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Triangle Calculator

Solve sides, angles, area, perimeter, radii, and more from common input sets.
Opposite pairs: a ↔ A, b ↔ B, c ↔ C.

Table of Contents

🔹 Triangle Area Calculation Methods

The area of a triangle can be found in multiple ways depending on the data you have. Our calculator automatically selects the correct method, but here are the most common formulas:

Formula When to Use Expression
Heron’s Formula When all three sides a, b, c are known (SSS). Semi-perimeter: s = (a+b+c)/2
Area: √[s(s−a)(s−b)(s−c)]
Trigonometric Formula When two sides and the included angle are given (SAS). Area = ½·b·c·sin A
Right Triangle Formula When the triangle is right-angled. Area = ½·a·b (where a, b are legs)
Law of Sines Formula With ASA or AAS data. Area = ½·a·b·sin C

🔹 Example: Area by Heron’s Formula

Given: sides a = 7, b = 9, c = 12.

  • Semi-perimeter: s = (7+9+12)/2 = 14
  • Area: √[14(14−7)(14−9)(14−12)] = √(14·7·5·2) = √980 ≈ 31.3

The calculator will output 31.3 units² for the area in this example.

🔹 Triangle Calculator Overview

This Triangle Calculator solves a complete triangle from common input sets. Enter three sides (SSS), two sides and the included angle (SAS), one side and two angles (ASA/AAS), or right-triangle values. It returns all sides and angles, plus area, perimeter, inradius (r), circumradius (R), altitudes (h_a, h_b, h_c), and medians.

Notation: sides are a, b, c opposite angles A, B, C respectively. Angles can be in degrees or radians (use the toggle above).

🔹 What You Can Calculate

  • Complete dimensions: missing sides and angles for SSS, SAS, ASA/AAS, and right triangles.
  • Area & perimeter: via Heron’s formula (SSS) or Area = \tfrac12 bc \sin A (general).
  • Radii & heights: inradius r = \tfrac{Area}{s}, circumradius R = \tfrac{a}{2\sin A}, altitudes h_x = \tfrac{2\,Area}{x}.
  • Classification: equilateral / isosceles / scalene and acute / right / obtuse.

🔹 Input Modes & Key Formulas

Mode Required Inputs Main Formulas Used Notes
SSS Three sides: a, b, c Law of Cosines:
A = \cos^{-1}\!\left(\frac{b^2 + c^2 - a^2}{2bc}\right) (cyclic for B, C)
Area (Heron): s=\tfrac{a+b+c}{2}, Area = \sqrt{s(s-a)(s-b)(s-c)} 		Check triangle inequality: each pair of sides must sum > the third.
SAS Two sides & included angle: b, c, A Law of Cosines: a^2 = b^2 + c^2 - 2bc\cos A
Law of Sines for remaining angles 		Included angle must be between the two given sides.
ASA/AAS One side & two angles: e.g., a, B, C Angle sum: A = 180^\circ - (B + C)
Law of Sines: \tfrac{a}{\sin A} = \tfrac{b}{\sin B} = \tfrac{c}{\sin C} 		Angles must sum to less than 180° (or π rad).
Right Triangle Any 2 among a, b (legs) and c (hypotenuse) Pythagoras: c^2 = a^2 + b^2
Angle: A = \arctan \tfrac{a}{b}, C = 90^\circ 		Hypotenuse is the longest side. Area: \tfrac12 ab.

🔹 Quick Example

Given (SAS): b=4, c=6, A=40^\circ.

  • Find a via Law of Cosines: a=\sqrt{4^2+6^2-2\cdot4\cdot6\cos40^\circ}\approx 3.86.
  • Area: \tfrac12 bc\sin A = \tfrac12\cdot4\cdot6\cdot\sin40^\circ \approx 7.71.
  • Perimeter: a+b+c \approx 3.86+4+6=13.86.

Tip: Need trig values during homework? Use our Scientific Calculator to double-check sines, cosines, and arctangents.

🔹 Triangle Classification: By Sides & By Angles

Classifying a triangle helps you predict properties at a glance—useful for exams, CAD drawings, and quick design checks. Below are the standard classifications used in geometry and engineering contexts.

