Area of a Triangle

Area of a Triangle Calculator

The following calculator finds the area of a triangle based on the given parameters. Type in the required values and press the '»' button.

Calculate based on:

Side length and height
2 side lengths and angle between them
3 side lengths

height: area:

There are multiple ways of calculating area of a triangle. The most popular being multiplying the half of the length of the base by the height. Obviously this formula will be of any use only if both of the mentioned measures are knows. The following is the list of equations serving the same purpose but making use of different triangle properties.

Knowing length of one side of a triangle and the height of it:

Area of a triangle
\[ area = \frac{1}{2} base * height \]

Knowing the length of two sides and the angle between them:

Area of a triangle with sides and angle
\[ area = \frac{1}{2} side1 * side2 * sin(angle) \]

Knowing length of all sides of the triangle:

Area of a triangle with side lengths
\[ area = \sqrt{s(s-a)(s-b)(s-c)} \\ s = \frac{(a+b+c)}{2} \]

Calculation Background

The above are just 3 most popular of many more available formulas. To find out more complex ones, please consult the Wikipedia page on the subject. Depending on the data available a particular formula will do the job best for specific application. Also specific triangle configuration may make it easier to use one equation over the other. For instance a right triangle’a (the one having one of its internal angles measuring 900) area can be calculated if lengths of the two sides adjacent to the right angle are know. This is because the sides are equal to the heights measured from the respective bases.

Exercise 1: Calculate the area of a triangle knowing the length of the base is 12cm and the height of it is 8cm.

Solution: Knowing the length of the base and the height of the triangle we will use the standard equation:

\[ area = \frac{1}{2} base * height \]

Let’s replace the variables with the values:

\[ area = \frac{1}{2} base * height \\ base = 12 \\ height = 8 \\ area = \frac{1}{2} * 12 * 8 \\ area = 48cm² \]

Therefore, using the standard formula we’ve calculated that the area of the given triangle is 48cm².

Exercise 2: What is the area of a triangle knowing the length of the sides are as follows: 6cm, 8cm, 12cm.

Solution: As we don’t have any information of the height of the triangle we cannot use the formula we have applied previously. However the length of all the sides of the triangle are known. Therefore we will make use of the other formula mentioned earlier. To start with we need to calculate the half of the triangle’s perimeter.

\[ s = \frac{(a+b+c)}{2} \\ a = 6cm \\ b = 8cm \\ c = 12cm \\ s = \frac{(6+8+12)}{2} = \frac{26}{2} \\ s = 13 cm \]

Next we will use the calculated semi perimeter value: 13cm to solve the following equation:

\[ area = \sqrt{s(s-a)(s-b)(s-c)} \\ area = \sqrt{13 * (13 – 6) * (13 – 8) * (13 – 12)} \\ area = \sqrt{(13 * 7 * 5 * 1)} \\ area = \sqrt{455} \\ area = 21.33 cm² \]

Exercise 3: Two adjacent sides of a triangle have lengths of 8cm and 4cm respectively and form an angle of 40°. Calculate area of that triangle.

Solution: Here even different formula will be of use. Knowing the lengths of two adjacent sides and the angle they form we can make use of the following equation: \(area = \frac{1}{2}a*b*sin(α) \) The variables equal to:

\[ a = 8cm \\ b = 4cm \\ α = 40° \\ sin(α) = sin(40°) = 0.64 \text{ (using scientific calculator)} \\ area = \frac{1}{2} * 8 * 4 * sin(40°) = 16 * 0.64 \\ area = 10.24 cm² \]
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