# Area of a Triangle

A triangle is a simple closed polygon that is formed when you have three line segments or sides joined together. So a triangle has three sides or three line segments. It also has three vertices and three angles. There are six types of triangles in which three types are related to the sides and the other three types are related to the angles. Here we have mentioned that a triangle is a closed polygon, so what is a polygon? In simple terms, a polygon is a three or more sided figure.

This is how a triangle looks, but there are different types of triangles. Calculating the Area of a Triangle

There are different ways in which you can calculate the area of a triangle. The most common way of determining the area of a triangle is just taking half of the base then multiplying it by the height. In other words, it is half the base times the height. There also exists other formulas that may be used to get the area of a triangle. When calculating the area of a triangle, you will use the information you know and it will depend on the information that you already have regarding the angles and sides.

For example, if you are given the sides and angles, you may be able to calculate the areas even when you do not know the height.

So, in this tutorial, we will look at the different ways in which you can calculate the area of a triangle and we will make the calculations and formulas to be as simple as possible and easy to understand.

Using the Base and the Height

When you are given the base and height, you can find the area of the triangle in question. The base is just one side of a triangle. It doesn’t matter which you may want to call the base of the three sides. You can pick any side and say that it is the base. The height on the other hand is the measure that you give to the tallest point of the triangle. You can get this measure by drawing a line perpendicular from the base, and that line runs to the opposite vertex. You will find that this information will be provided to you, but sometimes, you might have to measure the lengths.

Let’s take for example; You have a triangle that has a base measuring 5 cm. You are also given the height, which measures 4 cm.

You now set up the formula which is:

A = ½ (bh) – where “b” stands for the length of the base of the triangle and “h” stands for the height of the triangle.

So, Area = ½ (5cm x 4cm)

Area =  ½ (20cm2)

Area =  10cm2

This formula works for a right triangle. In a right triangle, two sides are perpendicular meaning they are join at 90 degrees. So, one of the perpendicular sides may be taken as the height and the other as the base.

What about when if you are given one side length and the hypotenuse? When we talk of hypotenuse, it is the longest side within a right angle triangle. The hypotenuse is usually opposite to the right angle. In this case, you can use the Pythagorean Theorem to find the length of a missing side within a right angled triangle.

The Pythagorean Theorem states that (a2 + b2 = c2a2+b2=c2{\displaystyle a^{2}+b^{2}=c^{2}})

So, if the hypotenuse is denoted by the letter C, then it mean that the two perpendicular sides in a right angle are denoted by a and b in the Pythagorean Theorem.

Now, if the hypotenuse is 8 cm and the base is 4 cm, you can use Pythagorean Theorem to find the height.

a2 + b2 = c2

42 + b2 = 82

16 + b2 = 64

b2 = 64 – 16

b2 = 48

b = √48

6.9 cm

So, the height of a right triangle with an hypotenuse of 8 cm and a base length of 4 cm will have a height of 6.9 cm.

Now that you know the height, you can use the formula A = ½ (bh)  to calculate the area of the triangle.

So, A = ½ (bh)

A = ½ (4 x 6.9)

A = ½ (27.6)

A = 13.8 cm2

You can also use the Heron’s formula to find the area of a triangle. Usually, Heron’s formula is applied when you have the three sides provided, if it is not a right angled triangle.

Heron’s formula states that;

Area =√s(s-a)(s-b)(s-c) and in this case, S stands for the semi perimeter of the triangle and a, b, and c are the sides of the triangle.

So, if a is 6 cm, b is 5 cm, and c is 4 cm, it would mean that the perimeter is 6+5+4, which is 15 cm. The semi

perimeter is therefore half the perimeter, which is ½ x 15 = 7.5

Area =√s(s-a)(s-b)(s-c)

Area =√7.5(7.5-6)(7.5-5)(7.5-4)

Area =√7.5(1.5)(2.5)(3.5)

Area =√7.5(13.125)

Area =√98.43

Area = 9.92 cm2

There for area of a triangle that has three sides of 6cm, 5cm, and 4 cm will have an area of 9.92 cm2.