The world of geometry is filled with a multitude of shapes, each possessing its unique properties. Among these shapes, the triangle holds a prominent place. It is the simplest polygon, yet it forms the basis of more complex geometric forms. One of the key aspects of a triangle is its area. In this comprehensive guide, we will delve into the concept of the area of a triangle and understand the various methods on how to find the area of a triangle.
What is the Area of a Triangle?
The area of a triangle refers to the total space enclosed within the three sides of the triangle. It’s the region that the triangle covers in a two-dimensional plane. The area is always expressed in square units such as square meters (m²), square centimeters (cm²), square inches (in²), etc.
Area of a Triangle Formula
The most basic formula to find the area of a triangle involves its base and height. It is given by:
A = 1/2 × b × h
Here, ‘A’ denotes the area, ‘b’ the base, and ‘h’ the height of the triangle. This formula is applicable to all types of triangles, including scalene, isosceles, and equilateral triangles. It’s important to remember that in this formula, the base and the height are perpendicular to each other.
How to Find the Area of a Triangle?
Finding the area of a triangle involves various techniques, depending on the given dimensions. Here, we will explore several ways to calculate the area of a triangle.
Area of a Right-Angled Triangle
A right-angled triangle is a unique type of triangle that has one angle measuring 90 degrees. In this triangle, the height is the length of the perpendicular side. The formula to find the area of a right-angled triangle is:
A = 1/2 × Base × Height (Perpendicular distance)
Area of an Equilateral Triangle
An equilateral triangle is a triangle where all the sides are of equal length. The formula to find the area of an equilateral triangle is:
A = (√3)/4 × side²
Area of an Isosceles Triangle
An isosceles triangle has two of its sides equal, and the angles opposite the equal sides are also equal. The formula to find the area of an isosceles triangle is:
A = 1/4 × b × √(4a² – b²)
Here, ‘b’ is the base and ‘a’ is the length of one of the equal sides.
Perimeter of a Triangle
The perimeter of a triangle is the total distance around the triangle. It is calculated by adding all three sides of the triangle.
P = a + b + c
Here, ‘a’, ‘b’, and ‘c’ represent the sides of the triangle.
Area of Triangle with 3 Sides (Heron’s Formula)
When we know the lengths of all three sides of a triangle, we can use Heron’s formula to calculate the area. Heron’s formula includes two steps:
- Find the semi-perimeter of the triangle, which is half of the sum of the lengths of all sides.
- Apply the semi-perimeter value in the main formula, known as Heron’s formula.
A = √[s(s – a)(s – b)(s – c)]
Here, ‘s’ represents the semi-perimeter of the triangle, which is computed as s = (a+b+c) / 2.
Area of a Triangle Given Two Sides and the Included Angle (SAS)
When we know two sides of a triangle and the angle included between them, we can use the following formulas to calculate its area:
Area (∆ABC) = 1/2 × bc × sin(A) Area (∆ABC) = 1/2 × ab × sin(C) Area (∆ABC) = 1/2 × ca × sin(B)
For those exploring the fascinating realm of triangle geometry and area calculations, our angles page provides a complementary insight, offering essential knowledge on the measurement and relationships of angles that enhances the understanding and application of formulas in determining the area of a triangle.
Solved Examples on Area of a Triangle
Let us solve a few examples to understand the application of these formulas better.
Example 1: Find the area of a right-angled triangle with a base of 7 cm and a height of 8 cm.
A = 1/2 × b × h = 1/2 × 7 cm × 8 cm = 28 cm²
Example 2: Find the area of an equilateral triangle with a side of 5 cm.
A = (√3)/4 × side² = (√3)/4 × (5 cm)² = 10.83 cm²
Example 3: Find the area of a triangle with sides of 5 cm, 6 cm, and 7 cm using Heron’s formula.
Semi-perimeter, s = (5 cm + 6 cm + 7 cm) / 2 = 9 cm
A = √[s(s – a)(s – b)(s – c)] = √[9(9 – 5)(9 – 6)(9 – 7)] = 14.7 cm²
How Does Area of Triangle Formula Works?
The basic formula for the area of a triangle originates from the formula for the area of a rectangle, which is base times height (b × h). When a rectangle is divided into two congruent triangles by a diagonal, each triangle occupies half the area of the rectangle. Thus, the area of a triangle is half the product of its base and height.
How to Find the Area of a Triangle Using Vectors?
When vectors ‘u’ and ‘v’ form a triangle in space, the area of this triangle is equal to half of the magnitude of the cross product of these two vectors, computed as A = 1/2 |u × v|.
What is the Area when Two Sides of a Triangle and Included Angle are Given?
When two sides of a triangle and the angle included between them are known, the area is calculated as half the product of the two sides and the sine of the included angle.
How to Find the Area of a Triangle Given Three Sides?
When the lengths of all three sides of a triangle are known, Heron’s formula is used to calculate the area. The formula involves the semi-perimeter of the triangle and the lengths of the sides.
How to Calculate the Area of a Triangle?
The area of a triangle can be calculated by using the formula A = 1/2 × b × h, where ‘b’ is the base and ‘h’ is the height of the triangle.
How to Find the Area and Perimeter of a Triangle?
The area of a triangle is calculated using the formula A = 1/2 × b × h. The perimeter of a triangle is calculated by adding the lengths of all the sides.
How to Find the Base and Height of a Triangle?
When the area of the triangle and either the base or the height is given, the unknown dimension can be calculated using the formula for the area of a triangle. If the area ‘A’ and the base ‘b’ are known, the height ‘h’ is calculated as h = 2A/b. Similarly, if the height ‘h’ and the area ‘A’ are known, the base ‘b’ is calculated as b = 2A/h.