Mensuration Questions and Answers

Answer: C

The perimeter of the rectangle is 60 cm.

\[ \begin{aligned} \text{trapezium area} &= \tfrac{1}{2}(11 + 25) \times 12 = 216 \\ 3k \times 2k &= 6k^{2} = 216 \Rightarrow k = 6 \end{aligned} \]

The rectangle's sides are 18 cm and 12 cm, so its perimeter is 2(18 + 12) = 60 cm. 40, 50 and 70 do not correspond to a 3 : 2 rectangle of area 216 cm².

Answer: A

The longer parallel side is 18 cm.

Let the shorter parallel side be a cm, so the longer is a + 6. The area of a trapezium is half the sum of the parallel sides times the height:

\[ \begin{aligned} \tfrac{1}{2}(a + a + 6) \times 7 &= 105 \\ 2a + 6 &= 30 \\ a &= 12 \end{aligned} \]

The sides are 12 cm and 18 cm, and the longer is 18 cm. Check: \(\frac{1}{2}(12 + 18) \times 7 = 105\). 12 is the shorter side, the trap for anyone who stops at a, and 15 is the average of the two.

Answer: A

The area is 660 cm².

First find AB from the perimeter. Since ∠A = ∠B = 90°, AD and BC are both perpendicular to AB, so AD ∥ BC and ABCD is a trapezium with height AB:

\[ \begin{aligned} AB &= 120 - (48 + 17 + 40) = 15 \\ \text{Area} &= \tfrac{1}{2}(AD + BC) \times AB \\ &= \tfrac{1}{2}(40 + 48) \times 15 \\ &= 44 \times 15 = 660 \end{aligned} \]

Check: the parallel sides differ by 8 and the height is 15, and \(\sqrt{8^2 + 15^2} = 17\), which is CD, so the figure is consistent. 720 is what you get by multiplying 48 by 15 as if it were a rectangle.

Answer: A

The perimeter is 80 cm.

The 30 cm side is one leg, so the other leg h comes from the area, and the hypotenuse from Pythagoras:

\[ \begin{aligned} \tfrac{1}{2} \times 30 \times h &= 240 \Rightarrow h = 16 \\ \text{hyp} &= \sqrt{30^2 + 16^2} = \sqrt{1156} = 34 \\ \text{Perimeter} &= 30 + 16 + 34 = 80 \end{aligned} \]

Forgetting the half in the area formula gives h = 8 and a different, non-integer hypotenuse, so none of the other options fits.

Answer: A

The volume is 2304 m³ and the surface area is 1120 m². For a cuboid with length 18 m, breadth 16 m and height 8 m, the volume is \[ 18\times 16\times 8 = 2304 \] and the total surface area is \[ 2(lb + bh + hl) = 2(288 + 128 + 144) = 1120 \] The other options give different volumes (3304, 2004 and 2314), which do not equal the product of the three edges, so they are wrong.

The three face areas 288, 128 and 144 each appear twice in the surface area, and 288 x 8 gives the volume.

Answer: B

The area of the triangle is 119 cm². The area of a triangle equals its inradius multiplied by its semi-perimeter. The perimeter is 68 cm, so the semi-perimeter is 34 cm, and \[ \text{Area} = r\times s = 3.5\times 34 = 119 \] The options 152, 160 and 135 do not equal the product of the inradius and the semi-perimeter, so only 119 fits.

Answer: D

For a square of side \(a\), the circumcircle passes through the vertices and has the diagonal as its diameter, and the incircle touches the sides and has the side as its diameter.

\[ R = \frac{a\sqrt{2}}{2},\qquad r = \frac{a}{2} \]

\[ \frac{\pi R^2}{\pi r^2} = \frac{a^2/2}{a^2/4} = 2 \]

So the areas are in the ratio 2 : 1. The ratio of the radii is \(\sqrt{2} : 1\), so the ratio of the areas is the square of that. The ratios 1 : 2 and 1 : 3 are not correct, and 3 : 1 is not either. The answer is 2 : 1, option D.

Answer: B

In one round the wheel covers one circumference, so the circumference is the distance divided by the number of rounds.

\[ 2\pi r = \frac{44000\ \text{m}}{2000} = 22\ \text{m} \]

\[ r = \frac{22}{2\pi} \] \[ r = \frac{22 \times 7}{2 \times 22} = 3.5\ \text{m} \]

Here \(\pi\) is taken as 22/7. The value of 4.4 m is a diameter-like figure that comes from forgetting the factor of 2, and 3 m and 2.2 m do not result. The answer is 3.5 m, option B.

Answer: C

The perimeter is 40 cm. The diagonals of a rhombus bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs 8 cm and 6 cm, half of each diagonal.

The other options are not four times 10 cm.

Check by the area: the area is half the product of the diagonals, 16 × 12 / 2 = 96 square cm, and the side is 10 cm, so the height on a side is 96 / 10 = 9.6 cm. This is consistent with a rhombus of side 10 cm, so the perimeter is 4 × 10 = 40 cm.

Answer: B

The length and breadth satisfy l x b = 432 and l - b = 6, so l = 24 cm and b = 18 cm, since 24 x 18 = 432. The perimeter is 2(24 + 18) = 84 cm, and this is the diameter of the sphere, so the radius is 42 cm.

\[ V = \tfrac{4}{3}\pi r^{3} \]

\[ V = \tfrac{4}{3}\times\tfrac{22}{7}\times 42^{3} \]

\[ V = 310464\ \mathrm{cm}^{3} \]

So the volume is 310464 cm cubed, option (B). The other values do not follow from the radius of 42 cm. The perimeter of a rectangle of sides 24 and 18 is 84, and the diameter of the sphere is 84, so the radius is 42, which is used in the formula.