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Use a right triangle to find the apothem and the side length of the regular polygon. Finally, use the formula A= 12ap to find its area.
181.0square units
The area of a regular polygon is half the product of the apothem and the perimeter. We will first find the apothem and then the side length to obtain the perimeter. Finally, we will use this information to find the area.
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By drawing eight radii, we can divide the octagon into eight isosceles triangles. Since the triangles are congruent and a full turn measures 360^(∘), the central angles of the isosceles triangles measure 360^(∘)8=45^(∘).
Remember that an apothem bisects the central angle and the side of the regular polygon. Therefore, we obtain a right triangle with an acute angle that measures 22.5^(∘).
In order to find the length of the longer leg, which is the apothem, we can use trigonometric ratios. Recall that the cosine of an angle is the length of the adjacent side divided by the hypotenuse. We can use cos22.5^(∘) to find the apothem.
LHS * 8=RHS* 8
Use a calculator
Rearrange equation
Round to 1 decimal place(s)
The apothem of the regular octagon is about 7.4 units.
Consider the right triangle one more time.
Recalling that the sine of an angle is the length of the opposite side divided by the hypotenuse, we can use sin22.5^(∘) to find the length of the shorter leg, ZY.
LHS * 8=RHS* 8
Use a calculator
Rearrange equation
Round to 1 decimal place(s)
Since this polygon has eight congruent sides, to find its perimeter we will multiply the side length by 8.
ZY= 3.061467...
Multiply
Round to 1 decimal place(s)
a= 7.391036..., p= 48.983479...
Multiply
1/b* a = a/b
Calculate quotient
Round to 1 decimal place(s)
The area of the octagon is about 181.0 square units.