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Use a special right triangle to find the apothem and the side length of the regular polygon. Finally, use the formula A= 12ap to find its area.
162sqrt(3)m^2
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.
Let's do it!
By drawing the six radii, we can divide the hexagon into six isosceles triangles. Since the triangles are congruent and a full turn measures 360^(∘), the central angles of the isosceles triangles formed by the radii measure 3606=60^(∘).
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 30^(∘).
a/c* b = a* b/c
Commutative Property of Multiplication
sqrt(a)* sqrt(a)= a
Multiply
Calculate quotient
Consider the 30^(∘)-60^(∘)-90^(∘) triangle one more time.
In this type of triangle, the shorter leg is half the length of the hypotenuse. Shorter Leg: 6sqrt(3)/2=3sqrt(3)m As previously mentioned, the apothem bisects the side of the regular hexagon. Therefore, the length of the side of the given polygon is twice the length of the side of the above triangle.
Consequently, the side length of the regular hexagon is 6sqrt(3)m. Since this polygon has six congruent sides, to find its perimeter we will multiply the side length by 6. Perimeter: 6* 6sqrt(3)=36sqrt(3)m
a= 9, p= 36sqrt(3)
Multiply
1/b* a = a/b
Calculate quotient