Triangle Calculator | Sides, Angles, Area & Perimeter
Solve any triangle instantly. Calculate all sides, angles, area, and perimeter using SSS, SAS, ASA, AAS, or right triangle inputs. Free online triangle calculator.
Enter Values
Fill in the fields and press Calculate to see instant results.
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Triangle Calculator: Solve Any Triangle Instantly
The Triangle Calculator is the most versatile geometry tool you will find online. Whether you are a student tackling high-school geometry, an architect designing a roof pitch, a land surveyor computing plot boundaries, or an engineer verifying structural dimensions, this free calculator gives you all sides, all angles, area, and perimeter in one step — no protractor or trigonometry tables required.
A triangle is defined by three sides (a, b, c) and three interior angles (A, B, C), with the constraint that A + B + C = 180°. Knowing just three of these six values (at least one of which must be a side length) is sufficient to determine the rest. Our calculator supports the five most common input combinations:
- SSS — three sides known (uses Heron's formula for area and Law of Cosines for angles)
- SAS — two sides and the included angle (Law of Cosines)
- ASA / AAS — two angles and one side (Law of Sines)
- Right Triangle (two legs) — applies Pythagorean theorem + inverse tangent
- Right Triangle (leg + hypotenuse) — applies Pythagorean theorem + inverse sine
Enter your known values, select the appropriate case, and press Calculate — the results appear instantly with full precision.
💡 Quick Tip:If you only know the base and height, use Area = ½ × base × height. If you know all three sides, let Heron's formula do the work automatically.
When to Use This Triangle Calculator
This calculator is ideal whenever you need to solve a triangle and lack the time or tools for manual trigonometry:
📐 Geometry Homework
Verify your work on SSS, SAS, or ASA problems. Check angles sum to 180° and confirm area calculations.
🏗️ Construction & Architecture
Calculate roof pitch angles, rafter lengths, and triangular gable areas from two known dimensions.
🌍 Land Surveying
Compute plot areas and bearings using triangulation from measured distances and angles.
🎮 Game & Graphic Design
Determine polygon vertex positions, collision box angles, and 2D sprite dimensions.
⚙️ Engineering
Solve truss member forces, calculate bracket angles, and verify machined part dimensions.
🔭 Navigation & Astronomy
Use triangulation to determine distances to distant objects or calculate celestial angular separations.
Not ideal for: Spherical triangles (geodesy, celestial mechanics) where sides are arcs — use spherical trigonometry formulas instead.
Triangle Formulas Explained
Heron's Formula (SSS — Three Sides)
When all three sides are known, compute the semi-perimeter first, then apply Heron's formula:
s = (a + b + c) / 2
Area = √[s(s − a)(s − b)(s − c)]
Law of Cosines (SAS or SSS → Angles)
c² = a² + b² − 2ab · cos(C)
Rearranged: cos(C) = (a² + b² − c²) / (2ab)
Law of Sines (AAS / ASA)
a / sin(A) = b / sin(B) = c / sin(C)
Pythagorean Theorem (Right Triangles)
c² = a² + b² → c = √(a² + b²)
Variable Definitions
| Variable | Description | Units |
|---|---|---|
| a, b, c | Side lengths of the triangle | Any consistent unit (m, cm, ft, in) |
| A, B, C | Interior angles opposite to sides a, b, c | Degrees (°) |
| s | Semi-perimeter = (a + b + c) / 2 | Same unit as sides |
| Area | Enclosed surface area | Square units (m², ft², etc.) |
| Perimeter | Sum of all three sides: a + b + c | Same unit as sides |
Step-by-Step Calculation Guide
Example Type: SSS (Three Sides Known)
Identify your sides: a = 3, b = 4, c = 5
Compute semi-perimeter: s = (3 + 4 + 5) / 2 = 6
Apply Heron's formula: Area = √[6 × (6−3) × (6−4) × (6−5)] = √[6 × 3 × 2 × 1] = √36 = 6 square units
Compute perimeter: P = 3 + 4 + 5 = 12 units
Find angles using Law of Cosines: cos(C) = (9 + 16 − 25) / (2 × 3 × 4) = 0 → C = 90°. Triangle is a right triangle!
5 Worked Examples
Example 1: Classic 3-4-5 Right Triangle (SSS)
Scenario: A carpenter wants to ensure a corner is perfectly square. They measure sides of 3 ft, 4 ft, and 5 ft.
| Input | Value |
|---|---|
| Side a | 3 ft |
| Side b | 4 ft |
| Side c | 5 ft |
Results: Area = 6 ft², Perimeter = 12 ft, Angle C = 90° (confirmed right angle). The corner is perfectly square.
Example 2: Equilateral Triangle (SSS)
Scenario: An architect is designing a triangular skylight with all sides 8 m.
| Input | Value |
|---|---|
| a = b = c | 8 m |
Solution: s = 12, Area = √[12×4×4×4] = √768 ≈ 27.71 m², Perimeter = 24 m, all angles = 60°.
