Find the area of a regular hexagon with 8 feet sides. Round the answers to the nearest whole number.

Question

My teacher told me that I have to use radians instead of tan . I made a similar problem to this before with tan and it was this : 360 / 6 = 60

60 = top of one angle

180 - 60 = 120, 120 / 2 = 60

60 = side angle

Area = 1/2*Base*Height

tan (60) = h (10/2)

h = 5 * tan (60), h = 8.66

A = 1/2 (10) (8.66)

A = 43.3

43.3 * 6 = 259.8

Answer 1

draw a diagram

there are 6 congruent triangles

draw the height or apothem of one of the triangles

each central angle is 360/6=60°

the apothem forms two 30-60-90 right triangles

the side opposite the 30° angle is half the hypotenuse

you have a right triangle with an 8 foot hypotenuse and one leg being 4 feet

using the Pythagorean Theorem...

8^2=4^2+h^2

64=16+h^2

h^2=48

h=4√3

Area=(1/2)*apothem*perimeter

A=(1/2)(4√3)(48)

A=2√3*48

A=96√3

A=96*1.7320508

A=166.277 square feet

Answer 2

I would approach the problem like this:

This is a geometry problem, so I would draw a hexagram first, including the center point, and lines from the centerpoint out to each vertex.

This turns out to divide the hexagram into 6 equilateral triangles, which of course have all angles of 60 degrees (π/3 radians.)

Use Pythagoras to remind yourself that the height of one of these triangles is 8 * (1/2) * √3 = 4 * √3

So the area of the equilateral triangle is (1/2) * base * height, or (1/2) * 8 * (4 * √3) = 16 * √3

And there are 6 triangles in the hexagon, so the hexagon area is 6 * 16 * √3 = 96 * √3 , which is about 166.277 sq ft.

As a check, note that if we draw a circle with radius of 8', we get approximately 201 sq ft, so our answer for the hexagon seems about right, since the circle contains the hexagon.