Question

O is the center of the regular dodecagon below. Find its area. Round to the nearest tenth if necessary. (Square units)

Answer 1

389.1 square units

Explanation:

Area of the Dodecagon (12-sided polygon)

Area Formula

• Area (A) = number of sides (n) x length of side (l) x apothem (a) x 1/2

Finding the length (l)

Apothem Formula

• a = l / 2tan(180/n)°
• 11 = l / 2 tan15°
• l = 11 × 2tan(45 - 30)°
• l = 22 x (2 - √3)
• l = 44 - 22√3 units

Finding Area

• A = 12 × (44 - 22√3) x 11 x 1/2
• A = 66 x (44 - 22√3)
• A = 389.1 square units (approx.)

Answer 2

389.1 units² (nearest tenth)

Explanation:

Regular polygon: all side lengths are equal, all interior angles are equal.

Apothem: a line drawn from the center of any polygon to the midpoint of one of the sides

Radius: a line drawn from the center of the polygon to a vertex.

Therefore, we have been given the apothem of this regular dodecagon.

Formulae

`\textsf{Area of a regular polygon}=(n\:l\:a)/(2)`

where:

• n = number of sides
• l = length of one side
• a = apothem (the line drawn from the center of any polygon to the midpoint of one of the sides)

`\textsf{Length of apothem (a)}=(l)/(2 \tan\left((180^(\circ))/(n)\right))`

where:

• l = length of one side
• n = number of sides

Solution

First, calculate the length of one side of the regular dodecagon by substituting a = 11 and n = 12 into the apothem formula:

`\implies 11=(l)/(2 \tan\left((180^(\circ))/(12)\right))`

`\implies l=11 \cdot 2 \tan\left((180^(\circ))/(12)\right)`

`\implies l=44-22√(3)`

Now substitute n = 12, the found value of l, and a = 11 into the area formula:

`\implies \textsf{Area}=(12(44-22√(3))(11))/(2)`

`\implies \textsf{Area}=389.0622274...`

`\implies \textsf{Area}=389.1\: \sf units^2 \: (nearest\:tenth)`