10 cm. Find the area of the rhombus.

Accepted Answer

Let the lengths of the two diagonals of the rhombus be d1 and d2.
Given d1 = 24 cm and d2 = 10 cm.

The area of a rhombus is given by the formula:
Area = (1/2) × d1 × d2
Area = (1/2) × 24 cm × 10 cm
Area = 12 × 10
Area = 120 sq cm.

Question 5

A rectangular garden is 50 m long and 42 m wide. A path 2 m wide is built around it, outside the garden. Find the area of the path.

Accepted Answer

Length of the rectangular garden (L) = 50 m.
Width of the rectangular garden (W) = 42 m.
Area of the garden = L × W = 50 m × 42 m = 2100 sq m.

The path is 2 m wide and built *outside* the garden.

New length of the garden including the path (L') = L + 2 × (path width)
L' = 50 + 2 × 2 = 50 + 4 = 54 m.

New width of the garden including the path (W') = W + 2 × (path width)
W' = 42 + 2 × 2 = 42 + 4 = 46 m.

Area of the garden including the path = L' × W' = 54 m × 46 m.
54 × 46 = 2484 sq m.

Area of the path = (Area of garden with path) - (Area of garden)
Area of the path = 2484 sq m - 2100 sq m
Area of the path = 384 sq m.

Question 6

The length of a rectangle is increased by 10% and its breadth is decreased by 10%. What is the percentage change in its area?

Accepted Answer

Let the original length be L and the original breadth be B.
Original Area = L * B.
New length = L + 0.10L = 1.10L
New breadth = B - 0.10B = 0.90B
New Area = (1.10L) * (0.90B) = 0.99LB
Change in Area = New Area - Original Area = 0.99LB - LB = -0.01LB
Percentage Change = (Change in Area / Original Area) * 100
= (-0.01LB / LB) * 100 = -1%
So, there is a 1% decrease in the area.

Question 7

The perimeter of a square and a circle are equal. If the area of the square is 121 cm², then the area of the circle is:

Accepted Answer

Let the side of the square be 'a' and the radius of the circle be 'r'.
Area of square = a² = 121 cm²
So, a = √121 = 11 cm.
Perimeter of square = 4a = 4 * 11 = 44 cm.
Perimeter of circle (Circumference) = 2πr.
Given that perimeter of square = perimeter of circle.
44 = 2πr
44 = 2 * (22/7) * r
44 = (44/7) * r
r = 7 cm.
Area of circle = πr² = (22/7) * 7² = (22/7) * 49 = 22 * 7 = 154 cm².

Question 8

The ratio of the area of two squares is 16:9. The ratio of their perimeters is:

Accepted Answer

Let the sides of the two squares be s1 and s2.
The ratio of their areas is s1² : s2² = 16:9.
Taking the square root of both sides, we get the ratio of their sides:
s1 : s2 = √16 : √9 = 4:3.
The perimeter of a square is 4 times its side.
So, the ratio of their perimeters will be 4s1 : 4s2 = s1 : s2 = 4:3.

Question 9

A rectangular field has its length and breadth in the ratio 5:3. If its area is 135 m², find the cost of fencing it at the rate of ₹6 per meter.

Accepted Answer

Let the length be 5x and the breadth be 3x.
Area = length × breadth = (5x) × (3x) = 15x².
Given Area = 135 m².
15x² = 135
x² = 135 / 15
x² = 9
x = 3 (since length cannot be negative).
Length = 5x = 5 * 3 = 15 m.
Breadth = 3x = 3 * 3 = 9 m.
Perimeter of the field = 2(length + breadth) = 2(15 + 9) = 2 * 24 = 48 m.
Cost of fencing = Perimeter × Rate per meter = 48 m * ₹6/m = ₹288.

Question 10

The diagonals of a rhombus are 16 cm and 12 cm. Its area is:

Accepted Answer

The area of a rhombus is given by the formula: A = (1/2) * d1 * d2, where d1 and d2 are the lengths of the diagonals.
Given diagonals d1 = 16 cm and d2 = 12 cm.
Area = (1/2) * 16 cm * 12 cm
Area = 8 * 12 cm²
Area = 96 cm².

Question 11

A path of uniform width 2m runs around the inside of a rectangular garden 10m long and 8m wide. Find the area of the path.

