Question

Help with this problem

Find the area of an equilateral triangle with a side of 9.

Answer 1

@campbell_st

Answer 2

First, do you know the formula for the area of a triangle?

Answer 3

A=1/2bh

Answer 4

the area of an equilateral triangle with side length \(a\) is
\[\huge A=\frac{\sqrt3}{4}a^2\]

Answer 5

in your example \(a=9\)

Answer 6

ok so how do you put that in calculator? @satellite73

Answer 7

Correct.
We need a base and a height.

Answer 8

The base of our triangle is 9.
We need to find the height, h.

Answer 9

Both the left and right triangles are 30-60-90 triangles.

Answer 10

you don't put it in a calculator, you just write it

Answer 11

do you do that sin or cosine thing for 30 60 90 triangle?

Answer 12

A 30-60-90 triangle has this ratio of the lengths of the sides:

Answer 13

i also had a question is how do you know which side would be square root 3?

Answer 14

The long leg is \(\sqrt{3}\) times the length of the short leg.
In our triangle, the short leg is 4.5, so the long leg is \(4.5\sqrt{3} \). The long leg is the height.
Now we have the base and the height, so we can find the area.

Answer 15

In a 30-60-90 triangle, you have a 30 deg angle, a 60 deg angle and a 90 deg angle, right?

Answer 16

yaa

Answer 17

In any triangle, a larger angle has a larger side opposite the angle.

Answer 18

That means in a 30-60-90 triangle, the side opposite the 30 deg angle is the shortest side.
The side opposite the 60 deg angle is the middle sized side.
The side opposite the 90 deg angle is the longest side.
We already knew that last statement bec the side opposite the 90 deg angle is the hypotenuse, and the hypotenuse is always the longest side of a right triangle.

Answer 19

ohok i got that parrt

Answer 20

Once you realize which side is shortest, which side is middle size, and which side is longest, then the ratio of the sides is what I wrote above.
The easy way to remember is the hypotenuse is twice the short leg.
The long leg is \(\sqrt 3\) times the shgort leg.

Answer 21

In our case, we knew the short leg was 4.5
That makes the long leg \(4.5\sqrt 3\)
Since the long leg of the 30-60-90 triangle is the height of the equilateral triangle, we now have the base and height of the equilateral triangle. Now we can find teh area of the equilateral triangle.

Answer 22

ok why did you dived 9 by 2

Answer 23

Because we needed a 30-60-90 triangle to find the height of the equilateral triangle.
The equilateral triangle has all sides of length 9.
When you split the equilateral triangle into two 30-60-90 triangles, the base is split into two equal sized halves, or 4.5 units each.

Answer 24

for the height?