How to Calculate the Area of a Triangle: Formulas and Examples
Area of a Triangle:- The area of a triangle represents the space enclosed by its three sides on a two-dimensional plane. As a closed shape with three sides and three vertices, a triangle’s area is determined by the total region it occupies. The general formula for calculating this area involves half the product of the triangle’s base and height, where the base and height are perpendicular to each other.
The term "area" generally refers to the space inside the boundary of any flat figure and is measured in square units like square meters (m²). Different shapes such as squares, rectangles, circles, and triangles each have specific formulas to calculate their areas. In this article, we’ll explore various formulas for how to find the area of a triangle, along with illustrative examples.
Area of a Triangle
The area of a triangle is a fundamental geometric concept used to determine the space enclosed by its three sides. Common methods to calculate the area include the base-height formula, Heron’s formula, and trigonometric approaches.
Typically, the area of a triangle is calculated using its base and height. For a triangle with base b and height h, the formula is:
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Types of Triangle
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Equilateral Triangle: A triangle with all three sides equal in length and all angles measuring 60°.
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Isosceles Triangle: A triangle with two sides of equal length and two angles that are also equal.
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Scalene Triangle: A triangle with all sides of different lengths and all angles of different measures.
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How to Find the Area of a Triangle
Finding the area of a triangle is a fundamental concept in geometry, involving various methods based on the type of triangle and available measurements. To find the area of a triangle, the most common formula is Area=1/2×base×heigh, which applies to any triangle with a known base and height.
For specific types, like the area of a right triangle, this formula uses the two sides forming the right angle. Additionally, the triangle formula for cases where all three sides are known is Heron’s formula, and for other cases, trigonometry formulas may be applied. Understanding these areas of triangle formulas can simplify calculations across various triangle types. Check out how to find the area of a triangle for different triangles below:-
1. General Formula for Area of a Triangle
The area of any triangle can be calculated by finding half the product of its base and height:
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Base (b): The side of the triangle chosen as the reference line.
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Height (h): The perpendicular distance from the opposite vertex to the base.
2. Area of a Right Triangle
For area of a right triangle (one angle is 90°), it can be calculated as:
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The base and height are the two sides that form the right angle.
3. Equilateral Triangle
An equilateral triangle, with all sides equal and angles of 60°, has an area calculated by:
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This formula uses the length of one side.
4. Isosceles Triangle
For an isosceles triangle, with two equal sides, the area can be found using the base-height formula:
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The height is the perpendicular distance from the vertex opposite the base to the base.
5. Triangle with Three Sides (Heron’s Formula)
When only the three sides are known, Heron’s Formula is used:
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Step 1: Calculate the semi-perimeter s:
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Step 2: Apply the semi-perimeter to the area formula:
This formula does not require knowing the height, making it useful for triangles with three different side lengths.
6. Perimeter of a Triangle
The perimeter, or the total length around the triangle, is the sum of all three sides:
7. Area of Triangle With Two Sides and Included Angle (SAS)
The formula for finding the area of a triangle when two sides and the included angle (SAS) are known can be derived using trigonometry.
Consider a triangle ABC, where AD is the perpendicular dropped from A to the side BC.
Thus,
Examples of Area of a Triangle
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What are Triangle Formulas?
Two essential formulas for triangles are those used to calculate a triangle’s area and perimeter. Different triangle formulas are applied depending on the type of triangle. Below, we’ll explore a detailed overview of the key formulas associated with triangles.
Properties of Triangles Formula
Geometric shapes possess unique characteristics related to their sides and angles, which help in their identification. The key properties of triangles are as follows:
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A triangle consists of three sides, three vertices, and three interior angles.
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The angle sum property of a triangle states that the sum of its interior angles is always 180°. For example, in triangle PQR,
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According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
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The Pythagorean Theorem applies to right-angled triangles, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides:
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The longest side of a triangle is opposite the largest angle.
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The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it.
Read More: Altitude of a Triangle: Definitions, Methods, and Uses
Area of a Triangle FAQs
Q1. What is the formula to find the area of a triangle?
Ans. The area of a triangle is calculated using the formula:
Area=1/2×base×height
Q2. How do you calculate the area of a triangle with three sides?
Ans. When the lengths of all three sides are known, Heron's formula is used:
where s is the semi-perimeter, calculated as a+b+c/2.
Q3. What is the area of an equilateral triangle?
Ans. The area of an equilateral triangle, where all sides are equal, is given by:
Q4. How do you find the area of a right triangle?
Ans. For a right-angled triangle, the area is calculated using:
Area=1/2×base×height
Q5. What is Heron’s formula and when is it used?
Ans. Heron’s formula is used to find the area of a triangle when the lengths of all three sides are known. It is especially useful when the height of the triangle is not provided.
