Right Triangle Calculator — Find All Properties from Any 2 Sides
Hypotenuse (c)
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Area
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Angle A (opposite side a)
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Angle B (opposite side b)
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How a Right Triangle Works
A right triangle is a triangle containing exactly one 90-degree (right) angle. The side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides are called legs (or catheti). According to Wolfram MathWorld, the right triangle is one of the most fundamental shapes in all of mathematics, forming the basis of trigonometry, coordinate geometry, and countless engineering applications. The relationship between its sides -- described by the Pythagorean theorem -- has been known for over 4,000 years, with evidence of its use found in ancient Babylonian clay tablets dating to approximately 1800 BCE.
Right triangles appear everywhere in practical life. Builders use them to ensure walls are plumb and floors are level (the 3-4-5 method). Surveyors calculate distances using right triangle trigonometry. According to the National Council of Teachers of Mathematics, understanding right triangles is a prerequisite for algebra, geometry, physics, and engineering courses. Navigation systems, architecture, computer graphics rendering, and even smartphone accelerometers all rely on right triangle mathematics to convert between coordinate systems and calculate distances.
The Right Triangle Formulas
The core formula is the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse. To find a missing side: c = sqrt(a^2 + b^2), a = sqrt(c^2 - b^2), or b = sqrt(c^2 - a^2). The area = (1/2) x a x b. The perimeter = a + b + c.
The two acute angles are calculated using inverse trigonometric functions: Angle A = arctan(a/b) and Angle B = arctan(b/a). These two angles always sum to exactly 90 degrees because the three interior angles of any triangle sum to 180 degrees, and one angle is already 90 degrees.
Worked example: Given legs a = 5 and b = 12. Hypotenuse c = sqrt(25 + 144) = sqrt(169) = 13. Area = (1/2)(5)(12) = 30 square units. Perimeter = 5 + 12 + 13 = 30 units. Angle A = arctan(5/12) = 22.62 degrees. Angle B = arctan(12/5) = 67.38 degrees. Check: 22.62 + 67.38 = 90 degrees.
Key Terms You Should Know
- Hypotenuse -- The side opposite the right angle in a right triangle. It is always the longest side and is denoted as c in the Pythagorean theorem.
- Legs (catheti) -- The two sides that form the right angle. In the Pythagorean theorem, these are a and b.
- Pythagorean triple -- A set of three positive integers (a, b, c) that satisfy a^2 + b^2 = c^2. The smallest is (3, 4, 5). Any scalar multiple of a triple is also a triple.
- Trigonometric ratios -- sin(A) = opposite/hypotenuse, cos(A) = adjacent/hypotenuse, tan(A) = opposite/adjacent. These ratios are defined specifically for right triangles.
- Altitude to the hypotenuse -- A perpendicular line from the right angle vertex to the hypotenuse. Its length equals (a x b) / c and it creates two smaller right triangles similar to the original.
Common Pythagorean Triples Reference Table
Pythagorean triples are integer-sided right triangles. Memorizing the common ones speeds up calculations in construction, carpentry, and standardized tests. There are infinitely many Pythagorean triples, but these are the most frequently used.
| Side a | Side b | Hypotenuse c | Area | Angle A |
|---|---|---|---|---|
| 3 | 4 | 5 | 6 | 36.87 degrees |
| 5 | 12 | 13 | 30 | 22.62 degrees |
| 8 | 15 | 17 | 60 | 28.07 degrees |
| 7 | 24 | 25 | 84 | 16.26 degrees |
| 9 | 40 | 41 | 180 | 12.68 degrees |
| 20 | 21 | 29 | 210 | 43.60 degrees |
The most commonly used triple in construction is (3, 4, 5) and its multiples: (6, 8, 10), (9, 12, 15), (12, 16, 20), etc. Builders mark these distances on strings to verify 90-degree corners without any measuring instruments beyond a tape measure.
Practical Examples
Example 1 -- Ladder against a wall: A 20-foot ladder leans against a wall with its base 8 feet from the wall. How high does it reach? Using a = height, b = 8, c = 20: height = sqrt(20^2 - 8^2) = sqrt(400 - 64) = sqrt(336) = 18.33 feet. The angle with the ground = arccos(8/20) = 66.42 degrees. OSHA recommends a 75-degree angle for safe ladder placement.
Example 2 -- TV screen size: A 55-inch TV has a 16:9 aspect ratio. The diagonal is 55 inches. Width = 55 x cos(arctan(9/16)) = 47.94 inches. Height = 55 x sin(arctan(9/16)) = 26.97 inches. You can verify this: sqrt(47.94^2 + 26.97^2) = sqrt(2298.24 + 727.38) = sqrt(3025.62) = 55.01 inches.
Example 3 -- Roof pitch calculation: A roof has a 6:12 pitch (rises 6 inches for every 12 inches of horizontal run). The rafter length for a 12-foot horizontal span = sqrt(6^2 + 12^2) = sqrt(36 + 144) = sqrt(180) = 13.42 feet. The roof angle = arctan(6/12) = 26.57 degrees. Use the roof replacement cost calculator to estimate the cost for this roof size.
Tips and Strategies
- Always identify the hypotenuse first: The hypotenuse is opposite the right angle and is always the longest side. If you are given the hypotenuse, you solve for a leg; if given two legs, you solve for the hypotenuse.
- Use the 3-4-5 shortcut: If two sides are multiples of 3 and 4, the third is the corresponding multiple of 5. For example, sides 15 and 20 give a hypotenuse of 25 (multiply the 3-4-5 triple by 5).
- Check with angle sum: The two acute angles must add to exactly 90 degrees. If they do not, there is an error in your calculation.
- Remember special triangles: A 45-45-90 triangle has legs in ratio 1:1:sqrt(2). A 30-60-90 triangle has sides in ratio 1:sqrt(3):2. These are worth memorizing for quick mental math.
- Use trigonometry when you have an angle: If you know one leg and one acute angle, use sin, cos, or tan to find the other sides. The sine calculator can help with individual trig computations.
- For non-right triangles, use the Law of Cosines: The Pythagorean theorem only works for right triangles. For oblique triangles, use the Law of Cosines calculator instead.
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a^2 + b^2 = c^2. This theorem applies exclusively to right triangles and is attributed to the Greek mathematician Pythagoras (circa 570-495 BCE), though evidence shows it was known to Babylonian mathematicians over a thousand years earlier. It is the foundation of distance calculations in geometry, physics, and computer science.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers that satisfy the equation a^2 + b^2 = c^2. The most common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (9, 40, 41). Any scalar multiple of a triple is also a valid triple -- for instance, (6, 8, 10) is simply (3, 4, 5) multiplied by 2. There are infinitely many primitive triples (where a, b, and c share no common factor), and they can be generated using Euclid's formula: a = m^2 - n^2, b = 2mn, c = m^2 + n^2 for positive integers m > n.
How do you find the hypotenuse?
To find the hypotenuse, square each leg, add the results, and take the square root: c = sqrt(a^2 + b^2). For example, if the legs are 3 and 4, the hypotenuse = sqrt(9 + 16) = sqrt(25) = 5. If the legs are 5 and 12, the hypotenuse = sqrt(25 + 144) = sqrt(169) = 13. This formula only works when you know both legs. If you know the hypotenuse and one leg, rearrange to find the other leg: a = sqrt(c^2 - b^2).
How do you find the angles of a right triangle?
One angle is always 90 degrees. The other two acute angles can be found using inverse trigonometric functions. If you know sides a and b (the two legs), then Angle A = arctan(a/b) and Angle B = arctan(b/a). Alternatively, if you know a leg and the hypotenuse, use arcsin(leg/hypotenuse) or arccos(leg/hypotenuse). The two acute angles always sum to exactly 90 degrees, which provides a built-in check for your calculations.
What are the special right triangles?
The two special right triangles are the 45-45-90 triangle and the 30-60-90 triangle. In a 45-45-90 triangle, the two legs are equal and the hypotenuse is leg times sqrt(2). In a 30-60-90 triangle, the sides are in the ratio 1 : sqrt(3) : 2, where 1 is the side opposite the 30-degree angle. These appear frequently in geometry, architecture, and standardized math tests. For example, cutting a square diagonally produces two 45-45-90 triangles, and cutting an equilateral triangle in half produces a 30-60-90 triangle.
Can the Pythagorean theorem be used for non-right triangles?
No, the Pythagorean theorem (a^2 + b^2 = c^2) applies only to right triangles. For non-right (oblique) triangles, you need the Law of Cosines: c^2 = a^2 + b^2 - 2ab x cos(C), where C is the angle opposite side c. When the angle C is exactly 90 degrees, cos(90) = 0, and the Law of Cosines reduces to the Pythagorean theorem. Use our Law of Cosines calculator for oblique triangle problems.