Two Figures MCQ Quiz - Objective Question with Answer for Two Figures - Download Free PDF
Last updated on Jul 9, 2025
Latest Two Figures MCQ Objective Questions
Two Figures Question 1:
A circle of maximum possible area is inscribed in a square. The cost of painting the area inside the circle is ₹6/m², while the cost of painting the area outside the circle (but within the square) is ₹5/m². If the total cost of painting the entire square is ₹370.3, find the side length of the square.
Answer (Detailed Solution Below)
Two Figures Question 1 Detailed Solution
Let the side length of the square be s meters.
Area of square = s²
Radius of the inscribed circle = s/2
Area of the circle = π × (s/2)² = (π × s²) ÷ 4
Cost of painting:
Inside circle: ₹6/m²
Outside circle (within square): ₹5/m²
Total cost =
⇒ ₹6 × (π × s² ÷ 4) + ₹5 × (s² – π × s² ÷ 4)
⇒ ₹6 × (πs² ÷ 4) + ₹5 × [s² – (πs² ÷ 4)]
⇒ ₹6 × (πs² ÷ 4) + ₹5 × s² × (1 – π ÷ 4)
Given total cost = ₹370.3
Now substitute π ≈ 3.14:
⇒ 6 × (3.14 × s² ÷ 4) + 5 × s² × (1 – 3.14 ÷ 4) = 370.3
⇒ 6 × (0.785s²) + 5 × s² × (1 – 0.785)
⇒ 4.71s² + 5 × s² × 0.215 = 370.3
⇒ 4.71s² + 1.075s² = 370.3
⇒ 5.785s² = 370.3
⇒ s² = 370.3 ÷ 5.785 ≈ 64
⇒ s = √64 = 8 meters
Thus, the correct answer is 8m.
Two Figures Question 2:
The sum of area of square and area of rectangle filed is 4284. Side of square and breadth of rectangle is equal. Perimeter of rectangle is 204 m. Find the total expenditure to cut the grass of rectangular filed at rate of Rs. 2.5 per sq m?
Answer (Detailed Solution Below)
Two Figures Question 2 Detailed Solution
Given:
Sum of area of square and rectangle = 4284 sq m
Side of square = Breadth of rectangle = x
Perimeter of rectangle = 2(L + B) = 204
Formula used:
Area of square = x²
Area of rectangle = L × x
Calculations:
Total area = x² + Lx = 4284
L + x = 102 ⇒ L = 102 - x
x² + x(102 - x) = 4284
x² + 102x - x² = 4284 ⇒ 102x = 4284 ⇒ x = 42
⇒ L = 102 - 42 = 60
⇒ Area of rectangle = 60 × 42 = 2520 sq m
Cost = 2520 × 2.5 = ₹6300
∴ Total expenditure to cut the grass is ₹6300.
Two Figures Question 3:
The ratio of the perimeter of a rectangle to a square is 3:2, and the length and breadth of the rectangle are (y+5) cm and y cm respectively. The area of the square is P, and the area of the rectangle is 50 cm². Find the side length of a cube whose volume is Q, if P + Q = 89.
Answer (Detailed Solution Below)
Two Figures Question 3 Detailed Solution
Given:
Perimeter ratio (Rectangle : Square) = 3 : 2
Length of rectangle = (y + 5) cm
Breadth of rectangle = y cm
Area of rectangle = 50 cm²
Area of square = P
P + Q = 89
Formula used:
Area of rectangle = length × breadth
Perimeter of rectangle = 2 × (length + breadth)
Perimeter of square = 4 × side
Area of square = side2 = P
Volume of cube = side3 = Q
Calculations:
Area: (y + 5) × y = 50
⇒ y2 + 5y = 50
⇒ y2 + 5y - 50 = 0
⇒ y = 5 (solving quadratic)
⇒ Length = 10 cm, Breadth = 5 cm
Perimeter of rectangle = 2 × (10 + 5) = 30 cm
Let side of square = x, then perimeter = 4x
Ratio ⇒ 30 : 4x = 3 : 2
⇒ 30/4x = 3/2
⇒ 60 = 12x
⇒ x = 5
Area of square = x2 = 25 = P
P + Q = 89 ⇒ Q = 64
Volume of cube = 64 ⇒ side3 = 64
⇒ side = 4
∴ Side length of the cube is 4 cm.
Two Figures Question 4:
The perimeter of the rectangle is 120 cm and the ratio of the length to the breadth of the rectangle is 7:8. Area of square is 4 cm2 more than the area of rectangle. Find the side of square?
Answer (Detailed Solution Below)
Two Figures Question 4 Detailed Solution
Given:
Perimeter of the rectangle = 120 cm
Ratio of length to breadth = 7 : 8
Area of the square is 4 cm² more than the area of the rectangle.
Formula used:
Perimeter of rectangle = 2 × (Length + Breadth)
Area of rectangle = Length × Breadth
Area of square = Side²
Calculations:
Let the length = 7x and the breadth = 8x (from the ratio 7:8).