Accepted Answer

Outer dimensions of the garden: Length (L) = 10m, Breadth (B) = 8m.
Area of the garden (outer rectangle) = L * B = 10 * 8 = 80 m².
The path runs around the inside, and its width is 2m.
So, the inner rectangular area (garden without path) will have:
New Length (l) = L - 2 * (path width) = 10 - 2 * 2 = 10 - 4 = 6m.
New Breadth (b) = B - 2 * (path width) = 8 - 2 * 2 = 8 - 4 = 4m.
Area of the inner rectangle = l * b = 6 * 4 = 24 m².
Area of the path = Area of outer rectangle - Area of inner rectangle
Area of path = 80 m² - 24 m² = 56 m².

Wait, I made a mistake in calculation: 80 - 24 = 56. Let me recheck the options and my answer.
Oh, 56 is an option. My previous answer was 48. Let me redo the calculation carefully.
Area of outer rectangle = 10 * 8 = 80 m².
Inner length = 10 - (2+2) = 6m.
Inner breadth = 8 - (2+2) = 4m.
Area of inner rectangle = 6 * 4 = 24 m².
Area of path = 80 - 24 = 56 m².
So the answer should be 56 m². My previous thought for the answer was incorrect. I need to ensure the correct option is selected based on the detailed explanation.

Question 12

The base of a triangle is 15 cm and height is 12 cm. The height of another triangle of double the area having base 20 cm is:

Accepted Answer

For the first triangle:
Base (b1) = 15 cm, Height (h1) = 12 cm.
Area (A1) = (1/2) * b1 * h1 = (1/2) * 15 * 12 = 15 * 6 = 90 cm².
For the second triangle:
Base (b2) = 20 cm.
Area (A2) = 2 * A1 = 2 * 90 = 180 cm².
Let the height of the second triangle be h2.
Area (A2) = (1/2) * b2 * h2
180 = (1/2) * 20 * h2
180 = 10 * h2
h2 = 180 / 10 = 18 cm.

Question 13

The diameter of a wheel is 70 cm. How many revolutions will it make to travel 110 meters?

Accepted Answer

Diameter (D) = 70 cm.
Radius (r) = D/2 = 35 cm.
Circumference of the wheel (distance covered in one revolution) = 2πr
= 2 * (22/7) * 35 cm = 2 * 22 * 5 cm = 220 cm.
Total distance to be travelled = 110 meters = 110 * 100 cm = 11000 cm.
Number of revolutions = Total distance / Circumference
= 11000 cm / 220 cm = 1100 / 22 = 50 revolutions.

Question 14

If the side of an equilateral triangle is 6 cm, then its area is:

Accepted Answer

The formula for the area of an equilateral triangle with side 'a' is (√3/4) * a².
Given side (a) = 6 cm.
Area = (√3/4) * (6)²
Area = (√3/4) * 36
Area = 9√3 cm².

Question 15

The area of a trapezium is 34 cm² and the length of one of the parallel sides is 10 cm. If its height is 4 cm, find the length of the other parallel side.

Accepted Answer

The formula for the area of a trapezium is A = (1/2) * (sum of parallel sides) * height.
Given Area (A) = 34 cm².
Length of one parallel side (a) = 10 cm.
Height (h) = 4 cm.
Let the length of the other parallel side be 'b'.
34 = (1/2) * (10 + b) * 4
34 = 2 * (10 + b)
34 / 2 = 10 + b
17 = 10 + b
b = 17 - 10
b = 7 cm.
So, the length of the other parallel side is 7 cm.

Question 16

A rectangular field's length is increased by 20% and its width is decreased by 10%. What is the percentage change in its area?

Accepted Answer

Let the original length (L) be 10 units and original width (W) be 10 units. 
Original Area = L * W = 10 * 10 = 100 sq units.

New length = L + 20% of L = 10 + (0.20 * 10) = 10 + 2 = 12 units.
New width = W - 10% of W = 10 - (0.10 * 10) = 10 - 1 = 9 units.

New Area = New Length * New Width = 12 * 9 = 108 sq units.

Percentage change in area = ((New Area - Original Area) / Original Area) * 100
= ((108 - 100) / 100) * 100
= (8 / 100) * 100 = 8% increase.

Question 17

The base of a triangle is 15 cm and its height is 12 cm. The height of another triangle of double the area having the base 20 cm is:

Accepted Answer

Area of the first triangle (A1) = (1/2) * base1 * height1
= (1/2) * 15 cm * 12 cm = 90 sq cm.

The second triangle has double the area, so Area2 = 2 * A1 = 2 * 90 = 180 sq cm.

