Side-labeled triangle with incircle, circumcircle, and altitudeLive current inputs
Math & Statistics
Triangle Solver Calculator
Solve a valid triangle from three side lengths, report all angles, area, semiperimeter, altitude, inradius, and circumradius, and verify the triangle inequality before drawing it.
Angle A-
Angle B-
Angle C-
Triangle area-
Perimeter-
Inradius-
Circumradius-
Triangle validity-
SCALED TRIANGLE GEOMETRY
Side-labeled triangle with incircle, circumcircle, and altitude
The drawing uses the entered side ratios. The orange altitude, green incircle, and violet circumcircle connect the headline values to the actual geometry.
COMPLETE SSS REGISTER
All sides, opposite angles, and geometric checks
The register keeps opposite side-angle pairs aligned and reconciles the triangle through several identities.
| Element | Entered or solved value | Opposite partner | Formula or check | Result |
|---|
SIDE ENTRY
Measure three straight sides in one unit
- Assign each side to its opposite capital angle.
- Use one length unit throughout.
- Do not round a measured side before entry.
- Check that the two shorter sides exceed the longest.
- Use the scaled drawing to catch swapped labels, not as a measurement substitute.
GEOMETRIC CONSISTENCY
Three sides determine one triangle up to reflection
SSS has no ambiguous case: a valid set of three sides fixes all three angles. Reflection changes orientation but not lengths, angles, or area.
Near the triangle-inequality boundary, area approaches zero and angle calculations become sensitive to small measurement error.
SSS SOLUTION
Validate the sides before applying inverse cosine
The strict triangle inequality must hold. The Law of Cosines then recovers each angle, and Heron's formula supplies area without first choosing a base.
Detailed calculation process and general formulas
A = cos⁻¹((b² + c² - a²) / (2bc))B = cos⁻¹((a² + c² - b²) / (2ac))s = (a + b + c) / 2K = √(s(s-a)(s-b)(s-c))r = K/s; R = abc/(4K)Symbols, meanings, and units
- a,b,c
- side lengths opposite A, B, and Clength
- A,B,C
- interior anglesdegrees
- s
- semiperimeterlength
- K
- triangle arealength²
- r,R
- inradius and circumradiuslength
SHAPE DIAGNOSTICS
The same side set yields several useful geometric views
Each answers a different construction question.
Angle set
-The largest angle must lie opposite the longest side.
Interior clearance
-The inradius measures the largest centered tangent circle.
Circumscribed scale
-The circumradius links the sides to the common vertex circle.
Decision takeaway: Always report the length unit and square it for area.
QUALITY CHECKS
Four identities a valid solved triangle should satisfy
- A + B + C = 180°
- Longest side opposite largest angle
- K = rs
- abc = 4RK
- Strict triangle inequality
Applied decisions
How side balance changes triangle shape
Nearly equal sides
All three sides are close in length.
What the result clarifies: Angles cluster near 60° and the incenter and circumcenter lie close together.
Near-degenerate set
The longest side approaches the sum of the other two.
What the result clarifies: Area and altitude collapse while the largest angle approaches 180°.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The calculator assumes straight Euclidean sides measured in one unit. It does not account for curved surfaces, measurement covariance, surveying projection, or tolerances unless you enter adjusted lengths.
Triangle Solver Calculator | SSS Angles, Area, Radii, and Scaled Diagram FAQ
Why are my sides invalid?
For a nondegenerate triangle, every pair of sides must sum to more than the third side.
Which unit does the area use?
The square of whatever common length unit you enter.
Why can the circumradius be very large?
Near a flat triangle, area becomes small while abc remains finite.
Does SSS have two solutions?
Only mirror-image placements, which share all reported lengths, angles, and area.