Practice Question On Area And Perimeter Of All Shapes

Area and perimeter are the measuring parameters related to various two-dimensional figures like triangles, rectangles, squares, parallelograms etc. 

Area:-

The total space enclosed by the boundary of a plane shape is called the area of that particular shape. In other words, the area of a shape is a measure associated with the part of the plane enclosed in the shape. Unit of the area is square, like square meter, square centimetre etc.

For example, If the length (l) of a rectangle is 10 cm and breadth (b) is 5 cm, then the Area of the rectangle 
= 1xb =10×5 = 50 sq cm

Perimeter:-

The length of the border around any enclosed plane is known as the perimeter. Therefore, the sum of the sides of a plane shape is the perimeter of that particular shape.

The unit of the perimeter is the same as the unit of sides of a given shape, like metre, centimetre etc.

For example, If the sides of a triangle are 2 cm, 8 cm, and 4 cm, respectively, then 12 cm,  14 cm, 18 cm, then
Perimeter of the triangle = 12 cm + 14 cm + 18 cm = 44 cm

Different shapes:-
1. Triangle
2. Square
3. Circle
4. Regular polygon

Formulas of Area and perimeter of different shapes:-
1. Triangle:

Various types of triangles are shown below –

i) Equilateral Triangle:

All the angles of this triangle are 60° and all its sides are equal. If the side is a, then

Area = √3a²/4
Perimeter = 3a

ii) Isosceles triangle:

It has two sides equal. Let, each of the equal sides is a and the third side is b, then

Area = [b√(4a² – b²)] / 4
Perimeter = a + a + b = 2a + b

iii) Scalene Triangle:

It has three unequal sides. Let sides be a, b and c then

Area = √[s(s-a)(s-b)(s-c)            [ Heron’s formula]
   Where, s = (a+b+c)/2
Perimeter = a + b + c

iv) Right angle Triangle:

Area = 1/2 × Base × Height 
Perimeter = a + b + c

2. Quadrilateral:-

Various types of a quadrilateral are shown below

i) Square:

A square has equal four sides. Let the side is a, then

Area = side² = a²
Perimeter = 4 × side = 4a

ii) Parallelogram:

Area = Base × Height
Perimeter = 2 ( length + breadth)

iii) Rectangle:

Area = Length × Breadth
Perimeter = 2 (length + breadth)

iv) Trapezium:

Four sides are a, b, c and d.

Area = 1/2 (sum of the parallel sides)× height
Perimeter = a + b + c + d

v) Rhombus:

Sides are a and diagonals are d1 and d2

Area = 1/2 × d1 × d2
Perimeter = 4a

3. Circle:-

If the radius of a circle is r, then
Area = πr²
Perimeter = 2πr

4. Regular Polygon:-
Sides of a polygon be a, then

Area = 5a² × √3/4
The perimeter of n sided polygon = n × side

Problems related to area and perimeter:-

1. Question

Find the perimeter of a triangle with sides equal to 5 cm, 9 cm and 10 cm. 
A) 24 cm
B) 12 cm
C) 16 cm
D) 18 cm
E) 30 cm

Answer:- A
Sol. 
Required perimeter = Sum of the sides = (5 + 9 + 10) cm = 24 cm

2. Question

An equilateral triangle has a perimeter of 75 cm. Find its area.
A) 225√3/4 cm²
B) 110√3/4 cm²
C) 625√3/4 cm²
D) 729√3/4 cm²
E) None of these

Answer:- C
Sol. Given that, the perimeter of an equilateral triangle is 75 cm.
Let, side be ‘a’
We know,
3a = 75
a = 25
Required area = (√3 × 25²)/4 = 625√3/4 sq cm.

3. Question

A square has a perimeter of 48 cm. Find its side and area.
A) 12, 144
B) 16, 144
C) 20, 200
D) 15, 125
E) None of these

Answer:- A
Sol.
Let, its side = a
According to the question,
4a = 48
a = 12 cm
Side = 12 cm
Area = side² = 12² = 144 cm²

4. Question

If the length and breadth of a rectangular house are 25 m and 10 m, then find its area and perimeter.

A) 250, 70
B) 200, 50
C) 150, 40
D) 250, 25
E) none of these

Answer :- A
Sol.
Area = Length × Breadth = 25×10 = 250 m²
Perimeter = 2(length + Breadth) = 2(25+10) = 70 m

5. Question

If the base and area of a parallelogram are 10 cm and 250 sq cm, then find its height.

A) 15 cm
B) 18 cm
C) 22 cm
D) 25 cm
E) 30 cm

Answer:- D
Sol.
 Area = Base × Height
height = area/ base = 250/10 = 25 cm

6. Question 

If the area and height of a right angle triangle is 42 cm² and 14cm. then find its base.

