Calculate sides, angles, area, and perimeter of right triangles using the Pythagorean theorem. Find hypotenuse or missing leg.
Hypotenuse
5.0000
Side c (longest)
3-4-5 Pythagorean Triple
Classic Pythagorean triple: 3^2 + 4^2 = 5^2
Ratio: 3 : 4 : 5
Area
6.0000
Perimeter
12.0000
Inradius
1.0000
Circumradius
2.5000
Note: In a right triangle, the altitude to the hypotenuse creates two similar triangles.
Note: The median to the hypotenuse always equals half the hypotenuse (c/2).
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
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
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)
Isosceles right triangle
Half of equilateral triangle
Smallest Pythagorean triple
| 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 |
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)
Hypotenuse
5.0000
Side c (longest)
Quick-start with common scenarios
Test your skills with practice problems
Practice with 5 problems to test your understanding.
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.
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.
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.
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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Hypotenuse
5.0000
Side c (longest)