I need help with two geometry questions! Can someone help me?

Question

Ok can someone tell me what you get as your answer after solving these! Serious answers only- I will give 10 pts to best answer!

What is the apothem of a regular hexagon with the side= 12cm??

What is the area of a regular hexagon with side = 12 cm. Round your answer to the nearest tenth.??

Someone please let me know what you think the answers are!

Answer 1

Regular hexagon makes 6 equilateral triangles with sides 12 cm

Apothem
= sqrt(3) / 2 (length of 1 side)
= sqrt(3) / 2 (12)
= 6 sqrt(3) cm

Area of hexagon
= area of 6 equilateral triangles
= 6 x 1/2 x base x height (apothem)
= 3 x 12 x 6 sqrt(3)
= 216 sqrt(3) cm^2

Answer 2

the apotherm is the perpendicular distance for the geometric center of the figure to any side. A sketch will be helpful to follow the reasoning below.

if you divide the hexagon into six triangles by drawing lines from the center to the intersections of the sides (or vertices) you will have created 6 equilateral triangles. This is true becvause there are six equal angles at the center which must add up to 360. and, therefor each angle at the center is 60 degrees. Also the 2 base angles of the triangles are equal and there sum must be 180 - 60 = 120 degrees. This reuires that each base angle is 60 degrees.

Now working with one tyriangle, a perpendicular from the vertex to the base will intersect the base at its midpoint. Thus 2 right triangles are created with the base = 1/2 the length of a side of 6 cm (given 12 divided by two)

the height of the triangle is the apothem and that is equal to
6 times the tangent of 60 degrees.

remember the area of a triangle is 1/2 of the height times the base:

since there are 6 triangles the area is:

6(6tan(60))12/2 = area = 216*tan(60)

if you had of recognized that the six triangles are equilateral you could have used the pythagoran theorem.

Answer 3

The apothem is the "inradius".
When you picture a regular hexagon, you can picture it as six equilateral triangles. If you split the triangle into two right triangles, the base is one half of twelve of six and the height is sqrt of three divided by two times twelve. (The height is your apothem)
The area of the right triangle is 1/2 times 6 time 6 times sqrt 3 or
18 sqrt 3. There are 12 right triangles in the hexagon so the total area is 216 times sqrt 3 . Substituting 1.73 for sqrt 3 gives 373.7 cm squared

Answer 4

The apothem of a regular hexagon is the distance from one side to the center. The area of a regular hexagon is one half the product of the apothem and the perimeter. So you really can not solve with the information that you have given.