GMAT Math: How to Find the Height of a Triangle?


 
If you have a right triangle and are given two sides and would like to find the third, use the Pythagorean Theorem: \(a^2+b^2=c^2\).

 
Suppose you need to know how to find the height of a triangle ABC given 3 sides, {6,7,8}.

Triangle with sides 6, 7, 8

This is a question some GMAT test takers ask.  They know they would need the height to find the area, so they worry: how would I find that height. 

The short answer is: forget about it!

Height of a triangle: which height?

I don’t mean to be flippant. It’s just that first of all, the “height” of a triangle is it’s altitude. Any triangle has three altitudes, and therefore has three heights! Confusing? I know, sorry. 

You see, any side can be a base. From any one vertex, you can draw a line that is perpendicular to the opposite base — that’s the altitude to this base. 

Any triangle has three altitudes and three bases. 

You can use any one altitude-base pair to find the area of the triangle, via the formula \(A= \frac{1}{2}bh\).
 

Triangle diagram 1-magooshTriangle diagram 2-magooshTriangle diagram 3-magoosh

In each of the diagrams above, the triangle ABC is the same. The green line is the altitude, the “height”, and the side with the red perpendicular square on it is the “base.”  All three sides of the triangle get a turn.

Finding a height

Given the lengths of three sides of a triangle, the only way one would be able to find a height and the area from the sides alone would involve trigonometry, which is well beyond the scope of the GMAT. 

You are 100% NOT responsible for knowing how to perform these calculations.  This is several levels of advanced stuff beyond the math you need to know.  Don’t worry about that stuff.

In practice, if the GMAT problem wants you to calculate the area of a triangle, they would have to give you the height. 

The only exception would be a right triangle — in a right triangle, if one of the legs is the base, the other leg is the altitude, the height, so it’s particularly easy to find the area of right triangles.

What you need to know

You need to know basic geometry.  Yes, there is tons of math beyond this, and tons more you could know about triangles and their properties, but you are not responsible for any of that.  You just need to know the basic geometry of triangles, including the formula:

A = 12 bh

 
If the triangle is not a right triangle, you have absolute no responsibility for knowing how to find the height — it will always be given if you need it.  

Here’s a free practice question for you.

Two sides of a triangle have length 6 and 8. Which of the following are possible areas of the triangle?

2
12
24

Click here for the answer and video explanation!

Some “more than you need to know” caveats

  • If you don’t want to know anything about this topic that you don’t absolutely need for the GMAT, skip this section!
  1. Technically, if you know the three sides of a triangle, you could find the area from something called Heron’s formula, but that’s also more than the GMAT will expect you to know. 
  2.  

  3. If one of the angles of the triangle is obtuse, then the altitudes to either base adjacent to this obtuse angle are outside of the triangle. 
  4.  

  5. Super-technically, an altitude is not a segment through a vertex perpendicular to the opposite base, but instead, a segment through a vertex perpendicular to the line containing the opposite base.

Triangle DEF-magoosh

In the diagram above, in triangle DEF, one of the three altitudes is DG, which goes from vertex D to the infinite straight line that contains side EF.  That’s a technicality the GMAT will not test or expect you to know. 

If the three sides of a triangle are all nice pretty positive integers, then in all likelihood, the actual mathematical value of the altitudes will be ugly decimals. 

Many GMAT prep sources and teachers in general will gloss over that, and for the purposes of easy problem-solving, give you a nice pretty positive integer for the altitude also. 

Remember this triangle ABC from above?

Triangle with sides 6, 7, 8

For example, the real value of the altitude from C to AB in the 6-7-8 triangle is:

Altitude formula

Not only are you 100% NOT expected to know how to find that number, but also most GMAT practice question writers will spare you the ugly details and just tell you, for example, altitude = 5. 

That makes it very easy to calculate the area. 

Yes, technically, it’s a white lie, but one that spares the poor students a bunch of ugly decimal math with which they needn’t concern themselves. 

Actually, math teachers of all levels do this all the time — little white mathematical lies, to spare students details they don’t need to know.

So far as I can tell, the folks who write the GMAT itself are sticklers for truth of all kinds, and do not even do this “simplify things for the student” kind of white lying. 

They are more likely to circumvent the entire issue, for example, by making all the relevant lengths variables or something like that.

Takeaways

Still with me?

Here’s what you need to know about triangles on GMAT test day:

  • \(Area =\frac{1}{2}bh\)
  • You’ll only need to know the height of right triangles on the GMAT
  • If it’s not a right triangle, you’ll be given the height
  • Know all three angles and two sides? Use the Pythagorean theorem

If you like more free resources (who doesn’t?), or are simply wondering what the GMAT test content is going to be like, check out our Complete Guide.

You’ve got this.

If you have any questions, let me know in the comments below. And yes, I read every single one of them!

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