# A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? [Take π = 22/7]

**Solution:**

To find the total length of the spiral, we will use the formula S_{n} = n/2 [2a + (n - 1) d], if they are in AP

The sum of the first n terms of an Arithmetic Progression is given by S_{n} = n/2 [2a + (n - 1) d].

Here, a is the first term, d is the common difference and n is the number of terms.

Semi - perimeter of circle, l = πr with the given radius is

l_{1} = π × (0.5 cm) = 0.5π cm

l_{2} =π × (1 cm) = π cm

l_{3} = π × (1.5 cm) = 1.5π cm

Therefore, l_{1}, l_{2}, l_{3}, ... the lengths of the semi-circles are in an A.P

So, AP is 0.5π, π, 1.5π, 2π, ...

a = 0.5π

d = π - 0.5π = 0.5π

We know that the sum of n terms of an A.P. is given by

S_{n} = n/2 [2a + (n - 1) d]

= 13/2 [2 × (0.5π) + (13 - 1)(0.5π) ]

= 13/2 [ π + 6π]

= 13/2 × 7π

= 13/2 × 7 × 22/7 (by taking π = 22/ 7)

= 143

Therefore, the length of such a spiral of thirteen consecutive semi-circles will be 143 cm.

**Video Solution:**

## A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, . . .as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? [Take π = 22/7]

### Class 10 Maths NCERT Solutions - Chapter 5 Exercise 5.3 Question 18:

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, . . .as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? [Take π = 22/7]

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm as shown in the figure, then the length of such a spiral of thirteen consecutive will be 143 cm.