Triangle Calculator
Calculate area, perimeter, angles, and other properties of any triangle
Calculation Results
Enter triangle measurements to calculate all properties.
Type:
Area (A): square units
Perimeter (P): units
Semi-perimeter (s): units
Side a: units
Side b: units
Side c: units
Angle α: °
Angle β: °
Angle γ: °
Height ha: units
Height hb: units
Height hc: units
Inradius (r): units
Circumradius (R): units
Triangle Diagram
Triangle Formulas
Key Properties
- Area (A): Space enclosed by the triangle
- Perimeter (P): Sum of all three sides
- Semi-perimeter (s): Half of the perimeter
- Angles (α, β, γ): Three interior angles (always sum to 180°)
- Heights (ha, hb, hc): Perpendicular distances from vertices to opposite sides
- Inradius (r): Radius of inscribed circle
- Circumradius (R): Radius of circumscribed circle
Calculation Formulas
- Area (Heron's formula): A = √[s(s-a)(s-b)(s-c)]
- Area (base-height): A = ½ × base × height
- Area (SAS): A = ½ × a × b × sin(γ)
- Perimeter: P = a + b + c
- Law of Cosines: c² = a² + b² - 2ab cos(γ)
- Law of Sines: a/sin(α) = b/sin(β) = c/sin(γ) = 2R
- Inradius: r = A/s
- Circumradius: R = abc/(4A)
- Height: ha = 2A/a
Where: a, b, c = side lengths, α, β, γ = opposite angles, s = semi-perimeter
Example Calculation
For a triangle with:
Side a = 5 units, Side b = 6 units, Side c = 7 units.Calculations:
- Perimeter = 5 + 6 + 7 = 18 units
- Semi-perimeter = 18 / 2 = 9 units
- Area = √[9(9-5)(9-6)(9-7)] = √[9×4×3×2] ≈ 14.7 units²
- Angle α ≈ 44.4° (using Law of Cosines)
- Angle β ≈ 57.1°
- Angle γ ≈ 78.5°
- Height ha ≈ 5.88 units
- Inradius ≈ 1.63 units
- Circumradius ≈ 4.04 units
About Triangles
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. Triangles can be classified by their sides (equilateral, isosceles, scalene) or by their angles (acute, right, obtuse).
Real-World Applications
- Engineering: Truss bridges and roof structures
- Navigation: Triangulation in GPS systems
- Art & Design: Composition and visual balance
- Physics: Force diagrams and vector analysis
Types of Triangles
- By Sides:
- Equilateral - All sides equal, all angles 60°
- Isosceles - Two sides equal, two angles equal
- Scalene - All sides and angles different
- By Angles:
- Acute - All angles less than 90°
- Right - One 90° angle
- Obtuse - One angle greater than 90°
Triangle Components
Equilateral
All sides equal
Isosceles
Two sides equal
Scalene
All sides different
Right
One 90° angle
Acute
All angles less than 90°
Obtuse
One angle greater than 90°
How to Use the Triangle Calculator
The Triangle Calculator helps compute various properties based on input values. Here's how to use it:
1. Select Calculation Method
Choose what measurements you know:
- Three Sides (SSS) - Enter lengths of all three sides
- Two Sides & Included Angle (SAS) - Enter two sides and the angle between them
- Two Angles & Side (ASA or AAS) - Enter two angles and one side
- Right Triangle - Enter two sides of a right triangle
- Base & Height - Enter base length and height
- Vertex Coordinates - Enter (x,y) coordinates of all three vertices
2. Enter Your Values
Input positive numbers in the appropriate fields:
Example 1 (SSS): a=5, b=6, c=7
Example 2 (SAS): a=5, b=6, γ=60°
Example 3 (Right): a=3, b=4 (hypotenuse=5)
3. Get Results
The calculator will compute all properties:
- Triangle type (equilateral, isosceles, scalene, right, acute, obtuse)
- Area and perimeter
- All side lengths and angles
- Heights, inradius, and circumradius
Practical Applications
- Calculate land area for triangular plots
- Determine materials needed for triangular structures
- Solve trigonometry problems
- Analyze forces in physics problems
Triangle Formulas
| Parameter | Formula | Description |
|---|---|---|
| Area (Heron's) | √[s(s-a)(s-b)(s-c)] | When all three sides are known |
| Area (base-height) | ½ × base × height | When base and height are known |
| Area (SAS) | ½ × a × b × sin(γ) | When two sides and included angle are known |
| Perimeter | a + b + c | Sum of all three sides |
| Law of Cosines | c² = a² + b² - 2ab cos(γ) | Relates sides and angles |
| Law of Sines | a/sin(α) = b/sin(β) = c/sin(γ) = 2R | Relates sides and opposite angles |
| Pythagorean Theorem | a² + b² = c² | For right triangles only |
| Inradius | A/s | Radius of inscribed circle |
| Circumradius | abc/(4A) | Radius of circumscribed circle |
Frequently Asked Questions
Use Heron's formula: First calculate the semi-perimeter s = (a+b+c)/2, then Area = √[s(s-a)(s-b)(s-c)].
The sum of interior angles in any triangle is always 180°.
Use the Pythagorean theorem: c² = a² + b² where c is the hypotenuse. For non-right triangles, use the Law of Cosines.