NO LINKS!! URGENT HELP PLEASE!!
O is the center of the regular dodecagon below. Find its area. Round to the nearest tenth.

Answer:

80.4 square units.

Step-by-step explanation:

solution Given:

apothem(a)=5

no of side(n)= 12

Area(A)-?

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

[tex]\boxed{\bold{Area =\frac{1}{2}* n * s * a}}[/tex]

where:

In this case, we have:

First, we need to find S.

We can find the length of one side using the following formula:

[tex]\boxed{\bold{s = 2 * a * tan(\frac{\pi}{n})}}[/tex]

substituting value:

[tex]\bold{s = 2 * 5 * tan(\frac{\pi}{12})=2.679}[/tex]   here π is 180°

To find the area substituting value in the above area's formula:

[tex]\bold{Area = \frac{1}{2}* 12 * 2.679 * 5=80.37\: sqaure\: units}[/tex]

in nearest tenth 80.4 square units.

Therefore, the area of the regular polygon is 80.4 square units.

Answer:

80.4 square units (nearest tenth)

Step-by-step explanation:

The given diagram shows a regular dodecagon (12-sided polygon) with an apothem of 5 units.

The apothem of a regular polygon is the distance from the center of the polygon to the midpoint of one of its sides.

We can calculate the side length of a regular polygon given its apothem using the following formula:

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

Substitute n = 12 and a = 5 into the equation to create an expression for s:

[tex]5=\dfrac{s}{2 \tan \left(\dfrac{180^{\circ}}{12}\right)}[/tex]

[tex]s=10\tan \left(15^{\circ}\right)[/tex]

Now we can use the standard formula for an area of a regular polygon:

[tex]\boxed{\begin{minipage}{6cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{n\cdot s\cdot a}{2}$\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the length of one side.\\ \phantom{ww}$\bullet$ $a$ is the apothem.\\\end{minipage}}[/tex]

Substitute the found expression for s, n = 12 and a = 5 into the formula and solve for A:

[tex]A=\dfrac{12 \cdot 10\tan \left(15^{\circ}\right) \cdot 5}{2}[/tex]

[tex]A=\dfrac{600\tan \left(15^{\circ}\right)}{2}[/tex]

[tex]A=300\tan \left(15^{\circ}\right)[/tex]

[tex]A=80.3847577...[/tex]

[tex]A=80.4\; \sf square\;units\;(nearest\;tenth)[/tex]

Therefore, the area of a regular dodecagon with an apothem of 5 units is 80.4 square units, rounded to the nearest tenth.

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