The leg of a right-angled triangle is 12 cm, with an angle of 30 degrees approaching it. Find the area of the triangle.

a and b = 12 cm – legs of the triangle, c – hypotenuse, angle A = 30 degrees.
By the Pythagorean theorem:
c ^ 2 = a ^ 2 + b ^ 2.
Also, from the properties of a right-angled triangle, it is known that opposite an angle of 30 degrees lies a leg 2 times smaller than the hypotenuse. Opposite the angle A = 30 degrees is leg a, then c = 2a.
(2a) ^ 2 = a ^ 2 + 12 ^ 2;
4a ^ 2 = a ^ 2 + 144;
4a ^ 2 – a ^ 2 = 144;
3a ^ 2 = 144;
a ^ 2 = 144/3;
a ^ 2 = 48;
a = √48;
a = 4√3 cm.
The area of a right-angled triangle is equal to half the product of its legs:
S = ab / 2;
S = 4√3 * 12/2 = 24√3 (cm square).
Answer: S = 24√3 cm square.