10.04 Area of triangles
Introduction
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.
Instead, we can use the base b and perpendicular height h (or simply the height) of a triangle to easily calculate its area. The height is the perpendicular distance from the base to the opposite vertex. It can be inside or outside a triangle depending on which side we select as the base.
Triangles and rectangles
Identifying the base and perpendicular height of a triangle is the first step in working out its area. This is because its area is directly related to the area of a particular rectangle. The area of the triangle is equal to half the area of the rectangle that has a length equal to the base of the triangle and a width equal to the height of the triangle.
Exploration
The applet below shows how we can break apart a triangle and rearrange its parts.
Slide the button to show how to break apart a triangle and rearrange its parts.
The area of the triangle is equal to the half of the area of the rectangle.
The same idea works if the perpendicular height lies outside the triangle.
The area of triangle is half of the area of a rectangle with the same height and base.
Area of a triangle
Exploration
In the applet below, we will experiment with changing the dimensions of a triangle to see what effect this has on its area. Try different types of triangles, or varying just the base. Which changes will influence the area? Which changes do not affect the area?
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 centimetres 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.
The base and height of a triangle influence the area.
By using the applet above, you may have noticed that the area of a triangle can be found by multiplying the length of its base with its perpendicular height and halving this product.
The area of a triangle is given by \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.
Example 2
Lisa has purchased a rectangular piece of fabric measuring 6\text{ m} in length and 9\text{ m} in width.
What is the area of the largest triangular piece she can cut out from it?
The area of a triangle is given by:
Unknown dimensions
We have found that the area of a triangle is given by half of the product of its base and height. If we already know the area, along with one of the dimensions, we can use this relationship to find the remaining dimension.
The triangle below has an area of 45\text{ km}^2, and a base of 9\text{ km}. How can we determine the height of the triangle?
From the formula, we know that \text{Area}=\dfrac12\times\text{base}\times\text{height}, which means that 45=\dfrac12\times9\times\text{height}. So we want to find the number that multiplies with \dfrac12 and 9 to give 45.
In other words, we can find the number of times \dfrac12\times9 fits into 45. This is given by
| \displaystyle \dfrac{45}{\frac12\times9} | \displaystyle = | \displaystyle \dfrac{2\times45}{9} | Multiply the numerator and denominator by 2 |
| \displaystyle = | \displaystyle \dfrac{90}9 | Evaluate the multiplication | |
| \displaystyle = | \displaystyle 10 | Evaluate the division |
So the height of the triangle is 10\text{ km}.
Examples
Example 3
Find the value of h if the area of this triangle is 48\text{ m}^2.
Using the formula of the area of a triangle is the only way to find the unknown dimension of a triangle.