## How to Find the Height of a Triangle Lesson

### Two Methods for Finding Height

There are two basic methods we can use to find the height of a triangle. They are given as:

**If we know the area and base of the triangle, the formula h =**^{2A}⁄_{b}can be used.**If we know side lengths and angles of the triangle, we can use trigonometry to find height.**

#### Using Area to Find the Height

The area formula for a triangle is A = ^{1}⁄_{2}bh. After rearranging the formula to isolate h, we end up with **h = ^{2A}⁄_{b}**. If we have the area and base, we simply plug them into this new formula to find height.

Example Problem:

Find the height of a triangle with a base of 10 and an area of 20.

Solution:

- Let's use the base and area formula to find the height.
- Plugging the values in to the formula, we get:

h =^{2A}⁄_{b}=^{2(20)}⁄_{(10)}= 4. **The height of the triangle is 4.**

We can check our solution by plugging the height in to the triangle area formula, A =^{1}⁄_{2}bh.

20 =^{1}⁄_{2}(10)(4), 20 = 20**✓**.

#### Using Trigonometry to Find the Height

Let's consider the image of the triangle above. There are three labeled sides and three labeled angles. The height is the vertical line labeled *H*.

The bottom side *AC* is perfectly horizontal. Since the height is vertical, *AC* and *H* are perpendicular. **Now, remember that the vertical projection of an angled line is its length times the sine of the line's angle off the horizontal.**

Using this relation, we can find the height of the triangle by using side *AB* or side *BC*. For the triangle above, the two possible trig formulas for height are given as:

**h = ABsin(α)****h = BCsin(γ)**

Example Problem:

In the triangle above, side BC is 7 long and angle γ is 45°. What is the height?

Solution:

- Plugging the given values into the formula, we get:

h = BCsin(γ)

h = (7)sin(45°) = (7)(.7071) = 4.950 **The height is 4.950.**