Question

An umbrella has 8 ribs which are equally spaced. Assuming the umbrella to be a flat
circle of radius 42 cm, find the area between the two consecutive ribs of the umbrella.

Answer 1

Hint: In the solution, first we have to assume the angle made by the two consecutive ribs of the
umbrella (suppose $\theta $). Since there are total 8 number of equally spaced ribs in the
umbrella, so we need to find the angle made by 8 by equating total angle ribs with $360^\circ $.

Complete step by step solution:
From that we get the value of $\theta $. After that we have to find the area between the two
consecutive ribs of the umbrella for the angle $\theta $.
Complete step by step solution:
It is known that the total angle of a circle is $360^\circ $.
Given that the umbrella has 8 ribs which are equally spaced as shown in the figure.

Let $\theta $ be the angle between each ribs.
Thus total angle made by 8 ribs $ = 8\theta $
Since the complete circle of the umbrella contains the equally spaced 8 ribs.
Thus the total angle made by 8 ribs is $360^\circ $.
Therefore we can say that $8\theta = 360^\circ $
Now solving for the angle $\theta $ we get
$\begin{array}{l}8\theta = 360^\circ \\ \Rightarrow \theta = \dfrac{{360^\circ }}{8}\\
\Rightarrow \theta = 45^\circ \end{array}$

Therefore, the angle between two consecutive ribs $ = 45^\circ $
It is known that the area of a circle can be defined as $\pi {r^2}$, where $r$ is the radius the circle
and $\pi = \dfrac{{22}}{7}$.
From that we can calculate the area of an arc with angle $\theta $.
Area of an arc $ = \dfrac{\theta }{{360^\circ }} \times \pi {r^2}$

Now we have to find the area between the two consecutive ribs of the umbrella.
Given that the radius of the circle $ = 42\;{\rm{cm}}$
Substituting $45^\circ $ for $\theta $, 42 cm for $r$ and $\dfrac{{22}}{7}$ for $\pi $.
Area $ = \dfrac{{45^\circ }}{{360^\circ }} \times \pi {\left( {42} \right)^2}$
$\begin{array}{l} = \dfrac{{45^\circ }}{{360^\circ }} \times \dfrac{{22}}{7} \times
{42^2}\\ = \dfrac{1}{8} \times \dfrac{{22}}{7} \times 1764\\ =
693\;{\rm{c}}{{\rm{m}}^2}\end{array}$
Hence, the required area between two areas between the two consecutive ribs of the umbrella is
693 cm 2.

Note: Here we have to find the area between the two consecutive ribs of the umbrella. Since the
number of ribs and the length of each rib are given. From that we can determine the angle
between two consecutive ribs. Thus we can determine our required area.