For the second triangle, base2 = 20 cm. Let its height be h2.
Area2 = (1/2) * base2 * h2
180 = (1/2) * 20 * h2
180 = 10 * h2
h2 = 180 / 10 = 18 cm.

Question 18

The ratio of the area of a circle to the area of a square whose side is equal to the radius of the circle is:

Accepted Answer

Let the radius of the circle be 'r'.
Area of the circle = πr^2.

The side of the square is equal to the radius of the circle, so the side of the square = 'r'.
Area of the square = side^2 = r^2.

Ratio of the area of the circle to the area of the square = (Area of circle) / (Area of square)
= (πr^2) / (r^2) = π/1 or π:1.

Question 19

A path of 2.5 m wide is built around a rectangular park of length 120 m and width 90 m. The area of the path is:

Accepted Answer

Original dimensions of the park:
Length (L) = 120 m
Width (W) = 90 m
Original Area of the park = L * W = 120 * 90 = 10800 sq m.

The path is 2.5 m wide and is built around the park, meaning it's outside.
New length (including path) = L + 2 * (path width) = 120 + 2 * 2.5 = 120 + 5 = 125 m.
New width (including path) = W + 2 * (path width) = 90 + 2 * 2.5 = 90 + 5 = 95 m.

Area of the park with path = New Length * New Width = 125 * 95 = 11875 sq m.

Area of the path = (Area of park with path) - (Original Area of park)
= 11875 - 10800 = 1075 sq m.

Question 20

The diagonals of a rhombus are 16 cm and 12 cm. What is the area of the rhombus?

Accepted Answer

The formula for the area of a rhombus given its diagonals (d1 and d2) is:
Area = (1/2) * d1 * d2

Given diagonals: d1 = 16 cm, d2 = 12 cm.
Area = (1/2) * 16 * 12
= 8 * 12
= 96 sq cm.

Question 21

The side of an equilateral triangle is 8 cm. Find its area.

Accepted Answer

The formula for the area of an equilateral triangle with side 'a' is:
Area = (√3 / 4) * a^2

Given side (a) = 8 cm.
Area = (√3 / 4) * (8)^2
= (√3 / 4) * 64
= 16√3 sq cm.

Question 22

If the radius of a circle is tripled, what is the ratio of the new area to the original area?

Accepted Answer

Let the original radius of the circle be 'r'.
Original Area (A_original) = πr^2.

If the radius is tripled, the new radius (r_new) = 3r.
New Area (A_new) = π(r_new)^2 = π(3r)^2 = π(9r^2) = 9πr^2.

Ratio of the new area to the original area = A_new / A_original
= (9πr^2) / (πr^2)
= 9/1 or 9:1.

Question 23

The parallel sides of a trapezium are 10 cm and 16 cm, and its height is 5 cm. Find its area.

Accepted Answer

The formula for the area of a trapezium is:
Area = (1/2) * (sum of parallel sides) * height

Given parallel sides (a and b): a = 10 cm, b = 16 cm.
Given height (h) = 5 cm.

Area = (1/2) * (10 + 16) * 5
= (1/2) * 26 * 5
= 13 * 5
= 65 sq cm.

Question 24

A rectangular hall is 15 m long and 12 m wide. The cost of flooring the hall at the rate of Rs. 80 per square meter is:

Accepted Answer

Area of the rectangular hall = Length * Width
= 15 m * 12 m = 180 sq m.

Cost of flooring = Area * Rate per square meter
= 180 sq m * Rs. 80/sq m
= Rs. 14400.

Question 25

The area of a square field is 441 sq m. The perimeter of the field is:

Accepted Answer

Let the side of the square field be 's'.
Area of the square = s^2.

Given Area = 441 sq m.
s^2 = 441
s = √441 = 21 m.

The perimeter of a square = 4 * side.
Perimeter = 4 * 21 m = 84 m.

Question 26

The area of a rectangular plot is 240 sq cm. If its length is increased by 10% and breadth is decreased by 10%, what is the new area of the plot?

Accepted Answer

Let the original length be L and breadth be B. Original Area = L * B = 240 sq cm.
New length = L + 0.10L = 1.1L
New breadth = B - 0.10B = 0.9B
New Area = (1.1L) * (0.9B) = 0.99 * (L * B)
New Area = 0.99 * 240 = 237.6 sq cm.

