Right Triangle Calculator
Calculate sides, angles, area, and perimeter of right triangles using the Pythagorean theorem. Find hypotenuse or missing leg.
Triangle Results
Hypotenuse
5.0000
Side c (longest)
Triangle Input
*Special Triangle Detected!
3-4-5 Pythagorean Triple
Classic Pythagorean triple: 3^2 + 4^2 = 5^2
Ratio: 3 : 4 : 5
Interactive Triangle Diagram
Complete Results
Sides
Angles
Area
6.0000
Perimeter
12.0000
Inradius
1.0000
Circumradius
2.5000
Altitudes & Medians
Altitudes (Heights)
Note: In a right triangle, the altitude to the hypotenuse creates two similar triangles.
Medians
Note: The median to the hypotenuse always equals half the hypotenuse (c/2).
All Six Trig Functions
For Angle A = 36.87 deg
sin(A) = opp/hyp = a/c
0.600000
cos(A) = adj/hyp = b/c
0.800000
tan(A) = opp/adj = a/b
0.750000
csc(A) = 1/sin(A)
1.666667
sec(A) = 1/cos(A)
1.250000
cot(A) = 1/tan(A)
1.333333
For Angle B = 53.13 deg
sin(B) = opp/hyp = b/c
0.800000
cos(B) = adj/hyp = a/c
0.600000
tan(B) = opp/adj = b/a
1.333333
csc(B) = 1/sin(B)
1.250000
sec(B) = 1/cos(B)
1.666667
cot(B) = 1/tan(B)
0.750000
SOHCAHTOA Reference
Pythagorean Theorem Verification
a^2 + b^2 = c^2
3.00^2 + 4.00^2 = 5.00^2
9.0000 + 16.0000 = 25.0000
25.0000 = 25.0000 (check)
Special Right Triangles Reference
45-45-90 Triangle
Isosceles right triangle
- Angles: 45 deg, 45 deg, 90 deg
- Side ratio: 1 : 1 : sqrt(2)
- If leg = x, hypotenuse = x * sqrt(2)
30-60-90 Triangle
Half of equilateral triangle
- Angles: 30 deg, 60 deg, 90 deg
- Side ratio: 1 : sqrt(3) : 2
- Short : Long : Hypotenuse
3-4-5 Triangle
Smallest Pythagorean triple
- Sides: 3, 4, 5 (or any multiple)
- Angles: 36.87 deg, 53.13 deg, 90 deg
- 3^2 + 4^2 = 9 + 16 = 25 = 5^2
Common Pythagorean Triples
| a | b | c | Verification |
|---|---|---|---|
| 3 | 4 | 5 | 3^2 + 4^2 = 9 + 16 = 25 |
| 5 | 12 | 13 | 5^2 + 12^2 = 25 + 144 = 169 |
| 8 | 15 | 17 | 8^2 + 15^2 = 64 + 225 = 289 |
| 7 | 24 | 25 | 7^2 + 24^2 = 49 + 576 = 625 |
| 20 | 21 | 29 | 20^2 + 21^2 = 400 + 441 = 841 |
| 9 | 40 | 41 | 9^2 + 40^2 = 81 + 1600 = 1681 |
Right Triangle Formulas
Pythagorean Theorem
c = sqrt(a^2 + b^2)
Area
A = (a * b) / 2
Angle from sides
angle = arctan(opposite/adjacent)
Altitude to hypotenuse
h = (a * b) / c
Inradius
r = (a + b - c) / 2
Circumradius
R = c / 2
Median to hypotenuse
m_c = c / 2
Median formula
m_a = sqrt((2b^2 + 2c^2 - a^2) / 4)
Triangle Results
Hypotenuse
5.0000
Side c (longest)
Common Right Triangle Examples
Quick-start with common scenarios
Right Triangle Practice Problems
Test your skills with practice problems
Practice with 5 problems to test your understanding.
?How Do You Solve a Right Triangle?
A right triangle has one 90-degree angle. Use Pythagorean theorem (a^2 + b^2 = c^2) to find sides, and trigonometry (SOHCAHTOA) for angles. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Given two values (sides or angles), you can find all others.
What is a Right Triangle?
A right triangle is a triangle containing one 90-degree (right) angle. The side opposite the right angle is called the hypotenuse and is always the longest side. Right triangles are fundamental to trigonometry, and their properties are used extensively in engineering, physics, architecture, and navigation.
Key Facts About Right Triangles
- One angle is always 90 degrees; other two sum to 90
- Pythagorean theorem: a^2 + b^2 = c^2 (c is hypotenuse)
- SOHCAHTOA: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj
- Hypotenuse is the longest side (opposite the right angle)
- Area = (1/2) * base * height = (1/2) * leg1 * leg2
- Special triangles: 30-60-90 (1:sqrt(3):2) and 45-45-90 (1:1:sqrt(2))
- Inverse trig functions find angles: arcsin, arccos, arctan
- Given any two sides, or one side and one angle, solve the whole triangle
Quick Answer
A right triangle has one 90-degree angle. Use Pythagorean theorem (a^2 + b^2 = c^2) to find sides, and trigonometry (SOHCAHTOA) for angles. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Given two values (sides or angles), you can find all others.
Frequently Asked Questions
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides: a^2 + b^2 = c^2. This only applies to right triangles (triangles with one 90 degree angle).
A right triangle is a triangle with one angle measuring exactly 90 degrees (a right angle). The two sides forming the right angle are called legs, and the side opposite the right angle is called the hypotenuse, which is always the longest side.
To find the hypotenuse when you know both legs: c = sqrt(a^2 + b^2). Square each leg, add them together, then take the square root. For example, if legs are 3 and 4: c = sqrt(9 + 16) = sqrt(25) = 5.
To find a missing leg when you know the hypotenuse and one leg: a = sqrt(c^2 - b^2). Square the hypotenuse, subtract the square of the known leg, then take the square root. Example: c=10, b=6, so a = sqrt(100 - 36) = sqrt(64) = 8.
Two special right triangles: 45-45-90 triangle (isosceles right triangle) with sides in ratio 1:1:sqrt(2), and 30-60-90 triangle with sides in ratio 1:sqrt(3):2. These ratios help solve problems quickly without calculation.
SOHCAHTOA is a mnemonic for trigonometric ratios in right triangles: SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), TOA (Tangent = Opposite/Adjacent). These let you find angles from sides or sides from angles.
Last updated: 2025-01-15
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Triangle Results
Hypotenuse
5.0000
Side c (longest)