# TrigonometryArea of an equilateral triangle

#### Peking Man

##### New member
Given an equilateral triangle ABC, P is any point inside it where PA = 3, PB = 4 and PC = 5.
Find area of the triangle using the Law of Sines or Law of Cosines.

#### Ackbach

##### Indicium Physicus
Staff member
What ideas have you had so far?

#### The Chaz

##### Member
Given an equilateral triangle ABC, P is any point inside it where PA = 3, PB = 4 and PC = 5.
Find area of the triangle using the Law of Sines or Law of Cosines.
I know you're new around here , so you might not know that it is preferable to show work. (That might be in the rules...)

So I'll get the ball rolling.

Letting a , b, c be the (equal) sides of the "big" triangle and A, B, C the angles formed by... oh dear, I have confused the letters. If the verteces of the outer triangle were renamed...
we have
c^2 = 3^2 + 4^2 - 2(3)(4)cos(C)
etc.
but c^2 = b^2 = a^2 and A + B + C = 360

Can you take it from here?

• Peking Man and Alexmahone

#### Prove It

##### Well-known member
MHB Math Helper
Heron's Formula may come in handy here, once you have used the Sine Rule and/or Cosine Rule to evaluate the side lengths of the triangle. If a, b, c are the side lengths of your triangle, then

$$Area = \sqrt{s(s - a)(s - b)(s - c)}$$

where $$s = \frac{a + b + c}{2}$$

#### Peking Man

##### New member
Heron's Formula may come in handy here, once you have used the Sine Rule and/or Cosine Rule to evaluate the side lengths of the triangle. If a, b, c are the side lengths of your triangle, then

$$Area = \sqrt{s(s - a)(s - b)(s - c)}$$

where $$s = \frac{a + b + c}{2}$$
- the teacher said, the Law of Cosines or Law of Sines is enough to solve the problem, but how?

---------- Post added at 11:29 PM ---------- Previous post was at 11:21 PM ----------

Area = (ab/2)sin C = (ac/2)sin B = (bc/2)sin A, and a = b = c. the only missing value is the side of the equilateral triangle .... the application of the Law of Sines and Cosines eludes me so far.

I followed CHAZ suuggestions, but three more internal angles remained unknown ... i have more problems to deal then.

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#### Prove It

##### Well-known member
MHB Math Helper
Can you evaluate ONE side of the equilateral triangle? If you have one side, you have them all. Then you can use Heron's Formula (much easier)...

#### Krizalid

##### Active member
This is really a classic problem.
Here's a Hint:

Construct an equilateral triangle on side AP, look for a congruent triangle and use Law of Cosines.