Formulas to Calculate the Area of Triangles
Suppose, you need to find out the expense of installing carpet over a triangular floor or you need to lay an extra layer of cheese on a triangular piece of pizza. In order to do so, you need to cover the surface of the triangular floor or pizza. The measurement of this triangular surface is called the area of the triangle.
Basically, the area of a triangle can be defined as the total space occupied by the 3 sides of a triangle in a 2D plane. The general formula for the area of a triangle is equal to half the product of its Height and Base, i.e., A = 1/2 × b × h. This formula is applicable to all types of triangles.
Note: Base & Height of a triangle are perpendicular to each other.
What is the Area of a Triangle?
What is an Area?
The area is the measurement or the quantitative value of the two-dimensional space occupied by an object. In other words, it is the size of the surface of any 2D figure like rectangle, square, triangle, circle, etc.
What is a Triangle?
A triangle is a closed 2D figure having three sides, three vertices, and three angles. It is the simplest form of Polygon. A triangle can be formed by joining any three dots such that the line segments connect each other end to end. Three line segments connecting the dots are the sides of the triangle, the point of intersection of two lines is known as vertex and the space between them is called an angle. It is also important to know that the sum of all the interior angles of a triangle is always 180 degrees.
How to Find the Area of a Triangle?
Given below are the methods to find the area of Triangle with pieces of information given. We can find the area of a triangle if :
The length of the two sides of a triangle is given.
The length of all the three sides of a triangle is given.
The length of any two sides of a right triangle is given.
The length of one side of the equilateral triangle is given.
The vertices of a triangle on the plane coordinate are given.
The vectors of the Triangle are given.
When the Three Sides of a Triangle are given
If the length of three sides of a triangle is given then how to calculate the area of a triangle by using Heron's Formula.
Heron's Formula
It states that the area of the triangle of sides a, b, and c is equal to:
\[A=\sqrt{s(s-a)(s-b)(s-c)}\]
Where 's' is the semi-perimeter of the triangle. This formula is used for triangles whose angles are not given and the calculation of height is complicated. Thus, Heron's formula helps us to find the area of a triangle having irregular sides.
Calculation of Semi-Perimeter
The measurement of the semi-perimeter of a triangle having sides a,b and c is important to find the area of the triangle using Heron's Formula.
Semi-perimeter, \[s = \frac{(a + b + c)}{2}\]
When any two sides of a Right-Angled Triangle are given
A right-angled triangle is a special triangle used as a base of trigonometry, calculus, etc.
One of the three sides of a right-angled triangle itself is height. Thus, there is no need for the projection of a perpendicular base from a vertex. Moreover, even if the length of height is not given instead, the length of any two sides is given then the length of height can be found by using the Pythagoras theorem.
Area of a right-angled triangle = 1/2 x base x height
The height of a right-angled triangle can be calculated by using the Pythagoras theorem that states: The square of the length hypotenuse (the longest side of a right triangle) is equal to the sum of the square of the other two sides (base and perpendicular)
Here, a = base, b = height and c = hypotenuse.
When the side of the Equilateral Triangle is given
An equilateral triangle is a triangle with all sides equal. To formula to find the area of an equilateral triangle is given below:
\[A = \sqrt{\frac{3}{4}}{s^{2}}\]
Here, ‘s’ is the length of the sides of an equilateral triangle.
When the Vertices of a Triangle on the Coordinate Plane are given
When three vertices of a triangle on the coordinate plane are known, then we can do the following check:
If the Triangle forms a right-angled triangle, then the basic formula of the Triangle can be used, which is half of the product of height and base.
If it's not a right triangle, then Heron's formula can be used after calculating the semi-perimeter by using the sides of the Triangle.
Note: It's not necessary for the Triangle to be right-angled to use the Pythagoras theorem. If we are able to find the height of the Triangle in the graph, then we can calculate the area. Similarly, we can use Heron's formula for any triangle whose three sides are known.
Conclusion
The area of a triangle is the region enclosed within the sides of the triangle. As discussed above, the area can vary from one triangle to another based on the length of the sides and the angles enclosed in it. The area of a triangle is expressed in square units, like, m2, cm2.
FAQs on Area of a Triangle
1. What is the Area of a Triangle?
Basically the area of a triangle can be defined as the total space occupied by the 3 sides of a triangle in a 2D plane. The general formula for the area of a triangle is equal to half the product of its Height and Base, i.e., A = 1/2 × b × h.
2. How to find the area of a triangle if you know three sides?
If you know the three sides of a triangle and you want to find the area of a triangle, then we can use Heron's Formula:
For an equilateral triangle, the formula is:
\[A=\sqrt{s(s-a)(s-b)(s-c)}\]
For an Isosceles triangle, the area formula is:
A = (base * height) / 2.
3. What is the Base of a Triangle?
The base of a triangle can be any selected side of a triangle, usually the bottom of a triangle is taken as base. For an isosceles triangle, the unequal side of the triangle is taken as a base. Even for the right-angled triangle one of the sides containing the right angle (90 degrees) can be taken as base.
4. What is the Height of a Triangle?
If the sides of an isosceles or equilateral triangle are given then by Pythagoras theorem, we can find the height of the triangle.