Question

NO LINKS!! URGENT HELP PLEASE!!

1. Find the area of a regular octagon. Each side is 12 m.

2. The perimeter of a regular polygon is 72 feet. An exterior angle of the polygon measures 40°. Find the length of each side.

3. If the perimeter of a regular pentagon is 50 in. Find the area. Show a drawing and work please.

Answer 1

Answer:

1)  695.3 m²

2)  8 ft

3)  172.0 in²

Step-by-step explanation:

Question 1

To find the area of a regular polygon, we can use the following formula:

$\boxed{\begin{minipage}{5.5cm}\underline{Area of a regular polygon}\\\\ A=\dfrac{s^2n}{4 \tan\left(\dfrac{180^{\circ}}{n}\right)} \\\\\\where:\\\phantom{ww} \bullet n is the number of sides.\\ \phantom{ww} \bullet s is the side length.\\\end{minipage}}$

Given the polygon is an octagon, n = 8.

Given each side measures 12 m, s = 12.

Substitute the values of n and s into the formula for area and solve for A:

$\implies A=\dfrac{(12)^2 \cdot 8}{4 \tan\left(\dfrac{180^{\circ}}{8}\right)}$

$\implies A=\dfrac{144 \cdot 8}{4 \tan\left(22.5^{\circ}\right)}$

$\implies A=\dfrac{1152}{4 \tan\left(22.5^{\circ}\right)}$

$\implies A=\dfrac{288}{\tan\left(22.5^{\circ}\right)}$

$\implies A=695.29350...$

Therefore, the area of a regular octagon with side length 12 m is 695.3 m² rounded to the nearest tenth.

$\hrulefill$

Question 2

The sum of an interior angle of a regular polygon and its corresponding exterior angle is always 180°.

If the exterior angle of a polygon measures 40°, then its interior angle measures 140°.

To determine the number of sides of the regular polygon given its interior angle, we can use this formula, where n is the number of sides:

$\boxed{\textsf{Interior angle of a regular polygon} = \dfrac{180^{\circ}(n-2)}{n}}$

Therefore:

$\implies 140^{\circ}=\dfrac{180^{\circ}(n-2)}{n}$

$\implies 140^{\circ}n=180^{\circ}n - 360^{\circ}$

$\implies 40^{\circ}n=360^{\circ}$

$\implies n=\dfrac{360^{\circ}}{40^{\circ}}$

$\implies n=9$

Therefore, the regular polygon has 9 sides.

To determine the length of each side, divide the given perimeter by the number of sides:

$\implies \sf Side\;length=\dfrac{Perimeter}{\textsf{ n }}$

$\implies \sf Side \;length=\dfrac{72}{9}$

$\implies \sf Side \;length=8\;ft$

Therefore, the length of each side of the regular polygon is 8 ft.

$\hrulefill$

Question 3

The area of a regular polygon can be calculated using the following formula:

$\boxed{\begin{minipage}{5.5cm}\underline{Area of a regular polygon}\\\\ A=\dfrac{s^2n}{4 \tan\left(\dfrac{180^{\circ}}{n}\right)} \\\\\\where:\\\phantom{ww} \bullet n is the number of sides.\\ \phantom{ww} \bullet s is the side length.\\\end{minipage}}$

A regular pentagon has 5 sides, so n = 5.

If its perimeter is 50 inches, then the length of one side is 10 inches, so s = 10.

Substitute the values of s and n into the formula and solve for A:

$\implies A=\dfrac{(10)^2 \cdot 5}{4 \tan\left(\dfrac{180^{\circ}}{5}\right)}$

$\implies A=\dfrac{100 \cdot 5}{4 \tan\left(36^{\circ}\right)}$

$\implies A=\dfrac{500}{4 \tan\left(36^{\circ}\right)}$

$\implies A=\dfrac{125}{\tan\left(36^{\circ}\right)}$

$\implies A=172.047740...$

Therefore, the area of a regular pentagon with perimeter 50 inches is 172.0 in² rounded to the nearest tenth.

Answer 2

Answer:

1.695.29 m^2

2.8 feet

3. 172.0477 in^2

Step-by-step explanation:

1. The area of a regular octagon can be found using the formula:

$\boxed{\bold{Area = 2a^2(1 + \sqrt{2})}}$

where a is the length of one side of the octagon.

In this case, a = 12 m, so the area is:

$\bold{Area = 2(12 m)^2(1 + \sqrt{2}) = 288m^2(1 + \sqrt2)=695.29 m^2}$

Therefore, the Area of a regular octagon is 695.29 m^2

2.

The formula for the exterior angle of a regular polygon is:

$\boxed{\bold{Exterior \:angle = \frac{360^o}{n}}}$

where n is the number of sides in the polygon.

In this case, the exterior angle is 40°, so we can set up the following equation:

$\bold{40^o=\frac{ 360^0 }{n}}$

$n=\frac{360}{40}=9$

Therefore, the polygon has n=9 sides.

Perimeter=72ft.

We have

$\boxed{\bold{Perimeter = n*s}}$

where n is the number of sides in the polygon and s is the length of one side.

Substituting Value.

72 feet = 9*s

$\bold{s =\frac{ 72 \:feet }{ 9}}$

s = 8 feet

Therefore, the length of each side of the polygon is 8 feet.

3.

Solution:

A regular pentagon has five sides of equal length. If the perimeter of the pentagon is 50 in, then each side has a length = $\bold{\frac{perimeter}{n}=\frac{50}{5 }= 10 in.}$

The area of a regular pentagon can be found using the following formula:

$\boxed{\bold{Area = \frac{1}{4}\sqrt{5(5+2\sqrt{5})} *s^2}}$

where s is the length of one side of the Pentagon.

In this case, s = 10 in, so the area is:

$\bold{Area= \frac{1}{4}\sqrt{5(5+2\sqrt{5})} *10^2=172.0477 in^2}$

Drawing: Attachment