Area of a Scalene Triangle Calculator

Enter all three side lengths — get area using Heron's formula, perimeter, and step-by-step working instantly.

Unit:
Side a (longest)
cm
Side b
cm
Side c
cm

Enter all three side lengths to calculate area.

Triangle Diagram

Heron's Formula — Key Formulas

Semi-perimeter

s = (a + b + c) / 2

Half the perimeter — the key intermediate value in Heron's formula.

Heron's Formula

A = √(s(s−a)(s−b)(s−c))

Area from three side lengths — no angles needed.

Perimeter

P = a + b + c

Sum of all three sides gives the perimeter.

Height

h = 2A ÷ a

Height from any base a, once area A is known.

Worked Example

Scalene triangle with sides 7 cm, 5 cm, and 4 cm:

Given: a = 7 cm, b = 5 cm, c = 4 cm
Step 1: P = 7 + 5 + 4 = 16 cm
Step 2: s = 16 ÷ 2 = 8 cm
Step 3: s−a = 1, s−b = 3, s−c = 4
Step 4: A = √(8 × 1 × 3 × 4) = √96
Answer: A ≈ 9.80 cm²

Common Scalene Triangle Measurements

Side a Side b Side c Area Perimeter
7 cm 5 cm 4 cm 9.80 cm² 16 cm
10 cm 8 cm 6 cm 24.0 cm² 24 cm
13 cm 12 cm 5 cm 30.0 cm² 30 cm
15 m 12 m 9 m 54.0 m² 36 m
25 m 20 m 15 m 150.0 m² 60 m

Real-World Applications

🏗️

Construction

Irregular plot measurement, roof trusses, and land boundaries often involve scalene triangles.

🌍

Surveying

Land area measurement for irregular plots and fields uses triangulation with Heron's formula.

🛶

Navigation

Triangulation for position finding uses scalene triangles formed by landmarks and observer.

📐

Engineering

Structural analysis of triangular frames and truss members with unequal dimensions.

🎨

Art & Design

Triangular composition in graphic design and textile patterns with asymmetric triangle shapes.

Frequently Asked Questions

What is Heron's Formula?
Heron's formula calculates triangle area from side lengths alone: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter. Named after Heron of Alexandria (~60 CE), it works for any triangle without needing angles.
Why is it called a scalene triangle?
Scalene means "unequal" in Greek. A scalene triangle has all three sides of different lengths, unlike isosceles (2 equal sides) or equilateral (3 equal sides). All angles are also different from each other.
When is a triangle invalid?
A triangle is invalid when one side is ≥ sum of the other two (the triangle inequality). For example, sides 1, 2, 10 cannot form a triangle because 10 > 1+2 = 3. This calculator checks all three inequalities and shows an error if violated.
Can Heron's formula be used for all triangle types?
Yes — it works for scalene, isosceles, equilateral, right-angled, and all valid triangles. It only needs the three side lengths. For a right triangle with legs 3 and 4 and hypotenuse 5: s = 6, A = √(6×3×2×1) = √36 = 6 cm², which matches ½×3×4 = 6 cm².
How do you find the height of a scalene triangle?
If area A and base a are known, height h = 2A ÷ a. A scalene triangle has three different heights — one for each base. The largest height corresponds to the shortest base, and the smallest height corresponds to the longest base.

Related Area Calculators

Circle Semicircle Square Rectangle Parallelogram Rhombus Trapezium Scalene Triangle Right Angle Triangle Equilateral Triangle Isosceles Triangle Regular Hexagon Sector & Segment