Overlaid triangles with a vertex trajectory through scenario spaceLive current inputs
Math & Statistics
Triangle Scenario Calculator
Compare baseline and alternative three-side triangle designs, quantify changes in area, perimeter, angles, and compactness, and trace a continuously interpolated geometry between them.
Baseline area-
Alternative area-
Interpolated area-
Full area change-
Full perimeter change-
Largest angle shift-
Interpolated compactness-
Scenario validity-
DESIGN MORPHOLOGY
Overlaid triangles with a vertex trajectory through scenario space
All triangles share a baseline edge and scale. Baseline, selected blend, and alternative outlines reveal whether the change enlarges the shape, rotates the apex, or pushes it toward degeneracy.
SCENARIO LADDER
Geometry at five interpolation checkpoints
Every row resolves a complete SSS triangle instead of interpolating the final answers.
| Blend | Sides a / b / c | Area | Perimeter | Angles A / B / C | Compactness |
|---|
COMPARISON SETUP
Change only the design dimensions that belong to the scenario
- Enter three valid baseline sides.
- Enter three valid alternative sides in the same unit.
- Move the blend control to inspect an intermediate design.
- Compare angle movement as well as area change.
- Use the scenario ladder to detect an invalid intermediate geometry.
WHY RESULTS ARE RECOMPUTED
Area and angles do not move linearly with side lengths
A halfway change in three sides is not generally a halfway change in area. Each intermediate row must pass the triangle inequalities and be solved independently.
Compactness distinguishes a proportionally enlarged triangle from a shape change: uniform scaling changes area and perimeter but leaves Q unchanged.
SIDE-SPACE INTERPOLATION
Blend the independent inputs before solving geometry
A blend fraction t changes each side linearly. Heron's formula and the Law of Cosines are then evaluated for that actual intermediate triangle, preserving nonlinear geometry.
Detailed calculation process and general formulas
a(t)=a₀+t(a₁-a₀)b(t)=b₀+t(b₁-b₀)c(t)=c₀+t(c₁-c₀)K(t)=√[s(t)(s(t)-a(t))(s(t)-b(t))(s(t)-c(t))]Q(t)=4πK(t)/P(t)²Symbols, meanings, and units
- t
- selected interpolation fractiondimensionless
- a₀,b₀,c₀
- baseline sideslength
- a₁,b₁,c₁
- alternative sideslength
- K(t)
- interpolated triangle arealength²
- P(t)
- interpolated perimeterlength
- Q(t)
- isoperimetric compactnessdimensionless
DESIGN CONSEQUENCES
Separate scale change from shape change
The comparison exposes consequences hidden by a single headline area.
Capacity effect
-Full area change quantifies the total two-dimensional gain or loss.
Boundary effect
-Perimeter change shows the corresponding edge-material consequence.
Shape effect
-Largest angle shift identifies the strongest redistribution of geometry.
Decision takeaway: A design can gain area while becoming less compact or creating a problematic acute or obtuse angle.
COMPARISON DISCIPLINE
Keep these conditions aligned
- Same length unit
- Same side naming convention
- Comparable construction intent
- Valid endpoints
- No hidden thickness change
- No unmodeled cutouts
Applied decisions
Two scenario comparisons with different decision stakes
Roof truss revision
A baseline member set is compared with a deeper alternative.
What the result clarifies: Angle movement can matter even when the area gain appears attractive.
Triangular planting bed
Two boundary layouts use different edge lengths.
What the result clarifies: Area and perimeter changes separate planting capacity from edging cost.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This calculator interpolates side lengths, not construction coordinates, loads, or material properties. Every scenario must independently satisfy the triangle inequalities. Structural design still requires force, connection, and code checks.
Triangle Scenario Calculator | Compare Two SSS Designs and Interpolate Geometry FAQ
Why can an intermediate scenario be invalid?
Linear side interpolation can cross a triangle-inequality boundary even if a careless endpoint-only review misses it.
What does compactness mean?
Q=4πK/P² compares enclosed area with boundary length; it is scale independent.
Does 50% mean half the area change?
No. It means the sides are halfway between the two designs; area is recomputed nonlinearly.
Can the two scenarios use different units?
No. Convert both sets to one common length unit first.