Question 27

The perimeter of a square is equal to the circumference of a circle. If the area of the square is 484 sq cm, what is the area of the circle? (Take π = 22/7)

Accepted Answer

Let the side of the square be 'a'. Area of square = a² = 484 sq cm. So, a = √484 = 22 cm.
Perimeter of square = 4a = 4 * 22 = 88 cm.
Let the radius of the circle be 'r'. Circumference of circle = 2πr.
Given, Perimeter of square = Circumference of circle.
88 = 2 * (22/7) * r
88 = (44/7) * r
r = (88 * 7) / 44 = 2 * 7 = 14 cm.
Area of circle = πr² = (22/7) * 14 * 14 = 22 * 2 * 14 = 44 * 14 = 616 sq cm.

Question 28

The base of a triangular field is three times its height. If the cost of cultivating the field at Rs. 36.72 per 100 sq meters is Rs. 495.72, find the height of the triangle.

Accepted Answer

Total cost of cultivating = Rs. 495.72
Cost per 100 sq meters = Rs. 36.72
Cost per sq meter = Rs. 36.72 / 100 = Rs. 0.3672
Area of the field = Total Cost / Cost per sq meter = 495.72 / 0.3672 = 1350 sq meters.
Let the height of the triangle be 'h' meters.
Base of the triangle 'b' = 3h meters.
Area of triangle = (1/2) * base * height
1350 = (1/2) * (3h) * h
1350 = (3/2) * h²
h² = (1350 * 2) / 3 = 2700 / 3 = 900
h = √900 = 30 meters.

Question 29

The ratio of the length and breadth of a rectangular plot is 5:3. If the area of the plot is 1500 sq meters, find the perimeter of the plot.

Accepted Answer

Let the length be 5x and the breadth be 3x.
Area = length * breadth = (5x) * (3x) = 15x²
Given Area = 1500 sq meters.
15x² = 1500
x² = 1500 / 15 = 100
x = √100 = 10.
Length = 5x = 5 * 10 = 50 meters.
Breadth = 3x = 3 * 10 = 30 meters.
Perimeter = 2 * (length + breadth) = 2 * (50 + 30) = 2 * 80 = 160 meters.

Question 30

A rectangular lawn 50 m long and 30 m wide has a path of 2 m wide running around it, outside. Find the area of the path.

Accepted Answer

Length of the lawn (L) = 50 m
Width of the lawn (B) = 30 m
Area of the lawn = L * B = 50 * 30 = 1500 sq m.
The path is 2 m wide running outside the lawn.
New length (L') = L + 2 * (width of path) = 50 + 2 * 2 = 50 + 4 = 54 m.
New width (B') = B + 2 * (width of path) = 30 + 2 * 2 = 30 + 4 = 34 m.
Area of lawn including path = L' * B' = 54 * 34 = 1836 sq m.
Area of the path = (Area of lawn including path) - (Area of lawn)
Area of the path = 1836 - 1500 = 336 sq m.

Question 31

If the length of a rectangle is increased by 20% and its breadth is decreased by 25%, what is the percentage change in its area?

Accepted Answer

Let the original length be L and breadth be B. Original Area = L * B.
New length = L + 20% of L = L + 0.20L = 1.2L.
New breadth = B - 25% of B = B - 0.25B = 0.75B.
New Area = (1.2L) * (0.75B) = (1.2 * 0.75) * (L * B) = 0.9 * (L * B).
Percentage change = ((New Area - Original Area) / Original Area) * 100
= ((0.9LB - LB) / LB) * 100 = (-0.1LB / LB) * 100 = -0.1 * 100 = -10%.
This indicates a 10% decrease.

Question 32

The diagonals of a rhombus are 24 cm and 10 cm. Find its area and perimeter.

Accepted Answer

Let the diagonals be d1 = 24 cm and d2 = 10 cm.
Area of rhombus = (1/2) * d1 * d2 = (1/2) * 24 * 10 = 12 * 10 = 120 sq cm.
The diagonals of a rhombus bisect each other at right angles.
Consider one of the four right-angled triangles formed by the half-diagonals and a side of the rhombus.
Half of d1 = 24/2 = 12 cm.
Half of d2 = 10/2 = 5 cm.
Let the side of the rhombus be 's'. Using Pythagoras theorem:
s² = (12)² + (5)² = 144 + 25 = 169
s = √169 = 13 cm.
Perimeter of rhombus = 4 * s = 4 * 13 = 52 cm.

Question 33

The parallel sides of a trapezium are 12 cm and 20 cm, and its height is 10 cm. Find its area.

