The total surface area of a cylinder is 462 cm² and its curved surface area is $\frac{1}{3}$​ of its total surface area. Find the radius of the cylinder.

Given, Total surface area = 462 cm²
Curved surface area = $\frac{1}{3}$​ x Total surface area
Now, Curved surface area = 2πrh

2πrh = $\frac{1}{3}$​× 462
2πrh = 154 cm²
Now, Total surface area = 2πrh + 2πr²
462 = 154 + 2πr²
462 − 154 = 2πr²
2πr² = 308
$2\times \frac{22}{7}\times r^2$ = 308
$r^2=\frac{308\times 7}{2\times 22}$
r² = 49
r = 7 cm
Hence, the radius of the cylinder = 7 cm