The area of a triangle is the amount of space that can fit within its outline. We could draw a grid of unit squares on top of a triangle and count the number of squares it contains, but this can be time consuming and inaccurate because of the slanted sides.

In the previous investigation , we discovered how to use the base ( b ), and the height ( h ) , to calculate the area using a formula.

Ideas

Area of a triangle

Area of a triangle

Let's look at how the base and height can affect the area of a triangle. In the applet below, we will experiment with changing the dimensions of a triangle. Try different types of triangles, or varying just the base.

Exploration

The following guide outlines the key features and concepts in the applet.

Click \text{Show Grid} (you can hide it again with \text{Hide grid}). If we assume this is a square centimeter grid, the number of squares that can fit within the triangle is its area. You may wish to count the squares for one triangle and check your answer with the one printed on the applet.
Click and drag the right vertex of the triangle left or right to change the length of its base. Click and drag the apex of the triangle up or down and left or right to change the length and position of its perpendicular height. This can also be done with the b, h, and "Apex" sliders.
The area of the triangle is being calculated with a formula as you change the dimensions of the triangle.

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Which changes will influence the area of the triangle?

Which changes do not affect the area of the triangle?

By using the applet above, you may have noticed the base and the height both influence the area, and that the height can be inside or outside a triangle depending on where the base and apex are.

You may have also noticed that the area of every triangle type could be found using the same formula, which is multiplying the length of the base by the length of the height and then halving that product.

Formula for the area of a triangle

\quad \text{Area}=\dfrac12\times\text{base}\times\text{height \quad or} \quad \text{A}=\dfrac12\times b\times h

Examples

Example 1

Find the area of the triangle shown.

Triangle with a height of 7 centimeters and base of 10 centimeters.

Worked Solution

Create a strategy

Use the area of a triangle formula.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times\text h	Use the formula
	\displaystyle =	\displaystyle \dfrac12\times10\times7	Substitute b=10 and h=7
	\displaystyle =	\displaystyle 35\text{ cm}^2	Evaluate

Idea summary

The area of a triangle is given by:

\displaystyle A=\dfrac12\times b\times h

\bm{A}

is the area of the triangle

\bm{b}

is the base of the triangle

\bm{h}

is the height of the triangle

Outcomes

6.G.A.1

Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

7.01 Area of triangles

Introduction