Find the Height of a Triangle
How to do it, given Area
Object of this page To practice applying the conventional area of a triangle formula to find the height, given the triangle's area and a base.
Example 1
In this triangle, the area is 17.7 square units, and its base is 4.
Let's find the height of this triangle, which is pictured below.

The red measurements are the ones that are relevant to this problem. Remember that each triangle has 3 base/height pairs. So whenever you are talking about the height, we have to make sure we know which of the 3 'bases' (or sides) of the the triangle we are talking about.
We can tell from the picture, that the height is perpendicular to the base whose measure is 4. That is why the side of length 4 is the base and the other sides do not affect this problem. 
Steps to Find Area
Step 1) Substitute known values into the area formula

Area = $$ \frac{1}{2} \cdot base \cdot height$$
17.7 = ½(4)(h)

Step 2) Find the height by solving for 'h' 
$$
17.7 = 2h
\\
h = \frac{17.7}{2} = 8.85
$$

Practice Problems
Problem 1)

If the area of the triangle on the left is 658.8square feet and its base is 24 inches, what is the height?

This problem is very similar to example 1 .
Step 1) Substitute known values into the area formula 
Area =$$ \frac{1}{2} \cdot base \cdot height$$
658.8 = ½(24)(h)

Step 2) Find the height by solving for 'h' 
$$
658.8 = 12h
\\
\frac{658.8}{12} = h
\\ h = 54.9
$$

Problem 2)

If the Area of the triangle on the left is 11.6 square units and its base is 4, what is its height?

Step 1) Substitute known values into the area formula 
Area =$$ \frac{1}{2} \cdot base \cdot height$$
11.6 = ½(4)(h)

Step 2) Solve for the height 
$$11.6 = 2h
\\ \frac{11.6}{2} =h
\\
h = 5.8
$$

Problem 3)

If the Area of the triangle on the left is 17.7 square units and its base is 15, what is its height?

This problem is very similar to example 1 .
Step 1) Substitute known values into the formula 
Area =$$ \frac{1}{2} \cdot base \cdot height$$
17.7 = ½(15)(h)

Step 2) Find the height by solving for 'h' 
$$
17.7 = 7.5h
\\ \frac{17.7}{7.5} = h
\\
h = 2.4
$$

Problem 4)

If the Area of the triangle on the left is 35.8. square units,find its height.

This problem is very similar to example 1 . In every triangle, there is 3 base/height pairs , but you can tell from the picture that the base is the side of length 12. Remember that the base and height are perpendicular.
Step 1) Substitute known values into the formula 
Area =$$ \frac{1}{2} \cdot base \cdot height$$
35.8 = ½(12)(h)

Step 2) Find the height by solving for 'h' 
$$
35.8 = 6h
\\
\frac{35.8}{6} =h
\\
h = 5.9
$$

Problem 5)

If the Area of the triangle on the left is 73.5. square units, find the height.

This problem is very similar to example 1 . In every triangle, there is 3 base/height pairs , but you can tell from the picture that the base is the side of length 11.Remember that the base and height are perpendicular.
Step 1) Substitute known values into the formula 
Area =$$ \frac{1}{2} \cdot base \cdot height$$
73.5 = ½(11)(h)

Step 2) Solve for the height. 
$$
73.5 = 5.5h
\\
\frac{73.5}{5.5} = h
\\
h = 13.4
$$

Problem 6)

If the Area of the triangle on the left is 141.9. square units, find its height.

This problem is very similar to example 1 . In every triangle, there is 3 base/height pairs , but you can tell from the picture that the base is the side of length 21. Remember that the base and height are perpendicular.
Step 1) Substitute known values into the formula 
Area =$$ \frac{1}{2} \cdot base \cdot height$$
141.9 = ½(21)(h)

Step 2) Solve 
$$
141.9 = 10.5 h
\\
\frac{141.9}{10.5} = h
\\
h = 13.5
$$

Some Challenge problems
All right, now let's try some more challenging problems involving finding the height of a triangle. You should be comfortable with the properties of equilateral and isosceles triangle types before attempting the following questions.
Challenge Problem 1)
If the perimeter of an equilateral triangle is 18 and its area is 15.6 square units, what is the measure of its height? Also, does it matter which side you choose as your base?
Since this is an equilateral triangle and we know its perimeter is 18, we can figure out that each side has a length of 6. (18/3 =6). It does not matter which base we choose, since all 3 heights ( or altitudes) in an equilateral triangle are congruent. From this point, it's just a matter of following the steps we have been using throughout this web page:
Step 1) Substitute known values into the formula 
Area =$$ \frac{1}{2} \cdot base \cdot height$$
15.6 = ½(6)(h)

Step 2) Solve for the height. 
$$
15.6 = 3h
\\
\frac{15.6}{3} = h
\\
h = 5.2
$$

