Trigonometry · Right Triangles
Right Triangle
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Solve any right triangle from any 2 known values — sides or angles. Get all properties, SOH-CAH-TOA ratios, altitude decompositions, and a live SVG diagram.
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Hypotenuse
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Area
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Angle α
Choose what you know:
Known: Legs a and b
Enter both legs of the right triangle. The right angle C = 90° is always known.
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Right Triangle
Hypotenuse c
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Area
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Angle α
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Angle β
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📐 Complete Right Triangle Solution
SOH-CAH-TOA (for angle α)
All 6 Trig Ratios
📋 Step-by-Step
Solving a Right Triangle
A right triangle has one 90° angle. With any 2 known values (2 sides, or 1 side + 1 angle), you can find all remaining sides and angles using the Pythagorean theorem and trigonometry.
SOH-CAH-TOA and Right Triangle Formulas
SOH: sin(α) = opposite/hypotenuse = a/c
CAH: cos(α) = adjacent/hypotenuse = b/c
TOA: tan(α) = opposite/adjacent = a/b
From 2 legs (a,b): c=√(a²+b²), α=arctan(a/b), β=arctan(b/a)
From a+α: b=a/tan(α), c=a/sin(α), β=90°−α
From b+α: a=b·tan(α), c=b/cos(α), β=90°−α
From c+α: a=c·sin(α), b=c·cos(α), β=90°−α
From a+c: b=√(c²−a²), α=arcsin(a/c), β=arccos(a/c)
From b+c: a=√(c²−b²), α=arccos(b/c), β=arcsin(b/c)
45-45-90: a=b, c=a√2, α=β=45°
30-60-90: b=a√3, c=2a, α=30°, β=60°
What is SOH-CAH-TOA?
SOH-CAH-TOA is a memory aid for the three basic trig functions in a right triangle. SOH: Sine = Opposite/Hypotenuse. CAH: Cosine = Adjacent/Hypotenuse. TOA: Tangent = Opposite/Adjacent. "Opposite" and "adjacent" are relative to the angle you're considering. For angle α opposite leg a: sin(α)=a/c, cos(α)=b/c, tan(α)=a/b. The inverse functions (arcsin, arccos, arctan) let you find angles from ratios.
What is the altitude from the right angle to the hypotenuse?
The altitude h from the right angle C to hypotenuse c divides the hypotenuse into segments p and q, where p = a²/c and q = b²/c. Key relationships: h = ab/c, h² = pq (geometric mean), a² = pc, b² = qc. This creates three similar right triangles — the whole triangle and the two sub-triangles formed by the altitude. This "geometric mean altitude" property is used in many geometry proofs.
How do special right triangles work?
The 45-45-90 triangle has legs in ratio 1:1. If leg = a, then hypotenuse = a√2. Angles are 45°, 45°, 90°. It's half a square cut diagonally. The 30-60-90 triangle has sides in ratio 1:√3:2. Short leg a (opposite 30°), long leg a√3 (opposite 60°), hypotenuse 2a. It's half an equilateral triangle. These triangles have exact trig values: sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2.
How do you find all angles given only two sides?
If you know both legs a and b: α = arctan(a/b), β = arctan(b/a) = 90°−α. If you know leg a and hypotenuse c: α = arcsin(a/c), β = arccos(a/c). If you know leg b and hypotenuse c: β = arcsin(b/c), α = arccos(b/c). If you only know one side and no angles (just c), you cannot determine the triangle — the two acute angles are undetermined (any right triangle with that hypotenuse is possible).