Austin_Rain 3 years ago How would I solve this question? All of the triangles in the figure below are congruent. What is the area of the figure? Note that all measurements are in centimeters. Note that the apothem shown is equal to 2 (sqrt)3

1. Austin_Rain

2. KingGeorge

A nice easy trick to this problem, is to notice that the area of the equilateral triangles is the same as the area of the hexagon. So you just have to find one to find the total area. The area of the hexagon is $A={1 \over 2}ap$where a is the apothem and p is the perimeter. We know the apothem to be $$2\sqrt3$$ and the perimeter to be $$6\cdot4=24$$ So the area of the hexagon is${1 \over 2}48\sqrt3 ={24\sqrt3}$Therefore, the total area is$2\cdot(24\sqrt3)=48\sqrt3$

3. KingGeorge

Wait, is the height of the triangle 3? Was I misreading the diagram?

4. Austin_Rain

But, that's not an answer. D: However, 24 (sqrt)3 ^2 is. WOuld that be it?

5. Austin_Rain

The height of the triangle is 3. Yes.

6. KingGeorge

If the height of the triangle is 3, then the area of each triangle is ${1\over2}\cdot3\cdot4=6$So then the answer should be $24\sqrt3+6\cdot6=24\sqrt3+36$

7. KingGeorge

There are 6 triangles, so that's why I multiplied the 6 by 6.

8. Austin_Rain

That is a possible answer. :D

9. Austin_Rain

Is that answer squared?

10. KingGeorge

No. That's the entire answer. $24\sqrt3+36 \;\;\;\text{cm}^2$

11. Austin_Rain

Oh, sweet. :D Thanks man!

12. KingGeorge

You're welcome.