Example 3: SAS — Two Sides and Included Angle
Scenario: A surveyor knows two boundary lengths of 120 m and 85 m meeting at a 55° angle.
| Input | Value |
|---|---|
| Side a | 120 m |
| Side b | 85 m |
| Angle C | 55° |
Results: c² = 14400 + 7225 − 2(120)(85)cos(55°) ≈ 21625 − 11688 = 9937 → c ≈ 99.7 m, Area = ½ × 120 × 85 × sin(55°) ≈ 4,179 m².
Example 4: ASA — Two Angles and Included Side
Scenario: A navigation problem where two observers 500 m apart each measure the angle to a landmark: 40° and 65°.
| Input | Value |
|---|---|
| Angle A | 40° |
| Angle B | 65° |
| Side c (between A & B) | 500 m |
Results: Angle C = 75°. By Law of Sines: a = 500 × sin(40°)/sin(75°) ≈ 332.9 m, b = 500 × sin(65°)/sin(75°) ≈ 469.1 m.
Example 5: Roof Rafter Length (Right Triangle)
Scenario: A roofer needs to cut rafters for a building 24 ft wide (so half-span = 12 ft) with a 9 ft rise.
| Input | Value |
|---|---|
| Leg a (horizontal run) | 12 ft |
| Leg b (vertical rise) | 9 ft |
Results: Hypotenuse = √(144 + 81) = √225 = 15 ft (rafter length). Pitch angle = arctan(9/12) = arctan(0.75) ≈ 36.87°.
Real-World Applications
Triangles appear in virtually every field. Here are six high-impact real-world scenarios:
🏠 Construction & Roofing
Calculate rafter lengths, roof pitches, and hip angle cuts. The 3-4-5 right triangle method ensures perfectly square foundations.
🗺️ Surveying & GPS
Triangulation is the backbone of land surveying and GPS systems. Three known positions determine any fourth point on Earth.
✈️ Aviation & Navigation
Pilots use the triangle of velocities to compute actual track, ground speed, and wind correction angle from airspeed and wind vectors.
🖥️ Computer Graphics & Gaming
All 3D rendering uses triangles as the fundamental polygon. Triangle calculators help compute UV coordinates, normals, and collision detection.
⚙️ Mechanical Engineering
Truss structures in bridges and cranes are composed of triangles for maximum rigidity. Engineers calculate member lengths and forces using triangle geometry.
📐 Interior Design & Landscaping
Compute triangular tile areas for flooring, lawn sections for irrigation planning, and fabric cuts for custom soft furnishings.
Common Mistakes to Avoid
❌ Using Slant Height Instead of Perpendicular Height
Consequence: Area = ½ × base × perpendicular height. Using a side length as the height gives a wrong result — often larger than correct.
✓ Fix: If you only have sides, use Heron's formula instead of the base × height formula.
❌ Violating the Triangle Inequality
Consequence: Entering sides like 1, 2, 10 produces an imaginary result because 1 + 2 < 10 — these lengths cannot form a triangle.
✓ Fix: Verify a + b > c, b + c > a, and a + c > b before calculating. Our calculator automatically flags invalid inputs.
❌ Confusing Degrees and Radians
Consequence: Entering 55 expecting degrees but the formula using radians produces a wildly incorrect angle.
✓ Fix: This calculator always uses degrees. If you have radians, convert: degrees = radians × (180/π).
❌ Forgetting that Angles Must Sum to 180°
Consequence: Entering three angles that add to 179° or 181° is mathematically impossible. The triangle cannot exist.
✓ Fix: Always verify A + B + C = 180°. If using AAS/ASA, the third angle is automatically computed as 180° − A − B.
Tips and Best Practices
- Use consistent units throughout: All side lengths must use the same unit. Mixing meters and centimeters causes a 100× error in area.
- Double-check the right triangle case: If c² = a² + b² exactly (or very close), your triangle is a right triangle and the simpler Pythagorean approach applies.
- For large-scale problems, carry extra decimal places: Intermediate rounding accumulates error. Keep 4–6 significant figures until the final step.
- Recognize Pythagorean triple shortcuts: 3-4-5, 5-12-13, and 8-15-17 are exact integer solutions — no calculator needed.
- Identify the ambiguous case (SSA): Two sides and a non-included angle can yield two valid triangles. Our calculator flags this ambiguity.
- Validate using the perimeter check: After computing all sides, confirm the perimeter matches your physical measurements. A large discrepancy indicates a data entry error.
Frequently Asked Questions
▶How do I find the area of a triangle without the height?
▶What is the Law of Cosines used for?
▶Can a triangle have two obtuse angles?
▶What are the types of triangles by angles?
▶What are the types of triangles by sides?
▶What is the ambiguous case in triangle solving?
▶How accurate is this triangle calculator?
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Conclusion
The Triangle Calculator is an indispensable tool for anyone working with geometry — from students verifying homework to engineers designing structural trusses. By supporting all standard cases (SSS, SAS, ASA, AAS, and right triangles), it covers the full breadth of triangle problems without requiring you to remember which law to apply. Whether you need a quick area for a landscaping project or a full angle breakdown for a construction plan, this calculator delivers accurate, instant results.
Remember: always use consistent units, verify the triangle inequality for SSS inputs, and keep extra decimal places in intermediate steps. Bookmark this page for your next geometry challenge!
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