Closing-side growth and area rise-and-fall across the angle sweepLive current inputs
Math & Statistics
Triangle Value Table Calculator
Hold two sides fixed, sweep their included angle across a chosen range, and compare how the closing side, area, altitude, and perimeter respond at every sampled angle.
Closing side at highlighted angle-
Area at highlighted angle-
Altitude at highlighted angle-
Perimeter at highlighted angle-
Maximum-area angle-
Maximum possible area-
Generated table rows-
Actual sweep range-
ANGLE RESPONSE LAB
Closing-side growth and area rise-and-fall across the angle sweep
Two labeled curves share the angle axis but use separate normalized scales. The live vertical marker shows the chosen angle, and the 90° area peak remains visible.
ANGLE VALUE TABLE
One complete SAS solution at every requested angle
The last row is always included even when the step does not land exactly on the end angle.
| Included angle A | Closing side a | Area K | Altitude h | Perimeter | Shape note |
|---|
SWEEP DESIGN
Choose a range that answers the actual design question
- Enter the two sides that remain fixed.
- Keep start and end strictly between 0° and 180°.
- Choose an increment fine enough for the decision.
- Use the highlighted angle for a current candidate.
- Export the table when exact sampled values matter more than the curve.
NONLINEAR RESPONSE
Area is symmetric around 90° but closing-side length is not
Angles A and 180°-A have equal sine, so they produce the same area with fixed b and c. Their cosines have opposite signs, so the closing sides differ.
A coarse table can miss a decision threshold even when the chart suggests its location. Reduce the increment around a critical range.
PARAMETER SWEEP
Sample one changing angle while holding both sides fixed
The closing side follows the Law of Cosines. Area and altitude follow the sine of the included angle, so they peak at 90° and fall toward zero at either flat limit.
Detailed calculation process and general formulas
a(A) = √(b² + c² - 2bc cos A)K(A) = ½bc sin Ah(A) = b sin AP(A) = a(A) + b + cK_max = ½bc at A = 90°Symbols, meanings, and units
- A
- swept included angledegrees
- b,c
- fixed enclosing sideslength
- a(A)
- angle-dependent closing sidelength
- K(A)
- angle-dependent arealength²
- h(A)
- altitude to side clength
DESIGN QUESTIONS
Three curves hidden inside one angle choice
The same input controls capacity, reach, and boundary length.
Capacity
-Area peaks at a right included angle.
Reach
-The closing side grows monotonically as A approaches 180°.
Boundary cost
-Perimeter inherits the closing-side growth.
Decision takeaway: Choose the angle for the required tradeoff; maximum area and minimum closing side are not the same objective.
TABLE INTERPRETATION
What to record with a chosen row
- Fixed side dimensions
- Selected angle tolerance
- Required minimum area
- Maximum allowed closing side
- Unit and rounding convention
- Whether the angle is constructible in practice
Applied decisions
Two angles with equal area but different closure
60° candidate
The included angle is acute.
What the result clarifies: It has the same sine as 120° but a shorter closing side.
120° candidate
The included angle is obtuse.
What the result clarifies: Area matches 60° while perimeter and closing span are larger.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The table is a deterministic Euclidean sweep. It does not include construction tolerance, material deformation, measurement uncertainty, or optimization constraints beyond the entered fixed sides and angles.
Triangle Value Table Calculator | Included-Angle Sweep and Response Curves FAQ
Why does area fall after 90°?
Area depends on sin A, which peaks at 90° and decreases toward zero by 180°.
Why does side a keep growing?
The cosine term becomes more negative as A grows, increasing a².
Can the end angle equal 180°?
No. That would be a degenerate straight line rather than a triangle.
Why is the final row included separately?
It preserves the requested boundary even when the increment does not divide the interval evenly.