find the area of a regular hexagon with an apothem of 9cm.give answer in simplified radical form?
the apothem of a regular polygon is the perpendicular distance from the centre to one of the sides.
A hexagon is made up 6 equilateral triangles, let's look at one of these
let its base be 2x, then its height is 9cm
and tan 60º = 9/x
x = 9/tan60, and the base of 2x would then be 18/tan60
the area of one triangle is
(1/2)(2x)(9)
= (1/2)(18/√3)(9)
= 81/√3
so the area of the hexagon is 6(81/√3)
= 486/√3
= 162√3 after rationalizing the denominator.
To find the area of a regular hexagon, you need to know its apothem. The formula to calculate the area of a regular hexagon is:
Area = (3 * √3 * apothem²) / 2
Given that the apothem is 9 cm, we can now substitute this value into the formula:
Area = (3 * √3 * 9²) / 2
Simplifying this expression:
Area = (3 * √3 * 81) / 2
Area = (243√3) / 2
Therefore, the area of the regular hexagon with an apothem of 9 cm is (243√3) / 2 square cm in simplified radical form.
To find the area of a regular hexagon, you can use the formula:
Area = (3 * √3 * s²) / 2
where s is the length of one side of the hexagon.
In this case, we are given the apothem, which is simply the distance between the center of the hexagon to any of its sides. The apothem, in relation to the side length, can be used to find the side length of the hexagon.
Since we don't have the side length directly, we can use the formula:
s = 2 * apothem * tan(π/6)
where tan(π/6) = √3/3.
Therefore, the side length can be calculated as follows:
s = 2 * 9 cm * (√3/3) = 6√3 cm
Now that we have the side length, we can substitute it in the formula to find the area:
Area = (3 * √3 * (6√3)²) / 2
Simplifying this expression:
Area = (3 * √3 * 36 * 3) / 2
= 324√3 cm²
Therefore, the area of the regular hexagon with an apothem of 9 cm is 324√3 square centimeters.