Hypotenuse, missing leg, both acute angles, area and perimeter of a right triangle from any two values, using Pythagoras' theorem and SOHCAHTOA.
By AdminDrafted with AI writing toolsUpdated Checked examples: 6
Legs of 3 and 4 cm meet at the right angle; the hypotenuse is 5 cm, the acute angles are 36.8699° and 53.1301°, and the triangle encloses 6 cm².
The circumcentre of a right triangle is the midpoint of its hypotenuse.
A right triangle is fixed by any two of its parts as long as one of them is a side. With both legs known, Pythagoras' theorem gives the hypotenuse c = √(a² + b²). With a side and an acute angle, the rest follows from sin α = a/c, cos α = b/c and tan α = a/b, and the two acute angles always add up to 90°.
The default legs of 3 cm and 4 cm make the 3-4-5 triangle: hypotenuse 5 cm, area 6 cm², perimeter 12 cm and angles of 36.8699° and 53.1301°. Builders use that ratio to square a corner, since marks 3 m and 4 m from it along two walls should be exactly 5 m apart. The results also include the height onto the hypotenuse (ab/c), the inradius ((a + b − c)/2) and the circumradius (c/2).
Angle α is opposite leg a and β is opposite leg b. Angles can be typed in degrees, radians, gradians or turns and must lie strictly between 0° and 90°.
In a right triangle, the legs a and b and the hypotenuse c (the side opposite the right angle) satisfy a² + b² = c². Legs of 4 m and 7 m give c = √(16 + 49) = √65 ≈ 8.062 m. The rule holds only when the angle between a and b is exactly 90°.
Whole numbers that satisfy a² + b² = c² are called Pythagorean triples. They make easy checks because the hypotenuse comes out exact.
| Leg a, cm | Leg b, cm | Hypotenuse c, cm | Area, cm² | Perimeter, cm |
|---|---|---|---|---|
| 3 | 4 | 5 | 6 | 12 |
| 5 | 12 | 13 | 30 | 30 |
| 6 | 8 | 10 | 24 | 24 |
| 8 | 15 | 17 | 60 | 40 |
| 7 | 24 | 25 | 84 | 56 |
For 8 and 15, c² = 64 + 225 = 289, so c = 17. Scaling a triple keeps its angles: 6-8-10 has the same shape as 3-4-5, with sides twice as long and four times the area.
The theorem also works backwards. If the three sides satisfy a² + b² = c², the angle opposite c is a right angle, which is why marks 3 and 4 units from a corner should sit exactly 5 apart.
Subtract instead of adding: b = √(c² − a²). A 5 m ladder whose foot stands 1.5 m from a wall reaches √(25 − 2.25) = √22.75 ≈ 4.770 m up the wall.
The hypotenuse is always the longest side, so a leg of 5 with a "hypotenuse" of 4 describes no right triangle.
The two legs are a base and its perpendicular height, so the area is ½ab. Legs of 3 cm and 4 cm enclose ½ × 3 × 4 = 6 cm², and the 4 m by 7 m triangle encloses 14 m².
The height from the right angle to the hypotenuse is ab/c: 3 × 4/5 = 2.4 cm for the 3-4-5 triangle. Half the hypotenuse times that height gives the same 6 cm².
The inscribed circle has radius (a + b − c)/2, and the circle through the three corners has radius c/2 because the hypotenuse is its diameter: 1 cm and 2.5 cm for the 3-4-5 triangle.
When one side and an acute angle are known, the trig ratios give the rest. For the angle α opposite leg a:
A hypotenuse of 10 cm at α = 30° gives a = 10 sin 30° = 5 cm and b = 10 cos 30° ≈ 8.660254 cm. The other acute angle is always 90° − α, here 60°.
Going the other way, α = arctan(a/b). For the 3-4-5 triangle, α = arctan(3/4) ≈ 36.869898° and β ≈ 53.130102°. The ladder above meets the ground at arccos(1.5/5) ≈ 72.54°.
The theorem is Proposition 47 in Book I of Euclid's Elements. For two points on a grid, the coordinate geometry calculator applies the same formula as the distance between them.
Checked against: 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2
Checked against: Independent calculation
Checked against: Side opposite 30° is half the hypotenuse
Checked against: tan 45° = 1 so b = a
Square both legs, add them and take the square root: c = √(a² + b²). Legs of 3 and 4 give √25 = 5, and legs of 6 in and 8 in give 10 in. If you know one leg and the angle opposite it instead, divide by the sine: c = a / sin α, so a 5 cm leg opposite 30° means a 10 cm hypotenuse.
For a missing leg, subtract the squares: b = √(c² − a²). A hypotenuse of 13 and a leg of 5 give √(169 − 25) = 12. The hypotenuse must be longer than either leg, or no right triangle exists. With one side and an angle, use SOHCAHTOA: opposite = hypotenuse × sin α and adjacent = hypotenuse × cos α.
A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so the side opposite 30° is half the hypotenuse; a hypotenuse of 10 gives legs of 5 and 8.6603. A 45-45-90 triangle has equal legs and a hypotenuse √2 ≈ 1.4142 times a leg, so legs of 7 give a hypotenuse of 9.8995.
It is a memory aid for the three trigonometric ratios in a right triangle: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. In the 3-4-5 triangle, the angle opposite the side of 3 has sin = 0.6, cos = 0.8 and tan = 0.75, and each inverse function returns the same 36.87°.
Mark 3 units along one side and 4 along the other, measured from the corner; the diagonal between the marks is exactly 5 when the angle is 90°. A longer diagonal means the angle is too wide: 5.05 m on 3 m and 4 m legs is 91.2°, and 4.95 m is 88.8°. Multiples such as 6-8-10 or 9-12-15 give a more precise check on large slabs.
This calculator includes 6 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.