The circumference of the base of the truncated cone is 4п and 10п. The height of the cone is exactly 4.

The surface area of ​​the truncated cone is calculated by the formula:

S = n * (r1 + r2) * l + n * r12 + n * r2 ^ 2.

Here r1 and r2 are the radii of the bases, l is the generator.

First, let’s calculate the base radii:

4 * n = 2 * n * r1;

r1 = 2;

10 * n = 2 * n * r2;

r2 = 5.

Now let’s lower the height from the extreme point of the smaller base to the larger one. We get a right-angled triangle, one of the legs of which is equal to the height, and the other – the difference in radii. Let’s find it:

5 – 2 = 3.

By the Pythagorean theorem, you can find the generator:

l = sqrt (9 + 16) = 5.

Then the total surface area of ​​the truncated cone will be equal to:

S = n * (2 + 5) * 5 + n * 4 + n * 25 = 64 * n.

Answer: the total surface area of ​​a truncated cone is 64 * p.