A) 8 cm
B) 6 cm
C) 9 cm
D) 12 cm
E) 15 cm

Answer:- B
Sol.
We know,
Area of a right angle triangle, 
Area = 1/2 × Base × Height
=> 42 = 1/2 × base × 14
Base = 6 cm.

7. Question

A wheel makes 200 revolutions in covering a distance of 44 km. Find the radius of the wheel.
A) 30 m
B) 35 m
C) 25 m
D) 40 m
E) 50 m

Answer:- B
Sol.
Distance covered in 1 revolution
= total distance/total revolution 
= 44 × 1000/200 = 220 m
Now, circumference ( perimeter) of the wheel,
2πr = 220
=> r = 220 × 1/2 × 7/22 
r = 35 m
Thus, the radius of the wheel= 35 m

Method 2:

By applying the logic that in 22/7, 7 should be a multiple of one of the given options. Here, 35 is the option.

8. Question

If the length of a rectangle is increased by 10% and the breadth of the rectangle is decreased by 5%, then find the percentage change in the area.

A) 4.5% increased
B) 5 % increased
C) 4% decreased
D) 5% decreased
E) None of these

Answer:-A
Sol.
Let, length =a and breadth = b
Required % change 
= (a + b + ab/100)%
= [10 – 5 – (10×5)/100] %
= 4.5 %
Thus, area increased by 4.5 %.

Method 2:

110/100×95/100

=1045/1000

=1045-1000/1000×100

=4.5%

9. Question

What percentage of its area will be increased when the Sides of a square are increased by 22%?
A) 24.80 %
B) 30.65 %
C) 35.60 %
D) 48.84 %
E) 50.74 %

Answer 😀
Sol:-
Required percentage 
= [ 22+22+ (22×22)/100]%
= (44 + 4.84)%
= 48.84 %

10. Question

The radius of a ring (in the form circle) is 28 cm. The ring is then moulded in the form of a square. Find the area of the square formed?

A.  2000 cm²

B.  1936 cm²

C.  2200 cm²

D.  1152 cm²

E.  1576 cm²

Answer: B

Explanation:

Radius of the ring (circle) (r) = 28 cm.

Length of the ring (Circumference/perimeter) = 2πr=2π(28)=176 cm

Let the side of the square be ‘a’ cm.

Perimeter of square (4a) = Circumference of the circle = 176 cm

or, 4a = 176 cm

or, a = 44 cm

Area of the square = a² = 44² = 1936 cm²

Thus, the area of the square is 1936 cm².

11. Question

The breadth of a rectangular park is 50% of its length. If the perimeter of the field is 900 m, then the area of the park is:

A.  65,000 m²
B.  62,000 m²
C.  45,000 m²
D.  68,500 m²
E. None of these

Answer: (C).
Explanation:
Let the length of the rectangular park be x m.
∴ breadth = 50% of x = x/2
Perimeter of rectangular park = 900
or, 2(length + breadth) = 900
or, 2(x+x/2)=900
or, 3x/2 = 450
or, x = 300
∴ Length = 300 m
Breadth= 300/2 = 150 m
∴ Area = length * Breadth = 300 × 150 = 45000 m²

12. Question

The diagonal of a square park is 20 m. What is the area of the park?
A.  200 m²
B.  500 m²
C.  450 m²
D.  60 m²

Answer: A
Explanation:
Area of square =1/2× (diagonal)² 
∴ Area=1/2 × 20 × 20 = 200 m²

13. Question

There is a rectangular ground in a school. If its perimeter and length is 120 m and 45 m respectively, find:
(i) it’s breadth
(ii) it’s area.

A) 15 m, 675 m²
B) 25 m, 575 m²
C) 15 m, 225 m²
D) 30 m, 300 m²
E) None of these

Answer :- A
Solution:
(i) Perimeter of the rectangular ground= 120 m
Now,
2(l + b) = 120 
l + b = 60 
45 + b = 60
b = 60 – 45 = 15 m
So, the breadth = 15 m.
(ii) Area of the rectangular ground = l × b = 45 m × 15 m = 675 m²
So, the required area = 675 m²

14. Question

88 cm long wire is first bent into a circle and then into a square. Which one will have more area?

A) circle > square

B) Square > circle

C) Square = circle

D) none of these

Answer :- A

Solution:
Length of the wire = 88 cm
Side of the square = 88 ÷ 4 cm = 22 cm
Area of the square = (Side)2 = (22)² cm² = 484 cm²
Circumference of the circle = 88 cm
2πr = 88
r = 88× 1/2×7/22 = 14 cm
Area of the circle = πr² = 22/7 × 14² = 616 cm²
Since, 616 cm² > 484 cm²

So, the circle have more area.