Question

Find the radius, central angle and perimeter of a sector whose arc length and area are
\[{\text{27}}{\text{.5 cm}}\] and $618.75{\text{ c}}{{\text{m}}^2}$ respectively.

Answer 1

Hint- Perimeter of a sector is the total length of the circumference of the circle subtended
within the angle\[\theta \]. The length of the arc of a circle is a part of the total circumference of the
circle given by$2\pi r$. In the given question, we have to find how much of the total circumference of
the circle is subtended by the sector. The formula used for the length of the arc of a circle of radius $r$
and subtending $\theta $ degrees at the centre of the circle is${P_{arc}} = 2\pi r \times \left(
{\dfrac{\theta }{{360}}} \right)$ .
The area covered by the arc having radius r and subtending an angle of $\theta $ degrees at the centre
of the circle is calculated by using the formula${A_{arc}} = \pi {r^2}\left( {\dfrac{\theta }{{360}}}
\right)$.
Complete step by step solution:
Substitute \[{\text{27}}{\text{.5 cm}}\]for the length of the arc in the formula ${P_{arc}} = 2\pi r \times
\left( {\dfrac{\theta }{{360}}} \right)$ and $618.75{\text{ c}}{{\text{m}}^2}$ for the area covered by
the arc in the formula ${A_{arc}} = \pi {r^2}\left( {\dfrac{\theta }{{360}}} \right)$.
$
27.5 = 2\pi r \times \left( {\dfrac{\theta }{{360}}} \right)..........{\text{(i)}} \\
618.75 = \pi {r^2}\left( {\dfrac{\theta }{{360}}} \right)..........(ii) \\
$
Divide equation (ii) by (i):
$
\dfrac{{27.5}}{{618.75}} = \dfrac{{2\pi r \times \left( {\dfrac{\theta }{{360}}} \right)}}{{\pi {r^2}\left(
{\dfrac{\theta }{{360}}} \right)}} \\
\dfrac{2}{{45}} = \dfrac{2}{r} \\
r = 45{\text{ cm}} \\
$
Hence, the radius of the arc is 45 cm.

Now, to calculate the central angle or the angle subtended by the arc at the centre of the circle,
substitute $r = 45{\text{ cm}}$ and ${P_{arc}} = 27.5{\text{ cm}}$ in the formula ${P_{arc}} = 2\pi r
\times \left( {\dfrac{\theta }{{360}}} \right)$.
\[
{P_{arc}} = 2\pi r \times \left( {\dfrac{\theta }{{360}}} \right) \\
27.5 = 2 \times \dfrac{{22}}{7} \times 45 \times \left( {\dfrac{\theta }{{360}}} \right) \\

\theta = \dfrac{{27.5 \times 7 \times 360}}{{2 \times 22 \times 45}} \\
= {35^0} \\
\]
Hence, the arc will subtend an angle of ${35^0}$ at the centre of the circle.
Now, to calculate the perimeter of the arc, add two of the radius arms to the length of the arc.
$
P = 27.5 + 2(45) \\
= 27.5 + 90 \\
= 117.5{\text{ cm}} \\
$
Hence, the perimeter of the arc is $117.5{\text{ cm}}$.
Note: Since the sector is just a part of the circle subtending an angle$\theta $ at the centre, first find out
by what factor of the full circle is covered by the sector using$\left( {\dfrac{\theta }{{360}}} \right)$.
The Addition of the two radii arms with the length of the arc results in the total perimeter of the sector
that is equivalent to${P_s} = 2\pi r \times \left( {\dfrac{\theta }{{360}}} \right) + 2r$. The formula can
be used to determine the area of any part of the circle (for all the sectors of a circle) depending on the
angle subtended in the centre.