Question

50 POINTS NEED HELP ASAP ANSWER ALL

Round to nearest hundredth 0.00
Apothem of the polygon=_ m
Perimeter= __M
Use the perimeter and apothem you
calculated
Area of the polygon=__m²

Answer 1

Answer:

Apothem of the hexagon = 24.25m

Perimeter of the polygon = 168m

Area of the polygon = 2,036.89m

Step-by-step explanation:

$apothem \: of \: a \: polygon\: = \frac{s}{2 \tan( \frac{180}{n} ) }$

where s = length of the side and

n = number of sides.

Apothem of the polygon =

$\frac{28}{2 \tan( \frac{180}{6} )} = \frac{28}{2 \tan(30) } = \frac{14}{ \tan(30) } = \frac{14}{ \frac{1}{ \sqrt{3} } }$

Apothem of the hexagon =

$14 \sqrt{3} \: m$ = 24.25m

Perimeter of the polygon = 6 × 28 = 168m.

Area of the hexagon =

$\frac{1}{2} \times apothem \times perimeter$

$= \frac{1}{2} \times 14 \sqrt{3 } \times 168 = 84 \times 14 \sqrt{3} = 1176 \sqrt{3} = 2036.89 \: {m}^{2}$

Area of the polygon = 2,036. 89m

Answer 2

Answer:

Apothem of the polygon = 24.25 m

Perimeter = 168 m

Area of the polygon = 2,036.89 m²

Step-by-step explanation:

Apothem

The apothem of a regular polygon is the distance from the center of the polygon to the midpoint of any side.

The formula to find the apothem of a regular polygon is:

$\boxed{\begin{array}{l}\underline{\textsf{Apothem of a regular polygon}}\\\\a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{ s is the side length.}\\ \phantom{ww}\bullet\;\textsf{ n is the number of sides.}\end{array}}$

In this case:

• s = 28 m
• n = 6

Substitute the values into the formula and solve for a:

$a=\dfrac{28}{2 \tan\left(\dfrac{180^{\circ}}{6}\right)}\\\\\\a=\dfrac{28}{2 \tan\left(30^{\circ}\right)}\\\\\\a=\dfrac{14}{\tan\left(30^{\circ}\right)}\\\\\\a=\dfrac{14}{\dfrac{1}{\sqrt{3}}}\\\\\\a=14\sqrt{3}\\\\\\a=24.25\; \sf m\;(nearest\;hundredth)$

Therefore, the apothem of the given polygon rounded to the nearest hundredth is 24.25 m.

$\dotfill$

Perimeter

The perimeter of a geometric shape is the sum of the lengths of all its sides.

The perimeter of a regular polygon can be calculated by multiplying the length of one side (s) by the number of sides (n):

$\textsf{Perimeter}=s \cdot n\\\\\textsf{Perimeter}=28 \cdot 6\\\\\textsf{Perimeter}=168\; \sf m$

Therefore, the perimeter of the given polygon is 168 m.

$\dotfill$

Area

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

$\boxed{\begin{array}{l}\underline{\textsf{Area of a regular polygon}}\\\\A=\dfrac{p\cdot a}{2}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{ p is the perimeter.}\\ \phantom{ww}\bullet\;\textsf{ a is the apothem.}\end{array}}$

In this case:

• p = 168 m
• a = 14√3 m

Substitute the values into the formula and solve for area:

$A=\dfrac{168 \cdot 14\sqrt{3}}{2}\\\\\\A=84\cdot 14\sqrt{3}\\\\\\A=1176\sqrt{3}\\\\\\A=2036.89\; \sf m\;(nearest\;hundredth)$

Therefore, the area of the given polygon rounded to the nearest hundredth is 2,036.89 m.