Accepted Answer

Let the parallel sides be a = 12 cm and b = 20 cm.
Height (h) = 10 cm.
Area of trapezium = (1/2) * (sum of parallel sides) * height
Area = (1/2) * (a + b) * h
Area = (1/2) * (12 + 20) * 10
Area = (1/2) * 32 * 10
Area = 16 * 10 = 160 sq cm.

Question 34

A square park has an area of 196 sq meters. A circular flower bed is at the center of the park with a radius of 7 meters. What is the area of the park excluding the flower bed? (Take π = 22/7)

Accepted Answer

Area of the square park = 196 sq meters.
Radius of the circular flower bed (r) = 7 meters.
Area of the circular flower bed = πr² = (22/7) * 7 * 7 = 22 * 7 = 154 sq meters.
Area of the park excluding the flower bed = (Area of square park) - (Area of circular flower bed)
Area = 196 - 154 = 42 sq meters.

Question 35

A farmer wants to fence his rectangular field of length 20m and width 15m. The cost of fencing is Rs. 25 per meter. He also wants to plough the field at Rs. 10 per sq meter. What is the total cost for fencing and ploughing?

Accepted Answer

Length of the field (L) = 20 m
Width of the field (B) = 15 m
Cost of Fencing:
Fencing is done along the perimeter.
Perimeter of the field = 2 * (L + B) = 2 * (20 + 15) = 2 * 35 = 70 meters.
Cost of fencing per meter = Rs. 25.
Total cost of fencing = Perimeter * Cost per meter = 70 * 25 = Rs. 1750.
Cost of Ploughing:
Ploughing is done over the area of the field.
Area of the field = L * B = 20 * 15 = 300 sq meters.
Cost of ploughing per sq meter = Rs. 10.
Total cost of ploughing = Area * Cost per sq meter = 300 * 10 = Rs. 3000.
Total Cost:
Total cost = Cost of fencing + Cost of ploughing = 1750 + 3000 = Rs. 4750.

Question 36

The circumference of a circle is 88 cm. What is the area of the circle? (Take π = 22/7)

Accepted Answer

The formula for the circumference of a circle is C = 2πr. Given C = 88 cm.
88 = 2 * (22/7) * r
88 = (44/7) * r
r = (88 * 7) / 44 = 2 * 7 = 14 cm.
The formula for the area of a circle is A = πr^2.
A = (22/7) * 14 * 14 = 22 * 2 * 14 = 44 * 14 = 616 sq cm.

Question 37

A rectangular park is 60 m long and 40 m wide. A path 2 m wide runs around the park on its outside. Find the area of the path.

Accepted Answer

Length of the park (L) = 60 m, Width of the park (W) = 40 m.
Area of the park = L * W = 60 * 40 = 2400 sq m.
The path is 2 m wide and runs outside the park. So, the new dimensions of the park including the path will be:
New length (L') = L + 2 * (path width) = 60 + 2 * 2 = 64 m.
New width (W') = W + 2 * (path width) = 40 + 2 * 2 = 44 m.
Area of the park with path = L' * W' = 64 * 44 = 2816 sq m.
Area of the path = Area of the park with path - Area of the park = 2816 - 2400 = 416 sq m.

Question 38

The area of a square is equal to the area of a circle. If the side of the square is 's' and the radius of the circle is 'r', what is the ratio of 's' to 'r'?

Accepted Answer

Area of square = s^2.
Area of circle = πr^2.
Given that the areas are equal: s^2 = πr^2.
To find the ratio s:r, take the square root of both sides:
s = √(πr^2)
s = r√π
Divide both sides by r: s/r = √π / 1.
So, the ratio of s to r is √π : 1.

Question 39

The diagonals of a rhombus are 16 cm and 12 cm. Find the area of the rhombus.

Accepted Answer

The formula for the area of a rhombus is (1/2) * d1 * d2, where d1 and d2 are the lengths of the diagonals.
Given d1 = 16 cm and d2 = 12 cm.
Area = (1/2) * 16 cm * 12 cm = 8 * 12 = 96 sq cm.

Question 40

The parallel sides of a trapezium are 14 cm and 18 cm and the distance between them is 9 cm. The area of the trapezium is:

Accepted Answer

The formula for the area of a trapezium is (1/2) * (sum of parallel sides) * height.
Given parallel sides = 14 cm and 18 cm.
Height (distance between them) = 9 cm.
Area = (1/2) * (14 + 18) * 9
Area = (1/2) * 32 * 9
Area = 16 * 9 = 144 sq cm.

