Area: half the product of the two legs
Because a right triangle is exactly half of a rectangle formed by its two legs, its area is (leg a × leg b) ÷ 2. For legs 3 and 4: (3 × 4) ÷ 2 = 6. For legs 5 and 12: (5 × 12) ÷ 2 = 30.
The two acute angles: arctangent of the opposite-to-adjacent ratio
The angle opposite leg a is found by taking the arctangent of a divided by b. For legs 3 and 4: arctan(3/4) ≈ 36.8699°, and the other angle is 90° − 36.8699° = 53.1301°. For legs 5 and 12: arctan(5/12) ≈ 22.6199°, and the other angle is 90° − 22.6199° = 67.3801°.
Why the two acute angles always sum to 90°
A triangle's three angles always sum to 180°, and a right triangle has one angle fixed at 90° by definition — that leaves exactly 90° to be split between the other two. This is why the second angle is always calculated as 90° minus the first, rather than with a second independent arctangent call.
Putting it all together
For the 3-4-5 triangle: hypotenuse 5, area 6, perimeter 12, angles 36.8699° and 53.1301°. For the 5-12-13 triangle: hypotenuse 13, area 30, perimeter 30, angles 22.6199° and 67.3801°. Every one of these values comes from the same two starting numbers — the two legs — since a right triangle is fully determined by them.