🔹 By Sides

Type Definition Quick Check Example
Equilateral All three sides equal; all angles 60°. a = b = c a=b=c=5 ⇒ perimeter 15, area ≈ 10.825
Isosceles At least two sides equal; base angles equal. a = b ≠ c (or any two equal) a=b=7, c=10 ⇒ equal base angles, altitude from apex bisects base
Scalene All sides different; all angles different. a ≠ b ≠ c a=6, b=8, c=11 ⇒ no symmetry; medians/angles all distinct

🔹 By Angles

Type Definition Side-Based Test Example
Acute All angles < 90°. Sort sides x ≤ y ≤ z; if x² + y² > z² ⇒ acute. 5, 6, 7 ⇒ 5²+6²=61 > 49
Right One angle = 90°. Sorted sides x, y, z; if x² + y² = z² ⇒ right. 3, 4, 5 ⇒ 9+16=25
Obtuse One angle > 90°. Sorted sides x, y, z; if x² + y² < z² ⇒ obtuse. 4, 7, 10 ⇒ 16+49=65 < 100

🔹 Fast Identification Workflow

  • Have all three sides? Check triangle inequality, then apply the side-based test above to label acute/right/obtuse.
  • Have two sides and included angle? Use Law of Cosines to find the third side; classify by sides and re-check angle sizes.
  • Have one side and two angles? Compute the remaining side with Law of Sines; labeling by sides follows immediately.

Practical tip: When tolerances matter (e.g., laser-cut parts), round consistently and avoid classifying solely by rounded values—small numeric errors can flip an acute/right/obtuse decision near 90°.

🔹 Law of Sines

The Law of Sines relates the ratio of a side length to the sine of its opposite angle. It is especially useful in ASA or AAS cases, and sometimes ambiguous in SSA cases (two possible triangles).

a / sin A = b / sin B = c / sin C = 2R (where R is the circumradius).

🔹 Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem and applies to any triangle. It’s essential for SAS (two sides and included angle) and SSS (three sides) cases.

  • a² = b² + c² − 2bc·cos A
  • b² = a² + c² − 2ac·cos B
  • c² = a² + b² − 2ab·cos C

🔹 Worked Example

Given (SSA): a=8, A=40°, B=65°.

  • Find C = 180° − (A+B) = 75°.
  • Use Law of Sines: b = a·sin B / sin A = 8·sin65° / sin40° ≈ 11.2.
  • Similarly, c = a·sin C / sin A = 8·sin75° / sin40° ≈ 12.3.

All sides and angles are now determined.

🔹 Common Pitfalls

  • Ambiguous SSA case: Two different triangles may satisfy the data (acute or obtuse solution). Our calculator automatically handles this by returning the valid case.
  • Rounding errors: When using decimals, ensure consistent precision. Slight rounding may cause the sum of angles to be slightly off from 180°.
  • Degree vs radian confusion: Always check your mode when inputting angles. If using radians, remember π rad = 180°.

For deeper trigonometric checks, see also our Degree–Radian Converter to avoid unit mistakes in manual calculations.

🔹 Triangle Centers at a Glance

Every non-degenerate triangle has four classical “centers” with powerful geometric meaning: the incenter, circumcenter, centroid, and orthocenter. These points arise from bisectors, perpendiculars, medians, and altitudes respectively, and they drive many workflows in surveying, CAD/CAM, finite-element meshes, and robotics path planning.

🔹 Definitions & Key Formulas

Center Construction Core Properties / Formulas Where It Lies
Incenter (I) Intersection of the three angle bisectors. Center of the incircle (tangent to all sides).
Inradius: r = Area / s where s is semiperimeter.
Distance to each side equals r. 		Always inside the triangle.
Circumcenter (O) Intersection of the three perpendicular bisectors of the sides. Center of the circumcircle (through all vertices).
Circumradius: R = a/(2\sin A) = b/(2\sin B) = c/(2\sin C). 		Inside (acute), on the hypotenuse midpoint (right), outside (obtuse).
Centroid (G) Intersection of the three medians (vertex to midpoint of opposite side). Balancing point of a uniform triangular plate.
Divides each median in a 2:1 ratio (vertex→centroid is 2 parts). 		Always inside the triangle.
Orthocenter (H) Intersection of the three altitudes (perpendiculars from vertices to opposite sides). Related to Euler line with G and O (and nine-point center).
In a right triangle, H is at the right-angle vertex. 		Inside (acute), on vertex (right), outside (obtuse).