Question 41

If the height of an equilateral triangle is 6√3 cm, then find its area.

Accepted Answer

For an equilateral triangle with side 'a', the height (h) is given by the formula h = (√3/2)a.
Given h = 6√3 cm.
6√3 = (√3/2)a
Multiply both sides by 2 and divide by √3 to find 'a':
a = (6√3 * 2) / √3 = 12 cm.
The formula for the area of an equilateral triangle is A = (√3/4)a^2.
A = (√3/4) * (12)^2 = (√3/4) * 144 = 36√3 sq cm.

Question 42

The perimeter of a rectangle is 60 cm and its length is twice its width. What is the area of a square whose side is equal to the length of the rectangle?

Accepted Answer

Let the width of the rectangle be 'w' cm. According to the problem, the length 'l' is twice its width, so l = 2w.
The perimeter of a rectangle = 2(l + w). Given perimeter = 60 cm.
2(2w + w) = 60
2(3w) = 60
6w = 60
w = 10 cm.
So, the width is 10 cm. The length l = 2w = 2 * 10 = 20 cm.
The side of the square is equal to the length of the rectangle, so the side of the square = 20 cm.
Area of the square = side^2 = (20 cm)^2 = 400 sq cm.

Question 43

A rectangular sheet of paper is 22 cm long and 14 cm wide. From this, the largest possible circle is cut out. What is the area of the remaining paper? (Take π = 22/7)

Accepted Answer

Length of the rectangular sheet = 22 cm, Width = 14 cm.
Area of the rectangular sheet = Length * Width = 22 * 14 = 308 sq cm.
For the largest possible circle to be cut from the sheet, its diameter must be equal to the shorter side of the rectangle, which is 14 cm.
So, the diameter of the circle = 14 cm.
Radius of the circle (r) = Diameter / 2 = 14 / 2 = 7 cm.
Area of the circle = πr^2 = (22/7) * 7 * 7 = 22 * 7 = 154 sq cm.
Area of the remaining paper = Area of rectangle - Area of circle = 308 - 154 = 154 sq cm.

Question 44

The base of a triangle is 15 cm and height is 12 cm. The height of another triangle of double the area with base 20 cm is:

Accepted Answer

For the first triangle:
Base (b1) = 15 cm, Height (h1) = 12 cm.
Area1 = (1/2) * b1 * h1 = (1/2) * 15 * 12 = 15 * 6 = 90 sq cm.
For the second triangle:
Area2 = 2 * Area1 = 2 * 90 = 180 sq cm.
Base (b2) = 20 cm.
Let the height of the second triangle be h2.
Area2 = (1/2) * b2 * h2
180 = (1/2) * 20 * h2
180 = 10 * h2
h2 = 180 / 10 = 18 cm.

Question 45

A rectangular plot has a perimeter of 48 meters and its length is twice its breadth. What is the area of the plot?

Accepted Answer

Let the length of the rectangular plot be L and the breadth be B.
Given, L = 2B.
The perimeter of the rectangle is 2(L + B).
So, 2(2B + B) = 48
2(3B) = 48
6B = 48
B = 8 meters
Then, L = 2 * 8 = 16 meters.
The area of the plot = L * B = 16 * 8 = 128 sq m.

Question 46

The diagonal of a square is 10√2 cm. What is the area of the square?

Accepted Answer

Let the side of the square be 'a' cm.
The diagonal of a square is given by a√2.
Given, diagonal = 10√2 cm.
So, a√2 = 10√2
This implies a = 10 cm.
The area of the square = a² = 10² = 100 sq cm.

Question 47

The circumference of a circle is 88 cm. What is the area of the circle? (Use π = 22/7)

Accepted Answer

Let the radius of the circle be 'r' cm.
The circumference of a circle is 2πr.
Given, 2πr = 88
2 * (22/7) * r = 88
(44/7) * r = 88
r = 88 * (7/44)
r = 2 * 7 = 14 cm.
The area of the circle = πr² = (22/7) * 14 * 14
= 22 * 2 * 14 = 44 * 14 = 616 sq cm.

Question 48

If the length of a rectangle is increased by 20% and its breadth is decreased by 10%, what is the percentage change in its area?

Accepted Answer

Let the original length be L and original breadth be B. Original area = L * B.
New length = L * (1 + 20/100) = 1.2L.
New breadth = B * (1 - 10/100) = 0.9B.
New area = (1.2L) * (0.9B) = 1.08LB.
Percentage change in area = ((New Area - Original Area) / Original Area) * 100
= ((1.08LB - LB) / LB) * 100
= (0.08LB / LB) * 100 = 0.08 * 100 = 8%.
Since the new area (1.08LB) is greater than the original area (LB), it is an 8% increase.

