Right Triangle Calculator · Practical guide
3 4 5 Right Triangle Angles
The most famous right triangle: sides 3, 4, and 5 with angles of about 36.87°, 53.13°, and 90°.
At a glance
The Complete Answer
A right triangle with legs 3 and 4 and hypotenuse 5 has three angles: 36.87° opposite the side of length 3, 53.13° opposite the side of length 4, and 90° at the right angle. Its area is (3 × 4) ÷ 2 = 6 square units and its perimeter is 3 + 4 + 5 = 12 units.
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Right Triangle Solution
Every Measurement at a Glance
| Property | Value | How It Is Found |
|---|---|---|
| Hypotenuse | 5 | √(3² + 4²) = √25 |
| Angle opposite side 3 | 36.87° | arctan(3 ÷ 4) |
| Angle opposite side 4 | 53.13° | 90° − 36.87° |
| Area | 6 | (3 × 4) ÷ 2 |
| Perimeter | 12 | 3 + 4 + 5 |
Where the Angles Come From
The smaller angle always faces the shorter leg. With legs 3 and 4, the angle opposite the leg of 3 is arctan(3/4) = 36.8699° ≈ 36.87°, and the two acute angles must sum to 90°, so the other is 53.13°. These are not round numbers — unlike the 30-60-90 and 45-45-90 triangles — which is why the 3-4-5 triangle is usually remembered by its sides rather than its angles.
Scaled Versions Keep the Same Angles
Any multiple of the triple — 6-8-10, 9-12-15, 1.5-2-2.5 — is a similar triangle with identical angles of 36.87° and 53.13°. Only the area and perimeter scale: a 6-8-10 triangle has an area of 24 and perimeter of 24. Carpenters and builders exploit this constantly, using a 3-4-5 layout to verify that a corner is exactly square.
Frequently Asked Questions
Is a triangle with sides 3, 4, and 5 always a right triangle?
Yes. Because 3² + 4² = 9 + 16 = 25 = 5², the converse of the Pythagorean theorem guarantees the angle opposite the side of 5 is exactly 90°.
Which angle is opposite the hypotenuse?
The 90° right angle. The hypotenuse is always opposite the right angle, and each acute angle sits opposite one of the legs — 36.87° opposite the leg of 3 and 53.13° opposite the leg of 4.
Why is the 3-4-5 triangle used in construction?
Measuring 3 feet along one line and 4 along another, then checking that the diagonal equals 5, confirms a perfect right angle without any angle-measuring tools.
What are the angles of a 5-12-13 triangle?
They are arctan(5/12) ≈ 22.62° and 67.38°, with the 90° right angle. Like 3-4-5, the angles are not round numbers, but the sides form a perfect integer triple.