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6.4: Area of Triangles

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Use formulas to find the area of triangles.


Example 1

How could we find the area of this triangle?

Make it into a parallelogram. This can be done by making a copy of the original triangle and putting the copy together with the original. As you can see in the diagram below, one triangle looks the same as the one above, and the other is upside down.

When you put the 2 triangles together, they make a parallelogram. The area of the parallelogram is bh, and since we used 2 triangles to make the parallelogram, the area of the triangle is half of \begin{align*}bh: \frac{bh}{2}\end{align*} or \begin{align*}\frac{1}{2} \ bh\end{align*}.

How many triangles are used to make up the parallelogram above? __________

Warning: Notice that the height \begin{align*}h\end{align*} (also often called the altitude) of the triangle is the perpendicular distance between a vertex and the opposite side of the triangle. This means that the altitude \begin{align*}h\end{align*} meets the base \begin{align*}b\end{align*} at a \begin{align*}90^\circ\end{align*} angle.

Reading Check

1. In a triangle, the height and the base in a must be ____________________________ because they form a \begin{align*}90^\circ\end{align*} angle.

2. Another name for the height of a triangle is the ___________________.

Area of a Triangle

If a triangle has base \begin{align*}b\end{align*} units and altitude \begin{align*}h\end{align*} units, then the area, \begin{align*}A\end{align*}, is \begin{align*}\frac{bh}{2}\end{align*} or \begin{align*}\frac{1}{2} \ bh\end{align*} square units.

\begin{align*}\text{Area} & = \frac{1}{2} \cdot \text{base} \cdot \text{height}\\ A &= \frac{bh}{2} \ \text{or} \ A = \frac{1}{2} \ bh\end{align*}

Reading Check

1. True or False: In a triangle, the base and the height intersect at an obtuse angle.

2. True or False: Multiplying by \begin{align*}\frac{1}{2}\end{align*} and dividing by 2 are the same thing.

3. Fill in the blanks:

In area formulas, the letter _____ usually stands for the base and the letter _____ usually stands for the height.

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