Mensuration
- The area of the square inscribed in a circle of radius 8 cm is
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Using Rule 10,
BD = Diagonal = 16 cm [∵ Radius = 8 cm]Area of square = 1 × BD² 2 Area of square = 1 × 16 × 16 = 128sq.cm. 2 Correct Option: C
Using Rule 10,
BD = Diagonal = 16 cm [∵ Radius = 8 cm]Area of square = 1 × BD² 2 Area of square = 1 × 16 × 16 = 128sq.cm. 2
- A circle is inscribed in an equilateral triangle and a square is inscribed in that circle. The ratio of the areas of the triangle and the square is
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Let AB = BC = CA = x units, thenAD = √ x² - x² = √3x 4 2 OD = 1 AD = x = radius of circle 3 2√3 ⇒ Diagonal of square = 2 × x = x 2√3 √3
∴ Triangle : Square= √3 x² : x² 4 2 × 3 = √3 : 1 = 3√3 : 2 2 3 Correct Option: C
Let AB = BC = CA = x units, thenAD = √ x² - x² = √3x 4 2 OD = 1 AD = x = radius of circle 3 2√3 ⇒ Diagonal of square = 2 × x = x 2√3 √3
∴ Triangle : Square= √3 x² : x² 4 2 × 3 = √3 : 1 = 3√3 : 2 2 3
- The ratio of the area of an equilateral triangle and that of its circumcircle is
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Let AB = BC = AC = 2x units
AD = √4x² - x² = √3xOD = 1 × √3x = x 3 √3 ∴ OB = √x² + x² 3 = 2x² = Circum radius √3 = √3 × 4x² : π × 4x² 4 3
= 3√3 : 4πCorrect Option: C
Let AB = BC = AC = 2x units
AD = √4x² - x² = √3xOD = 1 × √3x = x 3 √3 ∴ OB = √x² + x² 3 = 2x² = Circum radius √3 = √3 × 4x² : π × 4x² 4 3
= 3√3 : 4π
- Between a square of perimeter 44 cm and a circle of circumference 44 cm, which figure has larger area and by how much?
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Using Rule 10,
Area of a square = (side)²= 
Perimeter 
² = 44 ² 4 4
= (11)² = 121 cm²
Area of a cricle = π(radius)²= π Circumference ² = (Circumference)² 2π 4π = 44 × 44 4 × 22 7
= 22 × 7 = 154 cm²
Area of circle – Area of square
= 154 – 121 = 33cm²
∴ Area of the circle is larger than the area of the square by 33 cm².Correct Option: B
Using Rule 10,
Area of a square = (side)²= Perimeter ² = 44 ² 4 4
= (11)² = 121 cm²
Area of a cricle = π(radius)²= π Circumference ² = (Circumference)² 2π 4π = 44 × 44 4 × 22 7
= 22 × 7 = 154 cm²
Area of circle – Area of square
= 154 – 121 = 33cm²
∴ Area of the circle is larger than the area of the square by 33 cm².
- The perimeter of a square and a circular field are the same. If the area of the circular field is 3850 sq metres,what is the area (in m²) of the square?
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Using Rule 14,
Area of circular field = πr² = 3850 sq.m.
⇒ πr² = 3850⇒ r² = 3850 × 7 22
⇒ r² = 1225
∴ r = √1225 = 35m
Now, circumference of circle = 2πr= 2 × 22 × 35 = 44 × 5 = 220 m 7
∴ According to question,
Perimeter of Square = Circumference of circle
Let side of square be x m. then, 4x = 220m
⇒ x = 55m
∴ Area of square = x² = 55 × 55 m² = 3025 m²Correct Option: B
Using Rule 14,
Area of circular field = πr² = 3850 sq.m.
⇒ πr² = 3850⇒ r² = 3850 × 7 22
⇒ r² = 1225
∴ r = √1225 = 35m
Now, circumference of circle = 2πr= 2 × 22 × 35 = 44 × 5 = 220 m 7
∴ According to question,
Perimeter of Square = Circumference of circle
Let side of square be x m. then, 4x = 220m
⇒ x = 55m
∴ Area of square = x² = 55 × 55 m² = 3025 m²
