Does anyone know how to find apothem on a hexagon?

Question

Can anyone help me find the apothem of a regular hexagon with side = 12 cm

I know that the formula is A=(1/2)(a)(perimeter)
A=.5*a*60

Can anyone help please?

Answer 1

There may be an easier way, but I can do it with some geometry and trig.

First of all, the perimeter of a regular hexagon with side 12 would be 72, not 60. Six sides.

In hexagon ABCDEF, draw line segments AD and CE. The apothem will equal half of line segment CE, because it's perpendicular to AD, which bisects the hexagon.

The measure of each angle of a regular hexagon is 180(n-2)/n with n=6 sides, or 120 degrees. Use the Law of Cosines to find CE:

CE²=CD²+DE²-2*CD*DE*cos(angle D)
CE² = 144+144-2*12*12*cos 120°
= 288-288(-0.5)
= 288+144 = 432
CE=√432=√(144*3)
=√144√3=12√3

The apothem is half that, or 6√3.

Here's the general formula: The apothem of a regular hexagon is ½s√3. I didn't know it myself until I worked it out just now, which means I taught myself something today. lol

Answer 2

As others have stated, the apothem is the segment perpendicular to a side to the center. With a regular hexagon, if you draw in all of the diagonals you will get six equilateral triangles. If you draw the apothem in one of these triangles you will have two, 30-60-90 triangles. You know that the length of the short leg is 6 cm since the side of your hexagon is 12 cm and the apothem bisects the side. Knowing this and that the longer leg of a 30-60-90 triangle is sqrt(3) [√3] times the length of the shorter leg. Therefore, the apothem (the longer leg) is 6√3.

Answer 3

A hexagon with side 12 cm can be formed out of 6 congruent equilateral triangle with side 12 cm. So the apothem of the hexagon is equal to the height of one of the triangles.

apothem = (1/2)b√3 = (1/2)(12)√3 = 6√3 cm

Answer 4

The apothem goes from the center of the figure down to one side - creating a right angle with that side.

(this works for all figures - not just a hexagon)

1) Draw a line from the center to 2 adjacent corners of the figure to create a triangle.

2) The angle from the center is 360 divided by the number of sides of the figure
(in our case, 360/6 = 60)

3) Each of the other two angles of the triangle are going to be
(angle in step 2) + 2x = 180
in our case... 60 + 2x = 180... x=60

4)The apothem would be the altitude of that triangle - dropped from the angle in the center of the figure. That apothem is also going to cut the side of the figure in half.

If y = the side measure of the regular figure, and a = apothem. You can use trig to find the apothem.

tan x = a/[(1/2)(y)]

The x comes from step 3 above.

in our case...

tan 60 = a/(1/2 * 12)
tan 60 = a/6
6 * tan 60 = a
10.39

Answer 5

Divide the hexagon into 6 equilateral triangles. One side of an equilateral triangle is 5\/3. The area of an equilateral triangle is \/3 a^2 all over 4 where 'a' is a side. \/3(5\/3)(5\/3) / 4 = 75\/3 all over 4. Multiply 75\/3 / 4 by 6 because there are 6 triangles. The answer is 225\/3 / 2

Answer 6

Taking the center point of the hexagon and conecting it with two consecutive vertices you´ll form an equilateral triangle.
The height of this triangle is the apothem

In an equilateral triangle height = side*(sqrt3)/2

A = 6*12*12(sqrt3)/4 = 216 sqrt3 cm