🔹 Special Cases You Can Use in Practice

  • Right triangle: the circumcenter is the midpoint of the hypotenuse; the orthocenter is the right-angle vertex.
  • Equilateral triangle: all four centers coincide at the same point.
  • Isosceles triangle: all centers lie on the axis of symmetry.

🔹 Quick Coordinate Formulas (useful in CAD)

If vertices are A(x₁,y₁), B(x₂,y₂), C(x₃,y₃):

  • Centroid: G\,( (x₁+x₂+x₃)/3,\; (y₁+y₂+y₃)/3 )
  • Incenter: with side lengths a=BC, b=CA, c=AB, I\,( (ax₁+bx₂+cx₃)/(a+b+c),\; (ay₁+by₂+cy₃)/(a+b+c) )

🔹 Real-World Applications

  • Surveying & GIS: circumcenter helps with triangulation and evenly spaced reference points.
  • Mechanical/Structural design: centroid indicates mass center for plates, bracing, and load paths.
  • Robotics & path planning: incenter maximizes clearance to boundaries (largest inscribed circle).
  • Mesh generation (FEM/CFD): triangle centers inform quality metrics and refinement strategies.

Implementation note: our calculator already provides r, R, medians, heights, area, and perimeter. These quantities let you reconstruct centers and circles quickly in external tools.

🔹 Worked Triangle Examples

Let’s see how the calculator handles different cases: SSS, SAS, ASA/AAS, and Right Triangle. These step-by-step examples show what’s happening behind the scenes so you can replicate manually if needed.

🔹 Example 1: SSS Case

Given: a=7, b=8, c=9

  • Use Law of Cosines to find A: A = arccos((b²+c²−a²)/(2bc)) = arccos((64+81−49)/(2·8·9)) ≈ 48.2°.
  • Similarly, B≈57.1°, C≈74.7°.
  • Semi-perimeter: s=(7+8+9)/2=12.
  • Area: √[12(12−7)(12−8)(12−9)] = √(720) ≈ 26.8.

🔹 Example 2: SAS Case

Given: b=5, c=7, A=50°

  • a²=5²+7²−2·5·7·cos50°≈25+49−45≈29.1 ⇒ a≈5.4.
  • Use Law of Sines to solve for B: sinB/b=sinA/a ⇒ B=arcsin(b·sinA/a)≈44.4°.
  • C=180°−A−B≈85.6°.
  • Area: ½·5·7·sin50°≈13.4.

🔹 Example 3: ASA/AAS Case

Given: a=10, B=60°, C=45°

  • A=180°−(60+45)=75°.
  • Use Law of Sines: b=a·sinB/sinA=10·sin60°/sin75°≈9.33.
  • c=a·sinC/sinA=10·sin45°/sin75°≈7.32.
  • Area: ½·10·9.33·sin45°≈32.9.

🔹 Example 4: Right Triangle Case

Given: a=3, b=4

  • Hypotenuse: c=√(a²+b²)=√(9+16)=5.
  • Angle A: arctan(a/b)=arctan(3/4)≈36.9°.
  • Angle B: 90°−36.9°=53.1°.
  • Area: ½·3·4=6.

These worked examples mirror what the calculator does instantly. If you want to practice related calculations, see also our Standard Deviation Calculator for step-by-step numeric breakdowns in statistics.

🔹 Why Triangle Math Matters in the Real World

Triangles are the backbone of practical geometry. From roof pitches and bridge trusses to satellite positioning and robotics, triangle relations let you convert partial, noisy measurements into actionable dimensions. Below are high-value use cases where our Triangle Calculator saves time and reduces error.

