Mensuration Questions and Answers

Answer: B

The radian is the unit of angle defined by the circle itself: the angle subtended at the centre by an arc equal in length to the radius.

\[ 2\pi \text{ radians} = 360^{\circ}, \qquad 1 \text{ radian} \approx 57.3^{\circ} \]

Unlike the degree, it is a pure ratio of two lengths, which is why formulas such as \( s = r\theta \) hold only in radians.

Answer: D

Convert to degrees first.

\[ \frac{2\pi}{3} = \frac{2 \times 180^{\circ}}{3} = 120^{\circ} \]

\[ \sin 120^{\circ} = \sin(180^{\circ} - 120^{\circ}) = \sin 60^{\circ} = \frac{\sqrt{3}}{2} = 0.866 \]

Sine never exceeds 1, so any option above it can be discarded immediately.

Answer: D

\[ \pi \text{ radians} = 180^{\circ} \]

\[ 105^{\circ} = 105 \times \frac{\pi}{180} = \frac{7\pi}{12} \]

\[ = 1.832 \text{ rad} \]

Since a radian is about \( 57.3^{\circ} \), an angle of \( 105^{\circ} \) should come out a little under 2 radians.

Answer: A

A sector of \( 90^{\circ} \) is one quarter of the circle, and is called a quadrant.

Its area is \( \tfrac{1}{4}\pi r^{2} \) and its arc \( \tfrac{1}{2}\pi r \). Two perpendicular diameters divide a circle into four such quadrants.

Answer: A

\[ A = \frac{\theta}{360} \times \pi r^{2} \]

\[ 140 = \frac{\theta}{360} \times \pi \times 256 \]

\[ \theta = \frac{140 \times 360}{256\pi} = \frac{50400}{804.25} \]

\[ \theta = 62.66^{\circ} \]

The whole circle covers 804.2 cm², and 140 is a little over a sixth of it - consistent with an angle a little over \( 60^{\circ} \).

Answer: A

\[ \pi \text{ radians} = 180^{\circ} \]

\[ 3.5 \text{ rad} = 3.5 \times \frac{180}{\pi} = 3.5 \times 57.2958 \]

\[ = 200.54^{\circ} \]

Since \( \pi \approx 3.14 \) radians is \( 180^{\circ} \), 3.5 radians must be somewhat more than a straight angle - which rules out every option below \( 180^{\circ} \).

Answer: B

Angles in opposite segments of a circle - equivalently, the opposite angles of a cyclic quadrilateral - add to \( 180^{\circ} \), so they are supplementary.

\[ \angle A + \angle C = 180^{\circ} \]

This follows from the inscribed-angle theorem: each angle is half the arc it faces, and the two arcs together make the whole \( 360^{\circ} \).

Complementary would mean adding to \( 90^{\circ} \), a different relation altogether.

Answer: A

A sector whose central angle is \( 180^{\circ} \) covers half the circle, bounded by a diameter and the arc - a semicircle.

Its area is \( \tfrac{1}{2}\pi r^{2} \) and its perimeter \( \pi r + 2r \), the arc plus the diameter.

Answer: C

Convert the radian measure to degrees.

\[ 1.9 \text{ rad} = 1.9 \times 57.2958 = 108.86^{\circ} \]

\[ \sin 108.86^{\circ} = \sin(180^{\circ} - 108.86^{\circ}) = \sin 71.14^{\circ} = 0.946 \]

The angle is in the second quadrant, where the sine is positive and still close to its maximum - so a value near 1 is expected.

Answer: A

The cone is drilled out of the cylinder, so three surfaces remain exposed: the cylinder's curved side, its base, and the cone's curved inner surface. The cylinder's top face is gone, replaced by the cavity's opening.

With \( r = 8 \) cm and \( h = 15 \) cm, the cone's slant height is

\[ l = \sqrt{r^{2} + h^{2}} = \sqrt{64 + 225} = \sqrt{289} = 17 \text{ cm} \]

\[ \text{TSA} = 2\pi r h + \pi r^{2} + \pi r l \]

\[ = 2\pi(8)(15) + \pi(64) + \pi(8)(17) \]

\[ = 240\pi + 64\pi + 136\pi = 440\pi \text{ cm}^{2} \]

The cavity's wall is exposed and must be counted, while the flat top is not - the drilling removes it entirely.