How to calculate the area of a common triangle
A triangle is a flat geometric figure formed by three line segments that meet at three points called vertices. The area of a triangle is the measure of the surface it occupies on the plane. There are different ways to calculate the area of a triangle, depending on the data you have about it.
In this article, we will present four ways to calculate the area of a common triangle, i.e., one that does not have special characteristics such as being equilateral, isosceles, or right-angled. The ways are:
- Using the base and the height
- Using all three sides
- Using two sides and the angle between them
- Using one side and two adjacent angles
Let's look at each of them in detail, with examples and step by step.
Using the base and the height
The simplest and most well-known way to calculate the area of a triangle is by using the measurement of the base and height. The base is any one of the sides of the triangle, and the height is the segment perpendicular to the base that passes through the opposite vertex. The formula to calculate the area using the base and height is:
A = (b x h) / 2
Where:
- A is the area of the triangle
- b is the measurement of the base
- h is the measurement of the height
To use this formula, just replace the base and height values in the expression and perform the indicated operations. Here's an example:
Example: Calculate the area of the triangle below, knowing that its base is 12 cm and its height is 9 cm.
Solution: Using the area formula with base and height, we get:
- A = (b x h) / 2
- A = (12 x 9) / 2
- A = 108 / 2
- A = 54 cm²
Using all three sides
When you don't know the measurement of the triangle's height, but you know the measurement of its three sides, you can calculate its area using Heron's formula. This formula uses the semi-perimeter of the triangle, which is half of the perimeter (the sum of the three sides). Heron's formula is:
A = √(s x (s - a) x (s - b) x (s - c))
Where:
- A is the area of the triangle
- s is the semi-perimeter of the triangle
- a, b, and c are the measurements of the triangle's sides
To use this formula, follow these steps:
- Find the triangle's semi-perimeter by adding the three sides and dividing by two.
- Replace the semi-perimeter and side values into the expression.
- Perform the multiplications and the square root.
Here's an example:
Example: Calculate the area of the triangle below, knowing that its sides measure 10 cm, 13 cm, and 15 cm.
Solution: Using Heron's formula, we get:
First, we calculate the semi-perimeter:
- s = (a + b + c) / 2
- s = (10 + 13 + 15) / 2
- s = 38 / 2
- s = 19 cm
Then, we replace the values in the area formula:
- A = √(s x (s - a) x (s - b) x (s - c))
- A = √(19 x (19 - 10) x (19 - 13) x (19 - 15))
- A = √(19 x 9 x 6 x 4)
- A = √(4116)
- A ≅ 64.15 cm²
Using two sides and the angle between them
When you know the measurement of two sides of the triangle and the angle formed by them, you can calculate its area using the sine of the angle. The formula is:
A = (a x b x sin(θ)) / 2
Where:
- A is the area of the triangle
- a and b are the measurements of the two known sides
- θ is the angle formed by these two sides
Just replace the values in the expression and perform the operations. Here's an example:
Example: Calculate the area of the triangle below, knowing that two of its sides measure 8 cm and 11 cm, and the angle formed by them is 60°.
Solution: Using the formula for two sides and the angle between them, we get:
- A = (a x b x sin(θ)) / 2
- A = (8 x 11 x sin(60°)) / 2
- A = (8 x 11 x 0.866) / 2
- A = 76.176 / 2
- A = 38.088 cm²
Using one side and two adjacent angles
Finally, when you know the measurement of one side of the triangle and the two angles adjacent to it, you can calculate its area using the sine of these angles. This is a more complex method that involves two steps. First, you need to find the other two sides using the law of sines. Then, you can calculate the area as previously explained. The formula is:
A = (a x (a x sin(α) / sin(γ)) x (a x sin(β) / sin(γ))) / 2
Where:
- A is the area of the triangle
- a is the known side
- α and β are the two known angles
- γ is the unknown angle (γ = 180° - α - β)
Follow these steps:
- Calculate the unknown angle γ by subtracting the known angles from 180°.
- Calculate the other two sides using the law of sines.
- Replace the values in the area formula and perform the operations.
This method is less common and can be more laborious, but it can be useful in specific situations. For example, if you are working with triangle projections.