How to Find the Area of a Triangle: All Formulas Explained

The area of a triangle is the amount of space enclosed within its three sides. Whether you are a student learning geometry, a professional in construction or design, or just someone who wants to understand basic math, knowing how to calculate triangle area is essential.

Formula 1: Base and Height (Most Common)

Area = ½ × base × height

The height (or altitude) is the perpendicular distance from the base to the opposite vertex.

Example:

A triangle with base 8 cm and height 5 cm:

Area = ½ × 8 × 5 = 20 cm²

Why It Works

A triangle is exactly half of a parallelogram (or rectangle) with the same base and height. This is why the formula is ½ × base × height.

Formula 2: Side-Angle-Side (SAS)

When you know two sides and the included angle:

Area = ½ × a × b × sin(C)

Where C is the angle between sides a and b.

Example:

Two sides measure 7 cm and 9 cm with an included angle of 50°:

Area = ½ × 7 × 9 × sin(50°)

= 31.5 × 0.766

= 24.1 cm²

Formula 3: Heron's Formula (SSS — Three Sides)

When you know all three sides but not the height or any angle:

Step 1: Calculate semi-perimeter: s = (a + b + c) ÷ 2

Step 2: Area = √[s(s − a)(s − b)(s − c)]

Example:

A triangle with sides 5, 6, and 7 units:

s = (5 + 6 + 7) ÷ 2 = 9

Area = √[9(9 − 5)(9 − 6)(9 − 7)]

= √[9 × 4 × 3 × 2]

= √216

= 14.7 square units

Formula 4: Equilateral Triangle

For a triangle where all sides are equal:

Area = (√3 ÷ 4) × a²

Where a is the length of any side.

Example:

An equilateral triangle with side length 6:

Area = (1.732 ÷ 4) × 36

= 0.433 × 36

= 15.59 square units

Formula 5: Right Triangle

For a triangle with a 90-degree angle:

Area = ½ × leg₁ × leg₂

The two legs are perpendicular to each other, so one is the base and the other is the height.

Example:

A right triangle with legs 3 and 4:

Area = ½ × 3 × 4 = 6 square units

Comparison Table

Real-World Applications

Construction:

Calculate gable end area for siding: measure base (width of house) and height (roof peak to attic floor).

Land Surveying:

Surveyors break irregular plots into triangles and sum the areas using coordinates or SAS formula.

Art and Design:

Triangular elements in design require area calculations for material estimation.

Agriculture:

Calculating the area of triangular fields for seed and fertilizer distribution.

FAQ: Area of a Triangle

The basic formula is ½ × base × height. There are also formulas using two sides and an angle (SAS) or three sides (Heron's formula).

Use Heron's formula to find the area, then rearrange: height = (2 × Area) ÷ base.

Yes, but you need the perpendicular height, not the slant height. For obtuse triangles, the height may fall outside the triangle.

Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter: (a + b + c) ÷ 2. Named after Hero of Alexandria.

Use the shoelace formula: Area = ½ × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Square units: cm², m², in², ft², etc. The units of the input sides determine the area units.