The area of a regular hexagon is 580cm^2. Find the length of one of its sides?

Question

The question is a grade 11 geometry and trigonometry question. I know a hexagon has 6 sides. I know it canbe split up into 2 trapeziums (trapezoids in USA) or into a few triangles. I am not sure how to approach this question. They want working. Please help me. Thankyou. Could you please explain the processes etc to me as well. Thankyou.

Answer 1

Suppose the length of the side of a regular polygon of 'n' sides is 'a' units.
Then the formula for the area 'A' of the polygon is

A = 0.25*n*(a^2)*cot(180/n)

Here the polygon is a hexagon. So n = 6. The length of side i.e. the value of 'a' is unknown.

But given area, A = 580_________(1)

From the above formula,
A = 0.25*6*(a^2)*cot(180/6)
A = 1.5*(a^2)*cot(30)
A = 1.5*(a^2)*sqrt(3)
A = 1.5*(a^2)*1.732 = 2.598(a^2)______(2)

From (1) and (2)
2.598(a^2) = 580
a^2 = 580/2.598 = 223.248
a = sqrt(223.248) = 14.94

Thus the reqd. length of the side is 14.94 cm.

Answer 2

A hexagon can be composed into 6 equilateral triangles. The area of an equilateral triangle is ( (sqrt 3)/4)*s^2 where s=side, sqrt3 is square root of 3 and ^2 is squared.

So the area of a hexagon is 6 times that. Making the equation 6*( (sqrt 3)/4)*s^2 = Area.

Solve for S

Answer 3

The hexagon can be treated as 6 equilateral triangles. So, the total area of the hexagon is 6 times the area of a triangle, whose base is 'l' and whose height 'h' will be l.sin60. Find out l and the perimeter is 6 times l. I am leaving the maths to you. good luck.

Answer 4

You can use this formula :
Area of regular hexagon = [(3 * 3^1/2)/2] * s^2, where s represents the length of one side.

Your formula will become:
580 = [(3 * 3^1/2)/2] * s^2
580 = 2.59807621135 * s^2 simplify the constant
223.24210408694 = s^2 divide both sides
14.94128856849 = s take the square root

Therefore your perimeter will be 6 times that or
Perimeter = 6 * 14.94128856849 = 89.64773141094, approximately 90cm.

Good luck!

Answer 5

Divide hexagon into 6 equilateral triangles of side x cm.
Area of 1 triangle = 580/6 = 96.7cm²
Area of 1 triangle = (1/2)x.x.sin 60° = 0.433x²
0.433 x² = 96.7
x = √(96.7/ 0.433)
x = 14.9 cm

Answer 6

first find d measure of each interior angle of the regular haxagon using the formula:
[(n-1)180]/2, where n is the number of sides.

each interior angle measures 60 degrees
therefore each triangle formed by the diagonals of the hexagon is equilateral triangle. we now use the special triangle 30-60-90 triangle to get the area of a triangle and multiply it to six to find the area of the regular hexagon.

if one side of the triangle is x its height is x square root of 3.
therefore the area of each triangle is (x times x sqrt of 3)/2
multiplied by six = 3(x)(xsqrt of 3 ) = area of heaxagon.

3x^2 times sqrt of 3 = 580
(3x^2 times sqrt of 3 )/ 3 sqrt of 3= (580)/3 sqrt of 3
x^2 = 111.621052
x = 10.57 cm.
Therefore each side is 10.57 cm