NCERT Solutions Ganita Prakash Chapter 2 Figure it Out — Where are the Angles?
Book page 45 & 46 Updated on2026-09-05
Q1.
Angles in a clock: a. The hands of a clock make different angles at different times. At 1 o'clock, the angle between the hands is 30°. Why? b. What will be the angle at 2 o'clock? And at 4 o'clock? 6 o'clock? c. Explore other angles made by the hands of a clock.
Answer
a. The centre of the dial is one full turn of 360°, and the twelve hour marks divide it into 12 equal parts.
Angle between two neighbouring hour marks = 360° ÷ 12 = 30°
At 1 o'clock the hour hand is on 1 and the minute hand on 12 — exactly one part apart. So the angle is 30°.
At 1 o'clock the hands are one hour-part apart, so the angle is 30°.
b. Just multiply 30° by the number of hour-parts between the hands:
Time
Parts between the hands
Angle
Type
2 o'clock
2 × 30
60°
acute
4 o'clock
4 × 30
120°
obtuse
6 o'clock
6 × 30
180°
straight
At 6 o'clock the two hands lie in one straight line — a straight angle of 180°.
c. Other clock angles:
Time
Smaller angle
Reflex angle on the other side
3 o'clock
90° (right angle)
270°
5 o'clock
150°
210°
9 o'clock
90°
270°
10 o'clock
60°
300°
12 o'clock
0°
360°
Did you know? The minute hand moves 360° in one hour, that is 6° every minute. The hour hand is twelve times slower — it turns only 0.5° in a minute.
Q2.
The angle of a door: Is it possible to express the amount by which a door is opened using an angle? What will be the vertex of the angle and what will be the arms of the angle?
Answer
Yes. A door turns about its hinges, so an angle describes exactly how far it is open.
Vertex: the hinge — the line where the door is joined to the wall (the frame).
Arms: the edge of the door, and the wall (or the door frame) it is measured from.
Position of the door
Angle
Fully closed
0°
Slightly open
an acute angle
Open square to the wall
90°
Pushed flat against the wall
180°
Why the angle is the right measure: the door does not slide or stretch — it only turns about the hinge. And the amount of turn about a fixed point is precisely what an angle measures.
Q3.
Vidya is enjoying her time on the swing. She notices that the greater the angle with which she starts the swinging, the greater is the speed she achieves on her swing. But where is the angle? Are you able to see any angle?
Answer
Yes — the angle is at the top, where the ropes are tied.
Vertex: the point (or bar) from which the swing hangs.
Arm 1: the rope in its resting position, hanging straight down.
Arm 2: the rope at the moment Vidya is pulled back to start, or at the highest point she reaches.
Why the angle matters: a bigger starting angle means the swing is pulled back higher, so it falls through a longer path and picks up more speed at the bottom. The angle, not the length of the rope, decides how far back she starts.
Try This: hang a small stone from a thread to make a pendulum. Pull it aside by a small angle, then by a larger angle, and watch how much faster it swings the second time.
Q4.
Here is a toy with slanting slabs attached to its sides; the greater the angles or slopes of the slabs, the faster the balls roll. Can angles be used to describe the slopes of the slabs? What are the arms of each angle? Which arm is visible and which is not?
Answer
Yes, the slope of each slab is exactly an angle — the steeper the slab, the bigger the angle and the faster the ball rolls.
Vertex: the point where the slab meets the side of the toy.
Arm 1 (visible): the edge of the slanting slab itself.
Arm 2 (not visible): the horizontal line through that point — an imaginary level line, the direction the slab would take if it were flat.
Why one arm is invisible: a slope always has to be measured from something. We compare the slab with the horizontal, but nobody draws the horizontal on the toy — we imagine it. The same happens on a road: a "steep hill" means a large angle with the horizontal.
Tip: hold a ruler horizontally at the joint of the slab. The angle between the ruler and the slab is the slope you are talking about.
Q5.
Observe the images below where there is an insect and its rotated version. Can angles be used to describe the amount of rotation? How? What will be the arms of the angle and the vertex? (Hint: Observe the horizontal line touching the insects.)
Answer
Yes. The amount of rotation is exactly what an angle measures.
Vertex: the fixed point about which the insect has been turned (the centre of rotation).
Arm 1: the horizontal line touching the insect in its original position.
Arm 2: the corresponding line touching the insect in its rotated position.
Measure the angle between these two lines with a protractor — that number tells you how much the insect has been turned.
Why we need the hint line: the insect itself is a picture, not a ray. Drawing the same reference line (say, along its body or the surface it stands on) in both pictures turns the comparison into an ordinary angle between two rays.
Did you know? A quarter turn is 90°, a half turn 180° and a full turn 360° — the same language is used for turning a steering wheel, a screw or a dancer's spin.