Triangle Calculator
Calculate the area and perimeter of a triangle using Heron's formula. Determine triangle type and validate triangle inequality.
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💡 Heron's Formula: Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter.
What is a Triangle?
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. The sum of the interior angles of a triangle always equals 180 degrees.
Types of Triangles by Sides
| Type | Definition | Properties |
|---|---|---|
| Equilateral | All three sides equal | All angles are 60° |
| Isosceles | Two sides equal | Two angles equal |
| Scalene | All sides different | All angles different |
Types of Triangles by Angles
- Acute Triangle: All angles less than 90°.
- Right Triangle: One angle exactly 90°.
- Obtuse Triangle: One angle greater than 90°.
Heron's Formula for Area
When you know all three sides of a triangle, you can calculate the area using Heron's Formula:
Step 1: Calculate semi-perimeter: s = (a + b + c) / 2
Step 2: Calculate area: Area = √[s × (s-a) × (s-b) × (s-c)]
Step 2: Calculate area: Area = √[s × (s-a) × (s-b) × (s-c)]
The Triangle Inequality Theorem
Not any three lengths can form a triangle. The Triangle Inequality Theorem states that the sum of the lengths of any two sides must be GREATER than the length of the third side:
a + b > c
a + c > b
b + c > a
a + c > b
b + c > a
⚠️ Note: If these conditions are not met, the three lengths cannot form a triangle.
The Pythagorean Theorem
For right triangles (triangles with a 90° angle), the relationship between sides is:
a² + b² = c²
Where c is the hypotenuse (longest side, opposite the right angle)
Where c is the hypotenuse (longest side, opposite the right angle)
Real-World Applications
- Construction: Triangles are the strongest structural shape. Trusses in bridges and roofs use triangles.
- Navigation: Triangulation is used in GPS and surveying to determine locations.
- Computer Graphics: 3D models are built using millions of triangles (polygons).
- Architecture: The pyramids of Egypt are triangular structures.
References
- Khan Academy. "Triangle Properties." 2024.
- Math is Fun. "Heron's Formula." 2024.