Chapter Test
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Use the formula A= 12aP to find the area of the regular polygon, where P is the perimeter and a is the apothem.
907.2in^2.
We will find the area of a dodecagon, which is a regular polygon with 12 sides, with a side length of 9 inches. The area of a regular polygon is half the product of its apothem a and perimeter P. Note that we are only given the side length.
We will first need to find the perimeter and the apothem of the polygon. Then we will use the formula A= 12aP to calculate the area.
The perimeter of the given polygon is 108 inches.
Let's now find the apothem. To do so, we will start by drawing the radii of the dodecagon. Be aware that the radii divide a regular dodecagon into 12 congruent isosceles triangles.
Since corresponding angles of congruent figures are congruent, the vertex angles of the isosceles triangles formed by the radii are congruent. Moreover, since a full turn measures 360^(∘), we can divide 360^(∘) by 12 to obtain their measures.
360/12=30
The vertex angles of the isosceles triangles measure 30^(∘) each.
The apothem bisects both the vertex angle of the isosceles triangle and the opposite side of the vertex, which is a side of the dodecagon. As a result, a 15^(∘)-75^(∘)-90^(∘) triangle is created. The length of its shorter leg is 9÷ 2=4.5ft.
In order to find the length of the longer leg, which is the apothem, we can use trigonometric ratios. Recall that the tangent of an angle is the length of the opposite side divided by the length of the adjacent side. We can use tan15^(∘) to calculate the apothem.
LHS * a=RHS* a
.LHS /tan(15^(∘)).=.RHS /tan(15^(∘)).
Use a calculator
Round to 1 decimal place(s)
Therefore, the length of the apothem is 16.8 inches, correct to the nearest tenth.
The area of the given regular polygon is approximately 907.2 square inches.