Question 49

A rectangular garden is 90 m long and 70 m wide. A path 5 m wide is to be built outside around it. Find the area of the path.

Accepted Answer

Original length of the garden = 90 m, width = 70 m.
Area of the garden = 90 * 70 = 6300 sq m.
When a path of 5 m wide is built outside, the new length and width will increase by 2 * 5 = 10 m.
New length (with path) = 90 + 10 = 100 m.
New width (with path) = 70 + 10 = 80 m.
Area of the garden including the path = 100 * 80 = 8000 sq m.
Area of the path = (Area of garden with path) - (Area of garden)
= 8000 - 6300 = 1700 sq m.

Question 50

The perimeter of an equilateral triangle is 60 cm. What is its area?

Accepted Answer

Let the side of the equilateral triangle be 'a' cm.
The perimeter of an equilateral triangle is 3a.
Given, 3a = 60 cm.
So, a = 60 / 3 = 20 cm.
The area of an equilateral triangle is (√3/4) * a².
Area = (√3/4) * (20)²
= (√3/4) * 400
= 100√3 sq cm.

Question 51

If the perimeter of a square is equal to the circumference of a circle, what is the ratio of the area of the square to the area of the circle?

Accepted Answer

Let the side of the square be 'a' and the radius of the circle be 'r'.
Perimeter of square = 4a.
Circumference of circle = 2πr.
Given, 4a = 2πr.
From this, a = (2πr) / 4 = (πr) / 2.
Area of square = a² = ((πr)/2)² = (π²r²)/4.
Area of circle = πr².
Ratio (Area of square : Area of circle) = ((π²r²)/4) : (πr²)
= (π²/4) : π
To simplify, multiply both sides by 4: π² : 4π
Divide both sides by π: π : 4.

Question 52

A rectangular sheet of paper is 22 cm long and 14 cm wide. From one of its longer sides, a semi-circular portion is cut off such that the diameter of the semi-circle is equal to the width of the sheet. Find the area of the remaining sheet.

Accepted Answer

Area of the rectangular sheet = Length * Width = 22 cm * 14 cm = 308 sq cm.
The diameter of the semi-circle is equal to the width of the sheet, which is 14 cm.
So, the radius of the semi-circle (r) = Diameter / 2 = 14 / 2 = 7 cm.
Area of the semi-circle = (1/2) * πr² = (1/2) * (22/7) * 7 * 7
= (1/2) * 22 * 7 = 11 * 7 = 77 sq cm.
Area of the remaining sheet = Area of rectangular sheet - Area of semi-circle
= 308 sq cm - 77 sq cm = 231 sq cm.

Question 53

The two diagonals of a rhombus are 16 cm and 12 cm. Find its area.

Accepted Answer

The area of a rhombus is given by the formula (1/2) * d1 * d2, where d1 and d2 are the lengths of its diagonals.
Given diagonals are d1 = 16 cm and d2 = 12 cm.
Area = (1/2) * 16 * 12
= 8 * 12 = 96 sq cm.

Question 54

The cost of carpeting a room 15 m long with a carpet 75 cm wide at Rs. 60 per meter is Rs. 10800. Find the width of the room.

Accepted Answer

Total cost of carpeting = Rs. 10800.
Cost per meter of carpet = Rs. 60.
Length of carpet required = Total cost / Cost per meter = 10800 / 60 = 180 meters.
The width of the carpet is 75 cm, which is 0.75 meters.
Area of the carpet = Length of carpet * Width of carpet = 180 m * 0.75 m = 135 sq m.
The area of the room is equal to the area of the carpet.
Area of the room = Length of room * Width of room.
Given length of the room = 15 m.
So, 135 sq m = 15 m * Width of room.
Width of room = 135 / 15 = 9 meters.

Question 55

A rectangular field has its length and breadth in the ratio 5:3. If its perimeter is 128 m, what is its area?

Accepted Answer

Let the length of the rectangle be 5x and the breadth be 3x. The perimeter of a rectangle is 2(length + breadth).
Perimeter = 2(5x + 3x) = 2(8x) = 16x.
Given perimeter = 128 m, so 16x = 128.
x = 128 / 16 = 8.
Length = 5x = 5 * 8 = 40 m.
Breadth = 3x = 3 * 8 = 24 m.
Area of the rectangle = length * breadth = 40 * 24 = 960 sq m.

