10.05 Area of triangles
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 $b$b and perpendicular height $h$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. Use the applet below to see the different ways that we can label the base and height of a triangle.
The relationship between 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 applet below shows how we can break apart a triangle and rearrange its parts. We can see that 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.
The same idea works if the perpendicular height lies outside the triangle. The applet below shows one way that we can break up the triangle and rearrange the parts for this case. Can you think of any other ways?
Calculating the area of a triangle
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 $\editable{\text{Show grid}}$Show grid (you can hide it again with $\editable{\text{Hide grid}}$Hide grid). If we assume this is a square centimetre 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 $b$b, $h$h and "Apex" sliders.
- The area of the triangle is being calculated with a formula as you change the dimensions of the triangle. Can you work out the formula?
Formula for the area of a triangle
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
$\text{Area }=\frac{1}{2}\times\text{base }\times\text{height }$Area =12×base ×height , or
$A=\frac{1}{2}\times b\times h$A=12×b×h
Worked examples
Example 1
Find the area of the right-angled triangle below.
Think: We can identify the base as the side with length $5$5 mm, so that the perpendicular height is the side with length $8$8 mm. Each dimension is in mm, so the area will be in mm2.
Do: Use the area formula with the given side lengths.
| $\text{Area }$Area | $=$= | $\frac{1}{2}\times\text{base }\times\text{height }$12×base ×height | (Formula for area of triangle) |
| $=$= | $\frac{1}{2}\times5\times8$12×5×8 | (Substitute the values for the base and the height) | |
| $=$= | $20$20 mm2 | (Perform the multiplication to find the area) |
So this right-angled triangle has an area of $20$20 mm2.
Reflect: Notice that we could have switched which side we called the base and which side we called the height, and we would still arrive at the same area for the triangle. Think about how the orientation of the triangle relates to how we choose to label the sides.
Example 2
Find the area of the scalene triangle below.
Think: This triangle has a base of $6$6 cm and a height of $4$4 cm. The area of this triangle will be in cm2.
Do: Use the area formula with the given side lengths.
| $\text{Area }$Area | $=$= | $\frac{1}{2}\times\text{base }\times\text{height }$12×base ×height | (Formula for area of triangle) |
| $=$= | $\frac{1}{2}\times6\times4$12×6×4 | (Substitute the values for the base and the height) | |
| $=$= | $12$12 cm2 | (Perform the multiplication to find the area) |
So this scalene triangle has an area of $12$12 cm2.
Reflect: Can you picture a rectangle that has twice the area of this triangle?
Example 3
Find the area of the oblique triangle below.
Think: In this case, the perpendicular height lies outside the triangle, but we know that the same formula still applies. The base is $3$3 m and the height is $5$5 m, so the area will be in m2.
Do: Use the area formula with the given side lengths.
| $\text{Area }$Area | $=$= | $\frac{1}{2}\times\text{base }\times\text{height }$12×base ×height | (Formula for area of triangle) |
| $=$= | $\frac{1}{2}\times3\times5$12×3×5 | (Substitute the values for the base and the height) | |
| $=$= | $7.5$7.5 m2 | (Perform the multiplication to find the area) |
So this oblique triangle has an area of $7.5$7.5 m2.
Reflect: Although we could change the order of the multiplication and still get the same area, in this case it is more appropriate to label the side with length $3$3 m as the base. Notice that the length of $5$5 m is not related directly to any side, so it is a perpendicular height.
Using the area to find a dimension
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$45 km2, and a base of $9$9 km. How can we determine the height of the triangle?
From the formula, we know that $\text{Area }=\frac{1}{2}\times\text{base }\times\text{height }$Area =12×base ×height , which means that $45=\frac{1}{2}\times9\times\text{height }$45=12×9×height . So we want to find the number that multiplies with $\frac{1}{2}$12 and $9$9 to give $45$45.
In other words, we can find the number of times $\frac{1}{2}\times9$12×9 fits into $45$45. This is given by
| $\frac{45}{\frac{1}{2}\times9}$4512×9 | $=$= | $\frac{2\times45}{9}$2×459 | (Multiply the numerator and denominator by $2$2) |
| $=$= | $\frac{90}{9}$909 | (Perform the multiplication in the numerator) | |
| $=$= | $10$10 | (Simplify the fraction) |
So the height of the triangle is $10$10 km.
Practice questions
Question 1
Find the area of the triangle shown.
Question 2
Find the value of $h$h if the area of this triangle is $48$48 m2.
An obtuse triangle, the base of the triangle sits horizontally at the bottom of the drawing. The obtuse angle is on the left of the bottom side. A dashed line extends the base to the left until it meets with another vertical dashed line at a right angle (indicated by a small square). The top of the vertical dashed line meets the top corner of the main triangle. The vertical dashed line is labeled "h m", this label is not included in the main triangle. The bottom side of the main triangle is labeled 4m. Two horizontal arrows next to the bottom label pointing away up to the corners of the main triangle indicate that the label is for only the bottom side and not the dashed line extension.
Question 3
Lisa has purchased a rectangular piece of fabric measuring $6$6 m in length and $9$9 m in width.
What is the area of the largest triangular piece she can cut out from it?