question about area below

Question

I have a question that has two equilateral triangles. looks like the Jewish star, and leaves a hexagon in the middle that isn't shaded but the triangles around are shaded with the side lengths of 6 I need to find the area of the shaded triangles.

Answer 1

I know it's hard to picture this in your head...

But you end up with 6 equilateral triangles

of side length 6 spaced around the nonshaded

hexagon in the center.

 

Take 1 triangle with side lengths of 6.

Draw a vertical line from the point at the

top to the center of the base, cutting the

triangle in half.

 

This divides the single triangle into two

right triangles.  Each has

a base of 3 making one leg, an angled side of length 6

representing the hypotenuse, and the other leg of

unknown length which will represent the height

of the original triangle.

 

     |\

  x |  \ 6

     |__\

       3

 

This is one half of one of the 6 triangles.

 

From the Pythagorean theorem..

a²+b² = c² where a,b are the legs,

and c is the hypotenuse of a right

triangle we can find x.

 

x² + 3² = 6²

x² + 9 = 36

x² = 36-9

x² = 27

x = √27

x = √[(9)(3)]

x = √[(3²)(3)]

x = 3√3

 

This is the height of the triangle.

 

Now we can find the area of one of the 6 triangles.

 

Area = (1/2)(base)(height)

Area = (1/2)(6)(3√3)

Area = 3(3√3)

Area = 9√3

 

This is the area of 1 of your 6 shaded triangles.

Since we have 6, multiply this by 6 for final total.

 

Area total shaded = 6(9√3) = 54√3