Question 56

If the side of a square is increased by 20%, then its area is increased by what percentage?

Accepted Answer

Let the original side of the square be 'a'.
Original area = a^2.
If the side is increased by 20%, the new side = a + 0.20a = 1.2a.
New area = (1.2a)^2 = 1.44a^2.
Increase in area = New area - Original area = 1.44a^2 - a^2 = 0.44a^2.
Percentage increase = (Increase in area / Original area) * 100
= (0.44a^2 / a^2) * 100 = 0.44 * 100 = 44%.

Question 57

The circumference of a circle is 88 cm. What is the area of the circle? (Use π = 22/7)

Accepted Answer

The formula for the circumference of a circle is C = 2πr.
Given C = 88 cm, so 2 * (22/7) * r = 88.
(44/7) * r = 88.
r = (88 * 7) / 44 = 2 * 7 = 14 cm.
The formula for the area of a circle is A = πr^2.
A = (22/7) * 14 * 14 = 22 * 2 * 14 = 44 * 14 = 616 sq cm.

Question 58

A rectangular garden is 90 m long and 75 m broad. A path 5 m wide is to be built outside around it. Find the area of the path.

Accepted Answer

Length of the garden (inner rectangle) = 90 m.
Breadth of the garden (inner rectangle) = 75 m.
Area of the garden = 90 * 75 = 6750 sq m.
The path is 5 m wide and built outside around the garden.
New length (outer rectangle) = 90 + 2*5 = 90 + 10 = 100 m.
New breadth (outer rectangle) = 75 + 2*5 = 75 + 10 = 85 m.
Area of the outer rectangle (garden + path) = 100 * 85 = 8500 sq m.
Area of the path = Area of outer rectangle - Area of inner rectangle
= 8500 - 6750 = 1750 sq m.

Question 59

The base of a triangle is twice its height. If the area of the triangle is 256 sq cm, find its height.

Accepted Answer

Let the height of the triangle be 'h' cm.
Given that the base is twice its height, so base = 2h cm.
The formula for the area of a triangle is (1/2) * base * height.
Area = (1/2) * (2h) * h = h^2.
Given area = 256 sq cm.
So, h^2 = 256.
h = √256 = 16 cm.

Question 60

The parallel sides of a trapezium are 20 cm and 12 cm, and the distance between them is 10 cm. Find its area.

Accepted Answer

The formula for the area of a trapezium is (1/2) * (sum of parallel sides) * height.
Given parallel sides (a and b) are 20 cm and 12 cm.
Height (h) = 10 cm.
Area = (1/2) * (20 + 12) * 10
= (1/2) * 32 * 10
= 16 * 10 = 160 sq cm.

Question 61

A square is inscribed in a circle whose radius is 7√2 cm. What is the area of the square?

Accepted Answer

When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
Radius of the circle (r) = 7√2 cm.
Diameter of the circle (d) = 2 * r = 2 * 7√2 = 14√2 cm.
Let the side of the square be 's'. The diagonal of a square is s√2.
So, s√2 = 14√2.
Dividing both sides by √2, we get s = 14 cm.
Area of the square = s^2 = 14^2 = 196 sq cm.

Question 62

The cost of tiling a rectangular floor at the rate of Rs. 25 per sq meter is Rs. 15000. If the length of the floor is 30 meters, what is its breadth?

Accepted Answer

Total cost of tiling = Rs. 15000.
Rate of tiling = Rs. 25 per sq meter.
Area of the floor = Total cost / Rate per sq meter = 15000 / 25 = 600 sq meters.
For a rectangular floor, Area = length * breadth.
Given length = 30 meters.
So, 600 = 30 * breadth.
Breadth = 600 / 30 = 20 meters.

Question 63

A circular garden has a radius of 14 meters. A 7-meter wide path runs around the garden on its outer side. What is the area of the path? (Use π = 22/7)

Accepted Answer

Radius of the circular garden (inner circle), r = 14 meters.
Width of the path = 7 meters.
Radius of the outer circle (garden + path), R = r + path width = 14 + 7 = 21 meters.
Area of the path = Area of the outer circle - Area of the inner circle
= πR^2 - πr^2 = π(R^2 - r^2).
= (22/7) * (21^2 - 14^2)
= (22/7) * (441 - 196)
= (22/7) * 245
= 22 * (245/7) = 22 * 35 = 770 sq m.