Use the fact that the diagonal is two times the length of a radius. Also, use trigonometric ratios to find the apothem and length of each side.
Yes, it is possible. Explanation: See solution.
Practice makes perfect
Let's consider a regular hexagon such that the length of a diagonal passing through the center is 12 cm long. We will mark the center of the polygon and one apothem, and we will let 2x be the length of each side of the hexagon.
Since GA is a radius, its length is half the length of the diagonal.
GA = 6 cm
A regular hexagon can be divided into 6 equilateral triangles and so, each central angle has a measure of 60^(∘). Remember that the apothem GH bisects the central angle.
m∠ HGA = 60^(∘)/2 = 30^(∘)
Next, we will use the trigonometric ratios to find the values of x and a.
By using a calculator, we find the value of sin( 30) and cos( 30).
sin( 30) &= x/6
&⇒ &
1/2 = x/6
cos( 30) &= a/6
&⇒ &
sqrt(3)/2 = a/6
By solving the equations above, we get x = 3 and a = 3sqrt(3). Next, we find the perimeter of the hexagon.
P = 6* 2x = 6 * 2* 3
⇒
P = 36
Finally, we apply the formula to find the area of a regular polygon.
Notice that a regular hexagon can be divided into 6 equilateral triangles.
This implies that the radii of the hexagon have a length of z, and therefore, the diagonals have a length of 2 z.
2 z = 12 ⇒ z = 6
By noticing this, we find the side length faster. Then, we just need to find the apothem as we did before, and then we can solve for the area of the given hexagon.
