Real-World Applications പ്രയോഗങ്ങൾ

Circles are everywhere: wheels, wells, tracks, tables, trees, clocks – even the Earth. Every lab below uses just three facts from this chapter. Move the sliders and watch the lengths and areas change.

1. Wheels: how far in one turn?

In one full turn a wheel rolls forward exactly its circumference, πd. So the number of turns needed for a journey is the distance divided by πd.

In the exam this looks like: "A bicycle wheel has diameter 70 cm. How many turns does it make to travel 1.1 km? (π = 22/7)" → circumference 220 cm, 110000 ÷ 220 = 500 turns.

2. Running track: why the starts are staggered

In a race round a bend, the outer runners run on bigger circles. Each lane is wider by w metres, so its radius is w more and one full circle is 2πw longer – whatever the inner radius.

In the exam this looks like: "Two concentric circular tracks are 1.5 m apart. How much longer is one round of the outer track?" → 2π × 1.5 = 3π ≈ 9.42 m.

3. A well and its parapet

A round well (കിണർ) is often surrounded by a wall or a cemented ring. The ring's area is the big circle minus the small one: π(R² − r²).

In the exam this looks like: "A well of radius 1 m has a 0.5 m wide cemented ring around it. Find the area of the ring." → π(1.5² − 1²) = 1.25π ≈ 3.93 sq m.

4. A tablecloth for a round table

A round cloth that hangs down d cm on every side has radius (table radius + d). Its area tells how much cloth is needed; its circumference tells how much lace goes around the edge.

In the exam this looks like: "A round table has diameter 1 m. A cloth hangs 20 cm below its edge all round. Find the length of lace for its border." → radius 70 cm, 2π × 70 = 140π ≈ 439.8 cm.

5. Pizza: one big or two small?

Area grows with the square of the radius. A 12-inch pizza is 1½ times as wide as an 8-inch one, so it has (1½)² = 2¼ times the area – more than two 8-inch pizzas.

In the exam this looks like: "The radius of one circle is three times that of another. How many times is its area?" → 3² = 9 times.

6. A path around a circular garden

A path of width w around a garden of radius r is a ring from r to r + w. Its outer edge needs a longer fence than its inner edge by 2πw.

In the exam this looks like: "A circular garden of radius 10 m has a 1 m wide path around it. Find the area of the path." → π(11² − 10²) = 21π ≈ 65.97 sq m.

7. The tip of a clock hand

The tip of the minute hand moves along a circle whose radius is the hand's length. In 60 minutes it goes all the way round (2πL); in m minutes it goes m/60 of that.

In the exam this looks like: "The minute hand of a clock is 7 cm long. How far does its tip move in 30 minutes? (π = 22/7)" → ½ × 2 × 22/7 × 7 = 22 cm.

8. A rope around the Earth

Wrap a rope tightly round the Earth's equator (radius about 6371 km). Now lift it 1 metre off the ground all the way round. How much longer must the rope be? Surprisingly, only 2π × 1 ≈ 6.28 m – the same as for a football! This is practice set 2, question 5 (page 8).

In the exam this looks like: "The radius of one circle is 1 cm more than that of another. How much longer is its circumference?" → 2π cm, whatever the radii.

9. Measuring a tree by its girth

Foresters can't measure a tree's diameter directly, but they can wrap a tape round the trunk. Diameter = girth ÷ π.

In the exam this looks like: "The circumference of a circle is 44 cm. Find its radius and area. (π = 22/7)" → r = 7 cm, area 154 sq cm.

10. Madhavan and the Kerala school

In the 14th century, Madhavan of Sangamagrama found that 1 − 1/3 + 1/5 − 1/7 + … comes closer and closer to π/4. Multiply the sums by 4 and watch them close in on π – slowly! Madhavan himself used faster methods and found π ≈ 3.14159265359.

In the exam this looks like: "Name the Kerala mathematician who found a way to calculate π to any accuracy." → Madhavan of Sangamagrama.

11. Try these word problems

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