139k views
2 votes
NO LINKS!! URGENT HELP PLEASE!!

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

NO LINKS!! URGENT HELP PLEASE!! O is the center of the regular dodecagon below. Find-example-1
User Anil Sidhu
by
8.0k points

2 Answers

2 votes

Answer:

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.

User Jayron
by
8.6k points
2 votes

Answer:

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.

User Karen Payne
by
8.4k points

Related questions

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories