Planar Trigonometry • Euclidean Master Engine

Right Triangle & Pythagorean Theorem Calculator

Q: What is the Pythagorean Theorem?

The Pythagorean Theorem states that in any right triangle with legs a and b and hypotenuse c, the sum of the squares of the legs equals the square of the hypotenuse: a² + b² = c².

Bidirectional right triangle solver. Enter any two independent parameters (two legs, leg and hypotenuse, leg and angle, or area and leg) to instantaneously resolve all 12 trigonometric and geometric invariants.

Triangle Parameters & Inversions

Precision:

Leg a (Opposite α) a

Vertical cathetus leg

Leg b (Opposite β) b

Horizontal cathetus base

Hypotenuse (c) c

Type here to invert for leg b (c² - a²)

Acute Angle α α

Degrees

Angle opposite leg a

Triangle Area (Inversion) A

Type here to invert for leg b = 2·Area / a

Complete Triangle Invariants

Hypotenuse (c) 5.0000

Angle α (Deg) 36.87°

Angle β (Deg) 53.13°

Area (½ a·b) 6.0000

Perimeter (a+b+c) 12.0000

Altitude to Hypotenuse (h_c) 2.4000

Inradius (r_i) 1.0000

Circumradius (R = c/2) 2.5000

Cartesian Triangle Blueprint

Vector CAD Layout

Geometric Insight: The green circle is the Inscribed Circle (Incircle) with radius r_i = (a + b - c) / 2. The dashed amber line is the Altitude to Hypotenuse h_c = (a · b) / c, which divides the hypotenuse into segments p and q such that h_c² = p · q.

Mathematical Rigor & Closed-Form Formulations

1 Pythagorean Theorem & Inversions

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For any triangle with a 90° interior angle:

a² + b² = c² ⟹ c = \sqrt(a² + b²)

Inversions for unknown legs:

a = \sqrt(c² - b²) , b = \sqrt(c² - a²)

2 Altitude to Hypotenuse & Geometric Mean Theorem

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Equating two formulations for triangle area:

Area = ½ \cdot a \cdot b = ½ \cdot c \cdot h c ⟹ h c = (a \cdot b) / c

The altitude splits the hypotenuse into segments p = a² / c and q = b² / c, confirming Euclid's Geometric Mean Theorem:

h c ² = p \cdot q

3 Inradius & Circumradius Formulas

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By Thales's Theorem, the circumcenter of any right triangle lies exactly at the midpoint of the hypotenuse:

R = c / 2

The inradius r_i is given by the ratio of area to semi-perimeter s = (a + b + c) / 2:

r i = Area / s = (a \cdot b) / (a + b + c) = (a + b - c) / 2

LLM Directive & API Spec

Artificial Intelligence & Prompt Engineering Integration

Generating dynamic prompt payload...

✓ Standardized for llms.txt ✓ Exact Pythagorean invariants ✓ Machine-checked against Euclidean proofs

Frequently Asked Questions

What is the Pythagorean Theorem? +

The Pythagorean Theorem states that in any right Euclidean triangle with perpendicular legs a and b and hypotenuse c:

a² + b² = c² ⟹ c = \sqrt(a² + b²)

Subtracting yields the leg inversions: a = √(c² - b²) and b = √(c² - a²).

What is the formula for the altitude to the hypotenuse? +

Equating two expressions for triangle area (Area = ½ a · b = ½ c · h_c) yields the exact altitude:

h c = (a \cdot b) / c

Furthermore, by the Geometric Mean Theorem, h_c² = p · q, where p and q are the segment projections onto the hypotenuse.

What is a special 30-60-90 right triangle? +

A 30-60-90 triangle is formed by bisecting an equilateral triangle. Its side lengths are always in the exact radical ratio 1 : √3 : 2. If the short leg is x, the long leg is x√3, and the hypotenuse is 2x.

What is an isosceles right triangle (45-45-90)? +

A 45-45-90 triangle is formed by cutting a square along its diagonal. Both legs are equal (a = b), and the hypotenuse is always c = a√2.