Respuesta :

semsee45

Answer:

389.1 units² (nearest tenth)

Step-by-step 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

[tex]\textsf{Area of a regular polygon}=\dfrac{n\:l\:a}{2}[/tex]

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)

[tex]\textsf{Length of apothem (a)}=\dfrac{l}{2 \tan\left(\frac{180^{\circ}}{n}\right)}[/tex]

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:

[tex]\implies 11=\dfrac{l}{2 \tan\left(\frac{180^{\circ}}{12}\right)}[/tex]

[tex]\implies l=11 \cdot 2 \tan\left(\frac{180^{\circ}}{12}\right)[/tex]

[tex]\implies l=44-22\sqrt{3}[/tex]

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

[tex]\implies \textsf{Area}=\dfrac{12(44-22\sqrt{3})(11)}{2}[/tex]

[tex]\implies \textsf{Area}=389.0622274...[/tex]

[tex]\implies \textsf{Area}=389.1\: \sf units^2 \: (nearest\:tenth)[/tex]

Answer:

389.1 square units

Step-by-step 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.)

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