Perimeter = 2 × (Length + Breadth) = 2 × (7x + 8x) = 120
⇒ 2 × 15x = 120
⇒ 30x = 120
⇒ x = 4
Length = 7x = 7 × 4 = 28 cm
Breadth = 8x = 8 × 4 = 32 cm
Area of rectangle = Length × Breadth = 28 × 32 = 896 cm²
Let the side of the square be 's'.
Area of square = s²
We are given that the area of the square is 4 cm² more than the area of the rectangle:
s² = 896 + 4
s² = 900
⇒ s = √900
⇒ s = 30 cm
∴ The side of the square is 30 cm.
Two Figures Question 5:
The perimeter of a square is equal the perimeter of a rectangle. The perimeter of the square is 40 m. If its breadth is two-thirds of its length, then what is the area (in m²) of the rectangle?
Answer (Detailed Solution Below)
Two Figures Question 5 Detailed Solution
Given:
Perimeter of square = Perimeter of rectangle
Perimeter of square = 40 m
Breadth of rectangle = 2/3 × Length of rectangle
Formula used:
Perimeter of square = 4 × side
Perimeter of rectangle = 2 × (length + breadth)
Area of rectangle = length × breadth
Calculations:
Perimeter of square = 40 m
⇒ 4 × side = 40
⇒ side = 10 m
Perimeter of rectangle = Perimeter of square = 40 m
Let the length of the rectangle be 'L' and breadth be 'B'.
Given B = (2/3)L
Perimeter of rectangle = 2 × (L + B)
⇒ 40 = 2 × (L + (2/3)L)
⇒ 20 = L + (2/3)L
⇒ 20 = 5L / 3
⇒ L = 60 / 5
⇒ L = 12 m
Now, calculate the breadth (B):
B = (2/3) × L
⇒ B = (2/3) × 12
⇒ B = 8 m
Area of the rectangle = L × B
⇒ Area = 12 × 8 = 96 m²
∴ The area of the rectangle is 96 m².
Top Two Figures MCQ Objective Questions
A wire is bent to form a square of side 22 cm. If the wire is rebent to form a circle, then its radius will be:
Answer (Detailed Solution Below)
Two Figures Question 6 Detailed Solution
Download Solution PDFGiven:
The side of the square = 22 cm
Formula used:
The perimeter of the square = 4 × a (Where a = Side of the square)
The circumference of the circle = 2 × π × r (Where r = The radius of the circle)
Calculation:
Let us assume the radius of the circle be r
⇒ The perimeter of the square = 4 × 22 = 88 cm
⇒ The circumference of the circle = 2 × π × r
⇒ 88 = 2 × (22/7) × r
⇒
⇒ r = 14 cm
∴ The required result will be 14 cm.
How many spherical lead shots each of diameter 8.4 cm can be obtained from a rectangular solid of lead with dimension 88 cm, 63 cm, 42 cm,(take
Answer (Detailed Solution Below)
Two Figures Question 7 Detailed Solution
Download Solution PDFGiven:
Diameter of each lead shot = 8.4 cm
Dimension of the rectangular solid = 88 × 63 × 42 (cm)
Concept used:
1. Volume of a sphere =
2. Volume of a cuboid = Length × Breadth × Height
3. The collective volume of all lead shots obtained should be equal to the volume of the rectangular solid.
4. Diameter = Radius × 2
Calculation:
Let N number of shots can be obtained.
Radius of each lead shot = 8.4/2 = 4.2 cm
According to the concept,
N ×
⇒ N ×
⇒ N = 750
∴ 750 lead shots can be obtained.
A rhombus has one of its diagonal 65% of the other. A square is drawn using the longer diagonal as side. What will be the ratio of the area of the rhombus to that of the square?
Answer (Detailed Solution Below)
Two Figures Question 8 Detailed Solution
Download Solution PDFGiven:
A rhombus has one of its diagonal 65% of the other.
A square is drawn using the longer diagonal as a side.
Concept used:
Area of rhombus = ½(diagonals product)
Area of square = side × side
Calculations:
Let diagonal(larger) of rhombus be 100 cm
Let the diagonal(smaller) diagonal be 65 cm (65% of larger diagonal)
Area of Rhombus = ½(100 × 65) = 3250
Side of square = 100 cm (equal to larger diagonal)
Area of square = (100 × 100) = 10000
Ratio,
⇒ Rhombus : Square = 3250 : 10000
⇒ 13 : 40
∴ The correct choice is option 3.
The volume of a cuboid is twice that of a cube. If the dimensions of the cuboid are (8 m × 8 m ×16 m), the total surface area of the cube is:
Answer (Detailed Solution Below)
Two Figures Question 9 Detailed Solution
Download Solution PDFGiven:
The volume of a cuboid is twice that of a cube.
Height = 16 cm
Breadth = 8 cm
Length = 8 cm
Formula Used:
The volume of the cuboid = Length × Breadth × Height
Volume of cube= (edge)3
Calculation:
The volume of the cuboid = 8 × 8 × 16
= 1024
The volume of a cuboid = 2 × volume of a cube.
