Question

NO LINKS!! URGENT HELP PLEASE!!

O is the center of the regular dodecagon below. Find its area. Round to the nearest tenth.

Answer 1

80.4 square units (nearest tenth)

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:

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

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

`5=(s)/(2 \tan \left((180^(\circ))/(12)\right))`

`s=10\tan \left(15^(\circ)\right)`

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

`\boxed{\begin{minipage}{6cm}\underline{Area of a regular polygon}\\\\$A=(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}}`

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

`A=(12 \cdot 10\tan \left(15^(\circ)\right) \cdot 5)/(2)`

`A=(600\tan \left(15^(\circ)\right))/(2)`

`A=300\tan \left(15^(\circ)\right)`

`A=80.3847577...`

`A=80.4\; \sf square\;units\;(nearest\;tenth)`

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

Answer 2

80.4 square units.

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:

`\boxed{\bold{Area =(1)/(2)* n * s * a}}`

where:

• n is the number of sides
• s is the length of one side
• a is the apothem

In this case, we have:

• n = 12
• s = ?
• a = 5

First, we need to find S.

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

`\boxed{\bold{s = 2 * a * tan((\pi)/(n))}}`

substituting value:

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

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

`\bold{Area = (1)/(2)* 12 * 2.679 * 5=80.37\: sqaure\: units}`

in nearest tenth 80.4 square units.

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