Mensuration Questions and Answers

Answer: C

Two perpendicular radii cut a circle into four equal parts, and each is a quadrant - a sector of \( 90^{\circ} \).

A semicircle is half the circle, a chord is a line segment, and a minor segment is the smaller region cut off by a chord rather than by radii.

Answer: A

Let the length and the breadth of the rectangle be 4x cm and 3x respectively.

(4x)(3x) = 6912

12x2 = 6912 

x2 = 576 = 4 * 144 = 22 * 122 (x > 0)

=> x = 2 * 12 = 24

Ratio of the breadth and the areas = 3x : 12x2 = 1 : 4x = 1: 96.

Answer: A

\[ 2\pi r = 82\pi \quad\Longrightarrow\quad 2r = 82 \]

\[ r = 41 \text{ cm} \]

The 82 cm on offer is the diameter, one step short of the answer.

Answer: A

\[ C = 2\pi r \]

A circumference is a length, so it must be proportional to \( r \) rather than \( r^{2} \) - which rules out the area formula \( \pi r^{2} \) on dimensional grounds alone.

Answer: D

Let the side of square be 2a.
So, the diameter of circle = 2a
Radius of circle = a

WKT, Area of square = 4a
Area of circle = π r²
Perimeter of circle = 2π r
Area of rectangle = l x b
Perimeter of rectangle = 2(l + b)
Given that difference between area of the square and circle is 262.5 cm².
Area of square - Area of circle = 262.5
4a² - (22/7)a² = 262.5
28a² - 22a² = 1837.5
6a² = 1837.5
a² = 306.25
a = 17.5 cm

Perimeter of circle = 2πa = 2 x (22/7) x 17.5 = 110 cm

Also given length of rectangle is 20% more than the breadth of rectangle.
Let the breadth of rectangle be X.
So, length of rectangle = (120/100)X = 1.2X
As per the question,
Perimeter of rectangle = Perimeter of circle
2(1.2X + X) = 110
2.4X + 2X = 110
4.4X = 110
X = 25 cm
Breadth of rectangle = 25 cm
Length of rectangle = 1.2(25) = 30 cm

Area of rectangle = 25 x 30 = 750 cm².

Answer: A

Correct answer: 220 sq. cm

\[\text{Curved surface area} = \pi r l = \frac{22}{7} \times 7 \times 10 = 220 \text{ sq. cm}\]

Adding the base, \(\pi r^2 = 154\) sq. cm, would give a total surface area of 374 sq. cm, so read carefully which is asked.

Answer: A

Same material means the same density, so weight is proportional to volume.

\[ V_{\text{solid}} = \frac{4}{3}\pi (3)^{3} = 36\pi \text{ cm}^{3} \;\longrightarrow\; 6 \text{ kg} \]

The hollow sphere has outer radius 9 cm and inner radius 6 cm:

\[ V_{\text{hollow}} = \frac{4}{3}\pi\left(9^{3} - 6^{3}\right) = \frac{4}{3}\pi (729 - 216) = \frac{4}{3}\pi (513) = 684\pi \text{ cm}^{3} \]

\[ \text{weight} = 6 \times \frac{684\pi}{36\pi} = 6 \times 19 = 114 \text{ kg} \]

The cavity must be subtracted; treating the object as a solid 18 cm sphere would give 162 kg.

Answer: D

Let the side of the square be a cm. Let the length and the breadth of the rectangle be l cm and b cm respectively. 

4a = 2(l + b) 

2a = l + b 

l . b = 480 

We cannot find ( l + b) only with the help of l . b. Therefore a cannot be found . 

Area of the square cannot be found.

Answer: B

\[ A = \pi r^{2} \]

An area must vary as the square of a length, which alone rules out \( \pi r^{2} + 2r \) as dimensionally inconsistent. The other two are real formulas from elsewhere: \( 4\pi r^{2} \) is a sphere's surface area and \( 2\pi r^{2} \) a hemisphere's curved surface.

Answer: A

Check the sides first: \( 10^{2} + 24^{2} = 100 + 576 = 676 = 26^{2} \), so this is a right-angled triangle with legs 10 cm and 24 cm.

\[ A = \frac{1}{2} \times 10 \times 24 = 120 \text{ cm}^{2} \]

Heron's formula agrees: with \( s = 30 \), \( \sqrt{30 \times 4 \times 6 \times 20} = \sqrt{14400} = 120 \). Spotting the 5–12–13 triple doubled saves that work.