An umbrella has 8 ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius , find the area between the two consecutive ribs of the umbrella.

Understand the Question The umbrella is modeled as a flat circle divided into equal sectors by its equally spaced ribs. The area between two consecutive ribs corresponds to the area of one sector of this circle. Total angle of a circle is , so the central angle for each sector is (or equivalently, each sector is of the total circular area ). Given radius . Step 1 · Find the Central Angle for Each Sector Given equally spaced ribs in a circle (): Step 2 · Calculate the Area Between Two Consecutive Ribs Radius of the circle, . Answer > Common Mistakes Decimal Conversion: Unless asked, leave the answer as the exact fraction rather than rounding to decimals. Using Diameter instead of Radius: The radius is already given as , do not divide it by . Missing Sector Count: Forgetting that ribs divide the circle into sectors, so each sector represents .

Areas Related to Circles | Exercise 11.1

Question 10

An umbrella has 8 ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius $45\text{ cm}$ , find the area between the two consecutive ribs of the umbrella.

Solution

Understand the Question

The umbrella is modeled as a flat circle divided into $8$ equal sectors by its $8$ equally spaced ribs.
The area between two consecutive ribs corresponds to the area of one sector of this circle.
Total angle of a circle is $360^\circ$ , so the central angle for each sector is $\theta = \dfrac{360^\circ}{8} = 45^\circ$ (or equivalently, each sector is $\dfrac{1}{8}$ of the total circular area $\pi r^2$ ).
Given radius $r = 45\text{ cm}$ .

Step 1 · Find the Central Angle for Each Sector

Given $8$ equally spaced ribs in a circle ( $360^\circ$ ): Diagram 1

\begin{aligned} \text{Angle of each sector } (\theta) &= \dfrac{360^\circ}{8} \\[0.6em] &= 45^\circ \end{aligned}

Step 2 · Calculate the Area Between Two Consecutive Ribs

Radius of the circle, $r = 45\text{ cm}$ .

\begin{aligned} \text{Area of sector} &= \dfrac{\theta}{360^\circ} \times \pi r^2 \\[0.6em] &= \dfrac{45^\circ}{360^\circ} \times \dfrac{22}{7} \times (45\text{ cm})^2 \\[0.6em] &= \dfrac{1}{8} \times \dfrac{22}{7} \times 45 \times 45 \\[0.6em] &= \dfrac{1}{8} \times \dfrac{22}{7} \times 2025 \\[0.6em] &= \dfrac{22 \times 2025}{8 \times 7} \\[0.6em] &= \dfrac{44550}{56} \\[0.6em] &= \dfrac{22275}{28}\text{ cm}^2 \end{aligned}

Answer

$\dfrac{22275}{28}\text{ cm}^2$

Common Mistakes

Decimal Conversion: Unless asked, leave the answer as the exact fraction $\dfrac{22275}{28}\text{ cm}^2$ rather than rounding to decimals.
Using Diameter instead of Radius: The radius is already given as $45\text{ cm}$ , do not divide it by $2$ .
Missing Sector Count: Forgetting that $8$ ribs divide the circle into $8$ sectors, so each sector represents $\dfrac{1}{8}\pi r^2$ .

Question 10

More questions in Exercise 11.1