🔹 Construction, Carpentry & Architecture

  • Roof design: Convert pitch to rafter lengths (right triangles). Area estimates help calculate shingle counts and loads.
  • Trusses & frames: Use SSS/SAS to confirm member lengths and gusset angles; check if a joint is acute/obtuse for connector choice.
  • Facade cladding: Irregular panels often resolve into triangles; Heron’s formula yields fast panel areas for ordering materials.

🔹 Surveying, Mapping & Navigation

  • Triangulation: Measure two angles from baseline points (ASA/AAS) to locate a target point without walking to it.
  • Range & bearing: Two known distances and included angle (SAS) fix the third side—useful for line-of-sight planning.
  • GPS/RTK checks: Cross-verify triangle closures to detect multipath or instrument setup errors.

🔹 Mechanical/Structural Engineering

  • Load paths: Member forces project along triangular frames; angles determine axial vs. shear components.
  • Sheet-metal & composites: Cut plans decompose into triangles; accurate areas minimize scrap and misfit.
  • Finite-element meshes: Triangle quality depends on angle bounds; circumradius/inradius inform mesh refinement.

🔹 Computer Graphics, Vision & Robotics

  • 3D models: Meshes are triangles; per-triangle areas feed surface integrals, shading, and collision detection.
  • Camera pose & depth: With feature matches, SSA/ASA constraints estimate scene geometry.
  • Robot reachability: Law of Cosines converts joint angles to end-effector distances (planar manipulators).

🔹 Input → Output Cheatsheet

Scenario Typical Inputs Use This Mode What You Get
Rafter sizing Rise, run (legs) Right Hypotenuse (rafter), angles, area for material estimates
Site triangulation Baseline + two angles ASA/AAS Unknown side distances, point location, closure check
Bracket design Two sides + included angle SAS Third side, remaining angles, bolt-hole layout
Panel takeoff Three edges SSS Area (Heron), medians/heights, perimeter
Robot link check Two link lengths + joint angle SAS Reach distance, angle complements for next joint

🔹 Pro Tips for Accuracy

  • Prefer included angles (SAS) when measuring in the field; it minimizes ambiguity.
  • When very obtuse/near-right angles are suspected, use side-based tests (x²+y² ? z²) to avoid angle-rounding traps.
  • Keep a consistent decimal precision across all inputs to prevent closure drift (sum of angles ≠ 180° by rounding).

🔹 Common Mistakes When Solving Triangles

Even with the right formulas, small errors can lead to big miscalculations. Here are the pitfalls to watch out for and how our calculator helps you avoid them.

🔹 Frequent Errors

  • Ignoring triangle inequality: Sides must satisfy a+b>c, a+c>b, b+c>a. Otherwise, no valid triangle exists.
  • Angle unit mismatch: Mixing degrees and radians is one of the most common mistakes. Always check the unit toggle before input.
  • SSA ambiguity: With two sides and a non-included angle, two different triangles may exist (or none). The calculator resolves this automatically.
  • Rounding too early: Premature rounding of intermediate results can cause closure issues (angles not summing to 180°).
  • Assuming a right triangle: Many real-world problems aren’t right-angled; applying a²+b²=c² incorrectly leads to false results.

🔹 Example Troubleshooting

Case: Entered sides a=4, b=2, c=7.

  • Check inequality: a+b=6 is NOT > c=7. Invalid triangle.
  • The calculator will show an error message instead of producing nonsense results.

🔹 Tips to Avoid Mistakes

  • Measure angles with a precision tool (like a digital protractor) if possible.
  • Use consistent units—don’t mix cm, m, and inches in the same problem.
  • Always verify that angles sum to 180° (±0.1° due to rounding) as a quick sanity check.

For similar error-prevention in trigonometric tasks, you can also try our Percentage Calculator to see how small changes in input scale to big percentage differences.

