Question

find the length of the ARC and area and perimeter (π3.14) centre angle is 45 radius is 16 cm​

Answer 1

Answer:

The length of the arc is 12.56 cm.

The area of the sector is 100.48 cm².

The perimeter of the sector is 44.56 cm.

Step-by-step-explanation:

We have given that,

• Central angle of an arc $\displaystyle{\sf\:(\:\theta\:)\:=\:45^{\circ}}$
• Radius of circle ( r ) = 16 cm
• $\displaystyle{\sf\:\pi\:=\:3.14}$

We have to find the length of the arc, area and perimeter of the sector.

Now, we know that,

$\displaystyle{\pink{\sf\:Length\:of\:arc\:=\:\dfrac{\theta}{360}\:\times\:2\:\pi\:r}\sf\:\:\:-\:-\:-\:[\:Formula\:]}$

$\displaystyle{\implies\sf\:Length\:of\:arc\:=\:\cancel{\dfrac{45}{360}}\:\times\:2\:\times\:3.14\:\times\:16}$

$\displaystyle{\implies\sf\:Length\:of\:arc\:=\:\dfrac{5}{\cancel{4}0}\:\times\:\cancel{2}\:\times\:3.14\:\times\:16}$

$\displaystyle{\implies\sf\:Length\:of\:arc\:=\:\dfrac{5}{\cancel{20}}\:\times\:3.14\:\times\:\cancel{16}}$

$\displaystyle{\implies\sf\:Length\:of\:arc\:=\:\cancel{\dfrac{5}{5}}\:\times\:3.14\:\times\:4}$

$\displaystyle{\implies\sf\:Length\:of\:arc\:=\:3.14\:\times\:4}$

$\displaystyle{\implies\boxed{\red{\sf\:Length\:of\:arc\:=\:12.56\:cm}}}$

Now, we know that,

$\displaystyle{\orange{\sf\:Area\:of\:sector\:=\:\dfrac{\theta}{360}\:\times\:\pi\:r^2}\sf\:\:\:-\:-\:-\:[\:Formula\:]}$

$\displaystyle{\implies\sf\:Area\:of\:sector\:=\:\cancel{\dfrac{45}{360}}\:\times\:3.14\:\times\:(\:16\:)^2}$

$\displaystyle{\implies\sf\:Area\:of\:sector\:=\:\dfrac{5}{\cancel{40}}\:\times\:3.14\:\times\:\cancel{16}\:\times\:16}$

$\displaystyle{\implies\sf\:Area\:of\:sector\:=\:\cancel{\dfrac{5}{10}}\:\times\:3.14\:\times\:4\:\times\:16}$

$\displaystyle{\implies\sf\:Area\:of\:sector\:=\:\dfrac{1}{2}\:\times\:4\:\times\:3.14\:\times\:16}$

$\displaystyle{\implies\sf\:Area\:of\:sector\:=\:\cancel{\dfrac{4}{2}}\:\times\:3.14\:\times\:16}$

$\displaystyle{\implies\sf\:Area\:of\:sector\:=\:2\:\times\:3.14\:\times\:16}$

$\displaystyle{\implies\sf\:Area\:of\:sector\:=\:6.28\:\times\:16}$

$\displaystyle{\implies\boxed{\blue{\sf\:Area\:of\:sector\:=\:100.48\:cm^2}}}$

Now, we know that,

$\displaystyle{\purple{\sf\:Perimeter\:of\:sector\:=\:2r\:+\:\dfrac{\theta}{360}\:\times\:2\:\pi\:r}\sf\:\:\:-\:-\:-\:[\:Formula\:]}$

$\displaystyle{\implies\sf\:Perimeter\:of\:sector\:=\:2\:\times\:16\:+\:\left(\:\dfrac{\theta}{360}\:\times\:2\:\pi\:r\:\right)}$

$\displaystyle{\implies\sf\:Perimeter\:of\:sector\:=\:2\:\times\:16\:+\:Length\:of\:arc}$

$\displaystyle{\implies\sf\:Perimeter\:of\:sector\:=\:2\:\times\:16\:+\:12.56}$

$\displaystyle{\implies\sf\:Perimeter\:of\:sector\:=\:32\:+\:12.56}$

$\displaystyle{\implies\boxed{\green{\sf\:Perimeter\:of\:sector\:=\:44.56\:cm}}}$

Answer 2

Answer:-

$\small \bf \underline{Given:}$

★★ Central angle of an arc (θ) = 45°

★★ Radius of circle (r) = 16cm

★★ Value of π = 3.14

$\small \bf \underline{To \: find:}$

We need to find

• Length of the arc
• Area and perimeter of the sector

$\small \bf \underline{Solution:}$

$\bf» \: length \: of \: the \: arc = \frac{θ}{360} \times 2\pi {r}$

$\bf = \frac{45}{360} \times 2 \times 3.14 \times 16$

$\bf = 2 \times 2\times 3.14$

$\bf = 12.56cm$

$\bf »\: area \: of \: the \: sector = \frac{θ}{360} \pi {r}^{2}$

$\bf = \frac{45}{360} \times 3.14 \times ( {16})^{2}$

$\bf = \frac{1}{8} \times 3.14 \times 16 \times 16$

$= \bf \: 3.14 \times 2 \times 16$

$\bf = 100.48c {m}^{2}$

» Perimeter = Length of arc + r + r

= 12.56 + 16 + 16

= 44.56cm