The volume of a cuboid = 2 × (edge)3
(edge)3 = 1024/2 = 512 m
edge = 8 m
The total surface area of the cube = 6 × 64
= 384 m 2
∴ The total surface area of the cube is 384 m 2.A cone of diameter 14 cm and height 24 cm is placed on top of a cube of side 14 cm. Find the surface area of the whole figure.
Answer (Detailed Solution Below)
Two Figures Question 10 Detailed Solution
Download Solution PDFGiven:
Cone diameter = 14 cm, Height = 24 cm
Cube of side = 14 cm
Formula used:
Curved surface area of cone = π × Radius × Slant height
Surface area of cube = 6 × side2
Slant height = √(height2 + radius2)
Area of circle = π × radius2
Calculation:
Slant height = √(height2 + radius2) = √(242 + 72) = 25
Curved surface area of cone = π × Radius × Slant height = (22/7) × 7 × 25 = 550 cm2
Surface area of cube = 6 × side2 = 6 × (14)2 = 1176 cm2
But some area covered by cone's base = π × radius2
⇒ (22/7) × 72 = 154 cm2
⇒ Total surface area = 550 + 1176 - 154 = 1,572 cm2
The sum of the length of the edges of a cube is equal to one eighth of the perimeter of a square. If the numerical value of the volume of the cube is equal to the numerical value of the area of the square, then the length of one edge of the cube is:
Answer (Detailed Solution Below)
Two Figures Question 11 Detailed Solution
Download Solution PDFLet the length of side of a cube and square be a and b units respectively
Now,
⇒ Sum of length of edges of a cube = (1/8) × perimeter of square
⇒ 12a = (1/8) × 4b
⇒ 24a = b
Also,
⇒ Volume of cube = area of square
⇒ a3 = b2
⇒ a3 = (24a)2
⇒ a = 576 unitsA circle of radius 21 cm is converted into a right angle triangle. If base and height of right angle triangle are in the ratio of 3 : 4, then what will be the hypotenuse of right angle triangle?
Answer (Detailed Solution Below)
Two Figures Question 12 Detailed Solution
Download Solution PDFGiven:
The radius of the circle = 21 cm
The ratio of base and height of the right-angled triangle formed = 3 : 4
Formulae Used:
In a right-angled triangle,
(Hypotenuse)2 = (Base)2 + (Height)2
The perimeter of the circle = 2πr, where r is the radius of the circle
Calculation:
Let the base and height of the given right angle triangle be 3x and 4x.
⇒ Hypotenuse = √{(3x)2 + (4x)2} = 5x
Radius of circle = r = 21 cm
According to the question,
The perimeter of circle = Perimeter of right angle triangle
⇒ 2πr = 3x + 4x + 5x
⇒ 2 × (22/7) × 21 = 12x
⇒ x = 11
∴ Hypotenuse of right-angle triangle = 5x = 5 × 11 = 55 cmFind the area of the circle whose circumference is equal to the perimeter of a square of side 11 cm.
Answer (Detailed Solution Below)
Two Figures Question 13 Detailed Solution
Download Solution PDFGiven:
Side of the square = 11 cm
Formula:
Perimeter of the square = 4a
Circumference of the circle = 2πr
Area of the circle = πr2
Calculation:
According to the question,
Circumference of the circle = Perimeter of the square
⇒ 2πr = 4a
⇒ 2πr = 4 × 11
⇒ 2 × (22 / 7) × r = 44
⇒ r = 7 cm
Area of the circle = πr2
⇒ (22 / 7) × 7 × 7 = 154 cm2
∴ Area of circle is 154 cm2.
Find the area of a square of the maximum size which can be inscribed in a right angle triangle of side 6 cm, 8 cm, 10 cm.
Answer (Detailed Solution Below)
Two Figures Question 14 Detailed Solution
Download Solution PDFGiven:
The side of the triangle is 6 cm, 8 cm, 10 cm.
Formula used:
Area of triangle = 1/2 × base × height
Area of square = side2
Calculation:
Let the side of the square be ‘a’
Area of Δ ABC = Area of Δ ADE + Area of Δ EFC + Area of square
⇒ 1/2 × 6 × 8 = 1/2 × a × (8 – a) + 1/2 × (6 – a) × a + a2
⇒ 24 = 7a – a2 + a2
⇒ a = 24/7
Area of square = side2 = a2
⇒ (24/7)2
⇒ 576/49
∴ area of square is 576/49 cm2A rectangular metal sheet is of length 24 cm and breadth 18 cm. From each of its corners a square of side x cm is cut off and an open box is made of the remaining sheet. If the volume of the box is 640 cubic cm, then what is the value of x ?
Answer (Detailed Solution Below)
Two Figures Question 15 Detailed Solution
Download Solution PDFGiven:
Length of a rectangular metal sheet = 24 cm
The breadth of a rectangular sheet = 18 cm
Formula used:
The volume of cuboid = lbh
Where, l = length, b = breadth and h = height
Calculation:
As shown in the above figures,
Length of box = (24 – 2x)
Breadth of box = (18 – 2x)
Height of box = x
The volume of the box = lbh
⇒ (24 – 2x)(18 – 2x)(x) = 640
From Option (3): If x = 4
⇒ 16 × 10 × 4 = 640
∴ The correct value of x is 4.