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Geometry

Triangle Theorems

Solve triangles with theorems. Free online Triangle Theorems. Calculate triangle theorems online — fast, accurate, mobile-friendly, no signup needed.

Generic triangle with sides a, b, c opposite to angles A, B, C.
Side c
6.244998
Area
15.155445

Derivation

  1. ├── 01Givena = 5, b = 7, C = 60
  2. ├── 02FormulaSide c: √(t²+a²-2 × t × a × Js(n))
  3. ├── 03Substitute√(t²+5²-2 × t × 5 × Js(n))
  4. ├── 04Compute Side c6.244998
  5. ├── 05FormulaArea: {let t=e.a,a=e.b,n=e.C;return.5 × t × a × Qs(n)}
  6. ├── 06Substitute{let t=e.5,5=e.7,n=e.60;return.5 × t × 5 × Qs(n)}
  7. └── 07Compute Area15.155445
Did you know?

Pythagoras of Samos (c. 570–495 BCE) likely wasn’t the first to find a²+b²=c² — Babylonian tablets from ~1800 BCE used it — but the first rigorous proof is credited to his school.

§01What is

Understanding the Triangle Theorems

The Triangle Theorems computes Side c from 3 inputs: side a, side b, angle c (°). Solve triangles with theorems.

Geometry is what turns raw measurements into useful answers about space — how much paint, how big a yard, how much material a project will need. Every craftsperson, architect, and DIYer reaches for these formulas regularly. The Triangle Theorems sits in that toolkit — it solve triangles with theorems. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

Side c = √(t²+a²-2 × t × a × Js(n)) | Area = {let t=e.a,a=e.b,n=e.C;return.5 × t × a × Qs(n)}

Where

a
Side a
b
Side b
C
Angle C (°)
Side c
Output value
Area
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Side a = 5, Side b = 7, Angle C (°) = 60.

  1. 01Start by noting the input — Side a: 5.
  2. 02Start by noting the input — Side b: 7.
  3. 03Start by noting the input — Angle C (°): 60.
  4. 04Substitute these values into the formula: Side c = √(t²+a²-2 × t × a × Js(n)) | Area = {let t=e.a,a=e.b,n=e.C;return.5 × t × a × Qs(n)}
  5. 05Compute Side c: the calculator returns 6.245.
  6. 06Compute Area: the calculator returns 15.1554.
  7. 07Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Triangle Theorems Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Side a halved

a = 2.5 (from 5)

Keep every other input at its default and halve the side a. See how side c responds.

  1. 01New Side a: 2.5
  2. 02Baseline Side c: 6.245
  3. 03New Side c: 6.1441
  4. 04Side c decreases by 1.6% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Side a doubled

a = 10 (from 5)

Keep every other input at its default and double the side a. See how side c responds.

  1. 01New Side a: 10
  2. 02Baseline Side c: 6.245
  3. 03New Side c: 8.88819
  4. 04Side c increases by 42.3% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Side b halved

b = 3.5 (from 7)

Keep every other input at its default and halve the side b. See how side c responds.

  1. 01New Side b: 3.5
  2. 02Baseline Side c: 6.245
  3. 03New Side c: 4.4441
  4. 04Side c decreases by 28.8% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Side b doubled

b = 14 (from 7)

Keep every other input at its default and double the side b. See how side c responds.

  1. 01New Side b: 14
  2. 02Baseline Side c: 6.245
  3. 03New Side c: 12.2882
  4. 04Side c increases by 96.8% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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