Planar Trigonometry • Euclidean Master Engine

Rätvinklig Triangel & Pythagoras Sats Kalkylator

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.

Lösare med 12 variabler för kateter, hypotenusa, höjd, inskriven radie och spetsiga vinklar.

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

Vanliga Frågor

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.