🔹 Key Takeaways

  • Inputs drive the method: SSS → Law of Cosines + Heron; SAS → Law of Cosines + Sines; ASA/AAS → Law of Sines; Right → Pythagoras.
  • Check feasibility first: enforce triangle inequality and angle sum = 180° (or π rad).
  • Be unit-aware: confirm angle unit (deg/rad) and keep a consistent decimal precision across all inputs.
  • Go beyond sides/angles: area, perimeter, inradius r, circumradius R, altitudes, and medians are instantly available.
  • Classify quickly: use side-based tests (x²+y² ? z²) for acute/right/obtuse when near 90° to avoid rounding traps.

🔹 Quick Formula Recap

Scenario Primary Formula(s) Notes
SSS
A = acos((b²+c²−a²)/(2bc)) (cyclic)
s = (a+b+c)/2, Area = √[s(s−a)(s−b)(s−c)]
 Best when all sides are measured.
SAS
a² = b² + c² − 2bc·cos A
then Law of Sines for the rest
 Use an included angle to avoid ambiguity.
ASA/AAS
A = 180° − (B+C)
a/sinA = b/sinB = c/sinC
 Angles must sum to less than 180° before solving.
Right
c² = a² + b²
Area = ½ab, A = arctan(a/b), C = 90°
 Hypotenuse is always the longest side.

🔹 Best Practices for Accurate Results

  • Measure included angles where possible (SAS) for robust solutions.
  • Delay rounding until the final step; keep 4–6 internal decimals to maintain closure.
  • For field work, remeasure at least one side/angle as a sanity check—tiny errors can flip a classification or area.

🔹 Next Steps

Enter your known values at the top, select the correct mode, and hit Calculate. Use the results (heights, radii, medians) to build incircles/circumcircles, check tolerances, and produce material takeoffs.

🔹 Frequently Asked Questions (FAQ)

▼ What information do I need to solve a triangle?
You need at least three pieces of data, with at least one of them being a side. Examples: three sides (SSS), two sides and an included angle (SAS), one side and two angles (ASA/AAS), or a right triangle with two values.
▼ How does the calculator choose which formula to use?
It automatically applies the correct formula: Law of Cosines for SSS/SAS, Law of Sines for ASA/AAS, and the Pythagorean theorem for right triangles.
▼ Can a triangle have more than one solution?
Yes. In SSA (Side-Side-Angle) cases, there may be two valid triangles (acute or obtuse). Our calculator detects and resolves the valid solution automatically.
▼ What is the triangle inequality rule?
For any triangle, the sum of the lengths of any two sides must be greater than the third side. If this rule is not met, a valid triangle cannot exist.
▼ What’s the difference between inradius and circumradius?
The inradius (r) is the radius of the inscribed circle tangent to all sides. The circumradius (R) is the radius of the circumscribed circle passing through all three vertices.
▼ Why don’t my angles add up exactly to 180°?
This is usually due to rounding. Use more decimal places in calculations to minimize closure error. The calculator rounds to your chosen precision, but internally uses higher precision.
▼ Can I use mixed units (cm, m, inches)?
No. All sides must be in the same unit. The calculator doesn’t convert between cm, m, or inches automatically. Convert values before entering for consistent results.
▼ How do I check if my triangle is right-angled?
Sort the sides so that c is the largest. If a² + b² = c² (within rounding), it’s a right triangle. The calculator also labels the triangle type automatically.
▼ Where can I use these calculations in real life?
Triangle calculations are used in construction, surveying, navigation, physics, computer graphics, and robotics. They help determine unknown distances, forces, and areas in both practical and digital applications.

🔹 References & Sources

The following references were used to build and verify the formulas, examples, and explanations on this page:

Source Details Link
MathWorld – Wolfram Research Comprehensive formulas on triangles, Heron’s formula, and laws of sines/cosines. mathworld.wolfram.com
Khan Academy Video tutorials and worked problems on triangle properties and trigonometry. khanacademy.org
CK-12 Foundation Geometry lessons on triangle centers, classifications, and problem solving. ck12.org
OpenStax – Geometry Textbook Open educational resource covering triangle types, congruence, and real-world applications. openstax.org
Calculator.net Competitor reference for UI/UX structure and coverage of triangle formulas. calculator.net