Area of Triangle Formula, Definition, Questions, Examples

Area of Triangle

Area of Triangle: A triangle is a closed figure with 3 angles, 3 sides, and 3 vertices. The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height. The base of the triangle is always at the bottom; it is the side that the triangle sits on. The height is the length between the base and the highest point of the triangle. 

Area of triangle = ½ × b × h. 

The area of a triangle is expressed in square units, as m², cm², in², and so on. This article will benefit students in studying mathematics, especially geometry. 

What is a Triangle?

Triangle is a 2-D figure and the Mensuration Formula for a triangle is ½ × Base × Height In Geometry, a triangle is a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180°. There are different types of triangles in mathematics namely Equilateral Triangle, Isosceles Triangle, and Right-Angled Triangle that are classified on the basis of their sides and angles. 

Area of Triangle Formula

As defined Area of triangle is ½ × base × height or it can be written as ½ x b x h. However, there are different types of triangles in mathematics that are classified on the basis of their sides and angles. The area of triangle formulas for all the different types of triangles like the equilateral triangle, right-angled triangle, and isosceles triangle are given below.

Derivation of Area of Triangle 

For a given triangle, where the base of the triangle is b and the height is h, the area of the triangle can be determined by the recipe, such as; A = ½ (b × h) sq. units. The Area of Triangle can be calculated using various formulas including Heron's Formula, and Area of a Triangle Given Two Sides and the Included Angle (SAS)

Area of Triangle with Three Sides (Heron’s Formula)

Given triangle sides, a, b, c, area A satisfies

16A²=4𝑎²𝑏²−(𝑐²−𝑎²−𝑏²)²

= 16A²=4a²b²−(c²−a²−b²)²

=(𝑎²+𝑏²+𝑐²)²−2(a^4+𝑏^4+𝑐^4)=(a²+b²+c²)2−2(a^4+b^4+c^4)

=(𝑎+𝑏+𝑐)(−𝑎+𝑏+𝑐)(𝑎−𝑏+𝑐)(𝑎+𝑏−𝑐)=(a+b+c)(−a+b+c)(a−b+c)(a+b−c)

=16𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)=16s(s−a)(s−b)(s−c) where 𝑠=(𝑎+𝑏+𝑐)/2s=(a+b+c)/2 is the semiperimeter.

Use Heron's formula with

sides a, b and c

s=a+b+c / 2

Area of Triangle= √s(s-a)(s-b)(s-c)

Area of a Triangle Given Two Sides and the Included Angle (SAS)

"SAS" which means "Side, Angle, Side", is the property of a triangle whose 2 sides and the angle between these sides is given.

Consider a,b, and c are the different sides of a triangle.

1. When sides 'b' and 'c' and included angle A is known, the area of the triangle is: 1/2 × bc × sin(A)
2. When sides 'b' and 'a' and included angle B is known, the area of the triangle is: 1/2 × ab × sin(C)
3. When sides 'a' and 'c' and included angle C is known, the area of the triangle is: 1/2 × ac × sin(B)

Triangle ACD is a right-angle triangle. Using trigonometry we get, 

⇒ sin(C) = h/b

⇒ h = b sin(C)

Height = h = AD = b sin(C) 

Base = Length of BC = a (as shown in figure above)

Therefore, area of a triangle ABC = (1/2)(base)(height) = (1/2)(length of BC)(height) = (1/2)(a)(b sin(C))

Perimeter of a Triangle

The perimeter of a triangle is the distance covered around the triangle and is calculated by adding all three sides of a triangle.

The perimeter of a triangle = P = (a + b + c) units

where a, b and c are the sides of the triangle.

Area of Triangle Related Questions

Practice some questions of the Area of Triangle to understand the concept more clearly. 

Question 1: Find the area of a triangle with a base of 20 cm and a height of 10 cm.

Solution:

Let us find the area using the area of triangle formula:

Area of triangle = ½ × b × h

A = ½ × 20 × 10

A = ½ × 200

Therefore, the area of the triangle (A) = 100 cm²

Question 2: A triangle has an area of 625 cm². One of its sides is given as 125 cm, and then determine the length of the altitude that is dropped on that particular side from the opposite vertex.

Solution: 

Area of triangle (A)= 625cm². Length of base = 123 cm.

We know,  Area of triangle  = ½ x Base x Height

∴ 625 = ½ x 125 x H

H = 1250/125

⇒ H = 10 cm.

Question 3:  The length of three sides of a scalene triangle are 5cm, 6cm and 7cm, and find the area of the triangle.

Solution: 

Three sides of the triangle are 5 m, 6 m and 7 m.

∴semi-perimeter = (5 + 6 + 7)/2 =  9cm.

Putting the values in the formula,

√s(s-a)(s-b)(s-c)

√9(9-5)(9-6)(9-7)

√9x4x3x2

6√6 m²

Question 4: In ∆ABC, angle A = 30°, side 'b' = 4 units, side 'c' = 6 units.

Solution: 

Area (∆ABC) = 1/2 × bc × sin A

= 1/2 × 4 × 6 × sin 30º

= 12 × 1/2 (since sin 30º = 1/2)

Area = 6 square units.