Question

The hour hand of a clock is 6 cm long. Find the area swept by it between 11:20 and 11:35 am (in cm2).

Answer 1

The hour hand of a clock completes one circle rotation in 12 hours.

So, the hour hand completes 1/12 part of the circle rotation in 1 hour.

Since the circle has a measurement of 2π radian, the hour hand completes 2π / 12 = π / 6 radian in 1 hour.

Now,

Measurement of angle completed by the hour hand in 1 hour = π/6 radian

⇒  Measurement of angle completed by the hour hand in 60 minutes = π/6 radian

⇒  Measurement of angle completed by the hour hand in 1 minute = π/6 × 1/60 = π / 360  radian

We're given to find the area enclosed by the hour hand between 11:20 am and 11:35 am.

The time difference between 11:20 am and 11:35 am is 15 minutes. Hence,

⇒  Measurement of angle completed by the hour hand in 15 minutes = 15π / 360 = π/24  radian

This  π/24  radian is the central angle of the sector swept by the hour hand.

Now we're going to find the area of the sector swept by the hour hand. Given that the length of the hour hand is 6 cm. This is the radius of the sector.

We know that area of the sector is  πr² · x/2π = r²x/2,  where 'r' is the radius and 'x' is the central angle of the sector in radian (If 'x' was in degrees, we would use  πr² · x/360°).

Here,

r = 6 cm

x = π/24 radian

So, area =

(6² · π/24) / 2  =  36π / 48  =  3π / 4  cm²

Hence the answer is  3π / 4  cm²,  and approximately we say,

3 · 3.14 / 4 = 2.36 cm²