The simplest way to find the area of a triangle is to take half of its base times the height. However, there are various other formulas to find the area of a triangle, depending upon the information we have. Furthermore, it is possible to calculate the area of a triangle just from the sides and angles of a triangle without knowing its height. In this article, we will see different methods of how to calculate area of triangle.
3 Methods to Calculate Area of Triangle
- Method 1: Using base and height
- Method 2: Using side lengths
- Method 3: Using trigonometry
Method 1: Using base and height
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By finding the base and height of the triangle
In a triangle, the base is one side of the triangle and height is the measure of the tallest point on a triangle. Moreover, we can find it by drawing a line perpendicular from the base to the opposite vertex. Such as, you have a triangle whose base is 5cm long and its height is 3cm long.
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Form a formula for the area of the triangle
So, the formula here is \( Area = \frac{1}{2}(bh)\). Also, here b is the base of the triangle and h is the height of the triangle.
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Now put the values in the formula
Firstly, multiple the value of base with height then multiplies the product by \(\frac{1}{2}\). In addition, this will give the area of the triangle in square units. In the above example the solution will be:
\( Area = \frac{1}{2}(bh)\)
\( Area = \frac{1}{2}(5)(3)\)
\( Area = \frac{1}{2}(15)\)
\( Area = 7.5cm^{2}\)
Hence, the area of a triangle is \(7.5cm^{2}\)
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How to find the area of a right triangle
In a right triangle, two sides are perpendicular to each other. So, one will be the height of the triangle and the other will be the base of the triangle. Thus, even, if the height or base is unstated, you are given them if you know the side lengths. So, you can use the \( Area = \frac{1}{2}(bh)\) formula to find the area.
Moreover, you can also use this formula if you know one side, plus the length of the hypotenuse. Furthermore, the hypotenuse is the longest side of the right triangle and is opposite to the right triangle using the Pythagorean Theorem (\(a^{2} + b^{2} = c^{2}\).
Suppose, if the hypotenuse of a triangle is side c, and the height and base would be the other two sides (a and b). Moreover, if you knew that the hypotenuse is 5cm and the base 4cm. so you can find the height by Pythagorean Theorem.
\(a^{2} + b^{2} = c^{2}\)
\(a^{2} + 4^{2} = 5{2}\)
\(a^{2} + 16 = 25\)
\(a^{2} + 16 – 16 = 25 – 16\)
\(a^{2} = 9\)
\(a = 3\)
So put the value in the area formula, and use two perpendicular side (a and b) to substitute the base and height:
\( Area = \frac{1}{2}(bh)\)
\( Area = \frac{1}{2}(4)(3)\)
\( Area = \frac{1}{2}(12)\)
\( Area = 6\)
So, the area of a triangle is \(6cm^{2}\).
Method 2: Using side lengths
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Calculate the semi-perimeter of the triangle
The semi-perimeter is equal to the half of its perimeter. So, to find the semi-perimeter we need to find its perimeter of the by adding up the length of its three sides then multiplying the product by \(\frac{1}{2}\).
E.g. suppose if a triangle has three sides that are 5cm, 4cm and 3cm long, then the semi-perimeter will be:
\(s = \frac{1}{2}(3 + 4 + 5)\)
\(s = \frac{1}{2}(12)\) = 6
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Apply Heron’s formula
Heron’s formula is \(Area = \sqrt{s(s-a)(s-b)(s-c)}\). Here ‘s’ is the semi-perimeter of a triangle and a, b, and c are the side lengths of a triangle.
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Put values in the formula and calculate it
While putting the values in the formula, make sure you substitute the semi-perimeter for each instance of s in the formula
\(Area = \sqrt{s(s-a)(s-b)(s-c)}\)
\(Area = \sqrt{6(6-3)(6-4)(6-5)}\)
\(Area = \sqrt{6(3)(2)(1)}\)
\(Area = \sqrt{6(6)}\)
\(Area = \sqrt{36}\) = 6
Therefore, the area of the triangle is \(6cm^{2}\).
Method 3: Using trigonometry
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Find the length of two adjacent sides and included angle
Firstly, the adjacent sides refer to those sides that meet at the vortex. Also, the angle is the angle between these two sides.
E.g. a triangle with two adjacent sides that measure 150cm and 231cm in length. And the angle between them is 123 degrees.
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Now use the trigonometry formula for the area of a triangle
The trigonometry formula is \(Area = \frac{bc}{2}sin A\). Here b and c are the adjacent sides and A is the angle between them.
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Put values in the formula
Firstly, validate that you substitute for the variables b and c. Also, multiply their values and divide them by 2.
E.g. \(Area = \frac{bc}{2}sin A\)
\(Area = \frac{(150)(231)}{2}sin A\)
\(Area = \frac{34650}{2}sin A\)
\(Area = 17325sin A\)
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Plug the values of sine in formula and multiply them
Calculate the value of sine using scientific calculator. After that, put the value into the formula.
The sine of 123 degree angle is 0.83867.
\(Area = 17325sin A\)
\(Area = 17325(0.83687)\) =14529.96
Hence the area of the triangle is \(14530cm^{2}\).
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