AI Math Assistant
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About This Calculator
Right triangles contain one 90-degree angle and obey special relationships that make them uniquely useful in trigonometry and practical measurement. This calculator solves for any missing side or angle using the Pythagorean theorem and trigonometric ratios, given at least two known values. Surveyors, architects, and navigators depend on right triangle calculations to determine heights, distances, and angles that cannot be measured directly.
Quick Tips
- 1 The hypotenuse is always the longest side, opposite the 90-degree angle.
- 2 Common Pythagorean triples (3-4-5, 5-12-13) let you skip calculations.
- 3 SOH-CAH-TOA: Sine=Opposite/Hypotenuse, Cosine=Adjacent, Tangent=Opp/Adj.
Example Calculation
A ladder reaches 12 feet up a wall, base 5 feet from the wall.
Hypotenuse (ladder): 13 ft | Angle at base: 67.38 degrees | Area: 30 ft2
Right Triangle Properties
A right triangle has one 90° angle. The side opposite the right angle is the hypotenuse — the longest side. The other two sides are called legs. The Pythagorean theorem (a² + b² = c²) relates the three sides and is one of the most fundamental relationships in mathematics.
Calculating Angles in a Right Triangle
Since one angle is always 90°, the other two must sum to 90° (complementary). Use trigonometry: angle A = arctan(a/b) and angle B = arctan(b/a). Alternatively, angle A = arcsin(a/c) or arccos(b/c). All methods give the same result.
Common Right Triangle Ratios
Special right triangles have memorized ratios. The 3-4-5 triangle (and multiples like 6-8-10, 5-12-13) are Pythagorean triples. The 45-45-90 triangle has sides in ratio 1:1:√2. The 30-60-90 triangle has sides in ratio 1:√3:2. These shortcuts speed up calculations.
Applications of Right Triangles
Right triangles are used in construction (roof pitch), navigation (bearing calculations), surveying (measuring heights), physics (force components), and computer graphics (distance calculations). Trigonometric functions (sine, cosine, tangent) are all defined using right triangles.
Frequently Asked Questions
The hypotenuse is the longest side of a right triangle, located opposite the 90° angle. Its length is c = √(a² + b²). For a 3-4-5 triangle, the hypotenuse is 5.
Pythagorean triples are sets of three positive integers where a² + b² = c². Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Any multiple of a triple is also a triple (e.g., 6,8,10).
If you know the hypotenuse (c) and one leg (a), the other leg is b = √(c² - a²). This calculator requires both legs and computes the hypotenuse.
Area = ½ × a × b, where a and b are the two legs. The legs serve as the base and height since they are perpendicular to each other.