What is the area of a regular hexagon if its perimeter is 60cm?

Question

your help is greatly appreciated!!

Answer 1

Someone has probably written out the formula somewhere out there on the internet; but I'm too lazy to look and the answer isn't that hard to derive, so I'll run through it here.

Hexagon has 6 sides. So each side will be 60 / 6 = 10 cm. If you draw lines from each vertex to the one directly opposite from it, you will divide the regular hexagon into 6 identical triangles. If we can figure out the area of each triangle, we can just multiply that be 6 and get the area for the whole hexagon. What's the area of a triangle? It's the 1/2 * base * height. You know the base (it's 10 cm). How do we find the height? First what kind of triangles are these? We know that the triangle is at least isosceles, since two of its sides are equal. What are its angles?

All the angles inside a hexagon add up to 720 degrees. The general rule for all convex polygons is:

the total number of degrees = (the number of sides - 2) * 180

So for a hexagon:

(6-2) * 180 = 720 degrees

Since it is a regular hexagon, all six of the vertices has the same angle:

720 / 6 = 120 degrees

When you divided the hexagon into triangles, you bisected each of the vertices, meaning each vertex of neighboring triangles got half of the 120 degrees--or 60 degrees. That means this isn't just an isosceles triangle, it is equilateral. You can determine the height of the triangle using either Pythagoras's theorem or using a trigonometric function (cosine or tangent) and you should get:

5 * sqrt(3)

where sqrt is "square root".

Now the area of each triangle is:

1/2 * 10 * 5 * sqrt(3) = 25 * sqrt(3)

The area of the hexagon is 6 times this, or

6 * 25 * sqrt(3) = 150 * sqrt(3) = 260

Answer 2

for a nicely-known hexagon P = 60, it follows the the section length = 10 formula : section = (n/2) s^2 tan (360/n) the place s is the section length and n is the kind of sides section = 300sqrt(3)

Answer 3

if it's a regular hexagon, that means that it has 6 sides, all of the same length. ... so how long is each side? Now figure out the area by making some triangles and quadrilaterals.

Answer 4

Side is 10cm .Apothem is =side*(sqrt3)/2 So

S = 60 *10*(sqrt3)/4 = 150sqrt3 cm^2