Area of a Triangle: Formulas, Derivations, and Examples

Area of Triangle

What is the Area of a Triangle? A triangle is a fundamental geometric polygon defined by three sides, three angles, and three vertices. The area represents the total two-dimensional space enclosed within these three sides. The standard formula is calculated as one-half of the base multiplied by the perpendicular height. The base is the side upon which the triangle rests, while the height (or altitude) is the vertical distance from that base to the opposite vertex.

Standard Formula: Area of a triangle = ½ × b × h.

The area of a triangle is always expressed in square units, such as m², cm², or in². This guide is designed to help students master core geometry concepts and apply them effectively in mathematical problem-solving.

What is a Triangle?

Triangles are 2D shapes, and the fundamental Mensuration Formula used to calculate their area is ½ × Base × Height. In geometry, a triangle is a three-sided polygon with three edges and three vertices. A key property of all triangles is that the sum of their internal angles always equals 180°. Triangles are categorized by their side lengths and interior angles, such as Equilateral, Isosceles, and Right-Angled triangles.

Area of Triangle Formula

The area of a triangle is mathematically expressed as ½ × b × h. Depending on the information provided, such as side lengths or angles, you can use specialized formulas to calculate the area for equilateral, isosceles, and right-angled triangles, which are outlined in the sections below.

Type of Triangle	Formula
Area of Equilateral Triangle	(√3/4)a²
Area of Isosceles Triangle	¼ b√4a²−b²
Area of Right-Angled Triangle	½ × Base × Height
Area of Scalene Triangle	√[s(s−a)(s−b)(s−c)

Derivation of Area of Triangle 

When the base (b) and height (h) are known, the area is determined by the formula: A = ½ (b × h) sq. units. For more complex scenarios, you can calculate the area using Heron’s Formula or by applying the Side-Angle-Side (SAS) trigonometry method.

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

For a triangle with side lengths a, b, and c, the area A can be derived using the following algebraic relationship:

16A² = 4a²b² − (c² − a² − b²)²

= 16A² = 4a²b² − (c² − a² − b²)²

= (a² + b² + c²)² − 2(a⁴ + b⁴ + c⁴) = (a² + b² + c²)² − 2(a⁴ + b⁴ + c⁴)

= (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)/2 is the semi-perimeter.

To find the area using Heron’s formula, follow these steps:

Identify side lengths a, b, and c.

Calculate the semi-perimeter: s = (a + b + c) / 2

Apply Heron’s Formula: Area = √s(s-a)(s-b)(s-c)

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

The "SAS" rule stands for "Side, Angle, Side" and is used to find the area of a triangle when the lengths of two sides and the included angle between them are known.

Let a, b, and c represent the three sides of the 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)

By treating triangle ACD as a right-angled triangle and applying trigonometry, we find the following:

⇒ sin(C) = h/b

⇒ h = b sin(C)

Height = h = AD = b sin(C)

Base = Length of BC = a

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

Perimeter of a Triangle

The perimeter of a triangle is the total distance around its boundary, calculated by summing the lengths of all three sides.

Perimeter of a triangle (P) = (a + b + c) units

Where a, b, and c are the individual side lengths.

Area of Triangle Related Questions

Work through these example problems to strengthen your understanding of the area of a triangle.

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

Solution:

We will calculate the area using the standard 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 and a base of 125 cm. Determine the length of the corresponding altitude.

Solution:

Given: Area (A) = 625 cm², Base = 125 cm.

Formula: Area = ½ × Base × Height

∴ 625 = ½ × 125 × H

H = (625 × 2) / 125

⇒ H = 10 cm.

Question 3: Find the area of a scalene triangle with side lengths of 5 cm, 6 cm, and 7 cm.

Solution:

Sides: a = 5 m, b = 6 m, c = 7 m.

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

Using Heron's formula:

Area = √s(s-a)(s-b)(s-c)

= √9(9-5)(9-6)(9-7)

= √9 × 4 × 3 × 2

= 6√6 m²

Question 4: In ∆ABC, angle A = 30°, side b = 4 units, and side c = 6 units. Calculate the area.

Solution:

Area = 1/2 × b × c × sin(A)

= 1/2 × 4 × 6 × sin(30°)

= 12 × 1/2 (Since sin 30° = 0.5)

Area = 6 square units.