About

Determining the base of a triangle is not a single formula but a branching decision tree that depends on which measurements are known. A miscalculated base propagates errors into area, centroid location, and moment-of-inertia computations used in structural engineering and land surveying. This tool implements six distinct resolution paths: the direct ratio b = 2Ah, the Law of Cosines for SAS configurations, a Law of Sines pipeline for AAS/ASA cases, perimeter subtraction with triangle inequality enforcement, and specialized isosceles/equilateral shortcuts. All angular inputs accept degrees and are internally converted via θ_rad = θ_deg × π180. Results are validated against the triangle inequality theorem: the sum of any two sides must strictly exceed the third.

Limitations: this calculator assumes Euclidean plane geometry. Results deviate on spherical or hyperbolic surfaces. Inputs near degenerate cases (angles approaching 0° or 180°) produce numerically unstable outputs. For obtuse triangles computed via the Law of Cosines, verify that the cosine term is negative as expected. Pro tip: when surveying irregular land parcels, measure at least one angle with a theodolite rather than deriving it from GPS coordinates to reduce propagation error.

Formulas

The primary formula for computing the base depends on the known measurements. Below are the six methods implemented in this calculator.

Method 1 - Area & Height:

b = 2 ⋅ Ah

where A = area of triangle (sq units), h = perpendicular height to base (units).

Method 2 - Two Sides & Included Angle (SAS, Law of Cosines):

b = √a² + c² − 2 ⋅ a ⋅ c ⋅ cos(B)

where a, c = the two known sides, B = the included angle between them.

Method 3 - One Side & Two Angles (AAS/ASA, Law of Sines):

C = 180° − A − B

b = a ⋅ sin(B)sin(A)

where a = known side, A = angle opposite side a, B = angle opposite the base.

Method 4 - Perimeter & Two Sides:

b = P − a − c

where P = perimeter, subject to triangle inequality: b > 0 and a + b > c (all permutations).

Method 5 - Isosceles (Equal Sides & Vertex Angle):

b = 2 ⋅ a ⋅ sin(θ2)

where a = length of each equal side, θ = vertex angle between the equal sides.

Method 6 - Equilateral Triangle:

b = a

All sides are equal. Input any one side length.

Angle Conversion:

θ_rad = θ_deg × π180

Reference Data

Triangle Type	Defining Property	Base Formula	Angle Constraint	Example Base
Scalene (SAS)	All sides different	b = √a² + c² − 2ac cos B	0° < B < 180°	7.81 cm
Scalene (AAS)	Two angles + non-included side	Law of Sines chain	A + B < 180°	9.24 cm
Isosceles	Two equal sides	b = 2a sin(θ÷2)	Vertex angle θ < 180°	6.18 cm
Equilateral	All sides equal	b = a	All angles = 60°	5.00 cm
Right (base as leg)	One 90° angle	b = √c² − a²	B = 90°	4.00 cm
From Area & Height	Area and perpendicular height known	b = 2A ÷ h	N/A	10.00 cm
From Perimeter	Perimeter and two sides known	b = P − a − c	Triangle inequality	8.00 cm
Obtuse Scalene	One angle > 90°	Law of Cosines (cos negative)	90° < B < 180°	12.53 cm
30-60-90	Special right triangle	Ratios 1 : √3 : 2	Fixed angles	5.00 cm (short leg)
45-45-90	Isosceles right triangle	Ratios 1 : 1 : √2	Fixed angles	5.00 cm (leg)
Heron's Formula (reverse)	All three sides known	A = √s(s−a)(s−b)(s−c)	All sides positive	Area verification
Golden Gnomon	Isosceles, apex 36°	b÷a = φ	θ = 36°	8.09 cm

Frequently Asked Questions

When the included angle B exceeds 90°, cos(B) becomes negative. This makes the −2ac cos(B) term positive, increasing the value under the radical. The result is a base longer than either known side, which is geometrically correct for obtuse triangles. The calculator handles this automatically; no special input is needed.

The triangle inequality theorem requires that any side must be strictly less than the sum of the other two. If you enter a perimeter P and two sides a and c such that the computed base b = P − a − c is zero, negative, or violates a + b > c, the triangle is degenerate. Reduce the length of one side or increase the perimeter.

In Euclidean geometry, the interior angles of a triangle sum to exactly 180°. If the two known angles A + B ≥ 180°, the third angle C would be zero or negative, which is impossible. The calculator validates this before computing and returns an error with explanation.

Both produce identical results. The isosceles formula b = 2a sin(θ÷2) is a simplified derivation of the Law of Cosines where a = c. The simplification avoids floating-point subtraction of nearly equal numbers, so it can actually be marginally more stable for very small vertex angles.

Yes. All computations use IEEE 754 double-precision floating-point arithmetic, supporting values from approximately 5 × 10⁻³²⁴ to 1.8 × 10³⁰⁸. Precision is limited to about 15 significant digits. For engineering applications requiring higher precision, verify the last two decimal places independently.

Yes. After computing all three sides (and all three angles when possible), the canvas renders the triangle using true proportional scaling fitted to the drawing area. Vertices are placed using trigonometric coordinate calculations, not schematic approximations. Side labels and angle arcs are drawn at their actual positions.