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

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:

$\boxed{\bold{Area =\frac{1}{2}* n * s * a}}$

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:

$\boxed{\bold{s = 2 * a * tan(\frac{\pi}{n})}}$

substituting value:

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

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

$\bold{Area = \frac{1}{2}* 12 * 2.679 * 5=80.37\: sqaure\: units}$

in nearest tenth 80.4 square units.

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

Answer 2

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:

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

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

$5=\dfrac{s}{2 \tan \left(\dfrac{180^{\circ}}{12}\right)}$

$s=10\tan \left(15^{\circ}\right)$

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

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

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

$A=\dfrac{12 \cdot 10\tan \left(15^{\circ}\right) \cdot 5}{2}$

$A=\dfrac{600\tan \left(15^{\circ}\right)}{2}$

$A=300\tan \left(15^{\circ}\right)$

$A=80.3847577...$

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

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