Triangle Calculator
Solve any triangle (SSS, SAS, ASA, AAS, SSA) for sides, angles, area, perimeter, and inradius with dynamic visual diagram.
Triangle Calculator
Enter any 3 values (sides or angles, with at least one side length) to solve for all remaining sides, angles, area, perimeter, inradius, circumradius, medians, and altitudes.
Input Parameters
Comprehensive Guide to Triangle Trigonometry & Geometry
A triangle is a closed polygon consisting of three straight line segments (edges) joining three non-collinear points (vertices). The sum of its three interior angles is always exactly 180 degrees (or π radians).
Triangle Classification by Sides
- Equilateral Triangle: All three sides are equal (a = b = c), and each interior angle is exactly 60° (π/3 rad).
- Isosceles Triangle: Exactly two sides are equal in length, and the angles opposite those sides are equal.
- Scalene Triangle: All three sides and all three interior angles have distinct, unequal measurements.
Triangle Classification by Angles
- Right Triangle: Contains exactly one 90° right angle (π/2 rad). Satisfies the Pythagorean Theorem a² + b² = c².
- Acute Triangle: All three interior angles are strictly less than 90° (θ < 90°).
- Obtuse Triangle: Contains exactly one interior angle strictly greater than 90° (θ > 90°).
Fundamental Geometric Theorems & Formulas
1. Law of Sines (Sine Rule)
Relates side lengths directly to the sines of their opposite angles and the circumradius R:
2. Law of Cosines (Cosine Rule)
Generalization of the Pythagorean theorem for any triangle:
3. Heron's Area Formula
Computes exact area from all three sides using semiperimeter s = (a+b+c)/2:
4. Inradius and Circumradius
Radii of inscribed and circumscribed circles:
Frequently Asked Questions
Why does AAA (Angle-Angle-Angle) not solve a unique triangle?
Three known angles establish geometric similarity, fixing the proportional shape and ratios of the triangle, but not its physical scale. An infinite number of scaled triangles share identical angles. At least one side length is required to fix the unique dimension.
What is the SSA ambiguous case?
When two side lengths and a non-included angle are known (SSA), the opposite side can potentially swing to create two different valid triangles (one acute and one obtuse), one right triangle, or no valid triangle if the opposite side is too short to reach the base line.
What is the Triangle Inequality Theorem?
The theorem states that for any geometrically valid triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining third side (a + b > c, a + c > b, b + c > a). If a + b ≤ c, the lines collapse into a flat segment.
What is the difference between an Incircle and a Circumcircle?
An incircle is the largest circle that fits completely inside the triangle, touching all three sides tangentially with radius r = Area/s. A circumcircle is the circle passing through all three triangle vertices with radius R = (abc)/(4×Area).
How do medians and the centroid work?
A median is a line segment connecting a vertex to the midpoint of the opposite side. The three medians always intersect at a single point called the Centroid, which divides each median in a 2:1 ratio and serves as the physical center of mass.
Understanding Triangle Calculator & Formula
The Triangle Calculator on CalculatorScale.com is engineered to deliver instant, mathematically verified calculations conforming strictly to standard institutional equations. Whether planning long-term investments, managing household budgets, tracking clinical fitness targets, analyzing academic genetics, calculating veterinary dosages, or converting dimensional units, every calculation updates dynamically in real time without lag.
Every formula used in this tool adheres to certified international standards, including World Health Organization biometric metrics, veterinary clinical guidelines, NIST physical conversion constants, and standard biological equations. Results are computed entirely client-side inside your browser with complete data privacy, ensuring your sensitive numbers and statistics never leave your device.
Our calculators are 100% free with no registration, subscriptions, or paywalls required. You can toggle between imperial and metric units, customize input parameters, export calculation summaries, or embed this calculator directly into your own website or blog.
How to Use the Triangle Calculator
- 1 Select your known triangle properties: SSS (3 sides), SAS (2 sides + included angle), SSA, ASA, or AAS.
- 2 Enter the known lengths and angles.
- 3 Click "Calculate ▶️" to solve all 3 side lengths, all 3 interior angles, area, perimeter, semi-perimeter, inradius, and circumradius with visual geometry diagram.
Triangle Calculator — Frequently Asked Questions
Q1 What information is needed to solve a triangle? ▾
You need at least three known properties with at least one side length: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), or AAS (Angle-Angle-Side).
Q2 What are the Law of Sines and Law of Cosines? ▾
The Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) relates side lengths to opposite angles. The Law of Cosines (c² = a² + b² - 2ab·cos(C)) solves triangles when two sides and the included angle or all three sides are known.
Q3 How do you calculate triangle area with Heron's formula? ▾
When all three side lengths (a, b, c) are known, calculate semi-perimeter s = (a+b+c)/2. The area is Area = √[s(s-a)(s-b)(s-c)].
Q4 What is the triangle inequality theorem? ▾
The triangle inequality theorem states that the sum of the lengths of any two sides of a valid triangle must always be strictly greater than the length of the remaining side.
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