The sides of a triangle are 13 cm, 20 cm and 21 cm. It revolves around a straight line containing

The sides of a triangle are 13 cm, 20 cm and 21 cm. It revolves around a straight line containing the largest of its sides. Find the volume and surface area of a body of revolution.

From the figure you can see that the rotation figure is two cones AOBS and AOBC.

AB perpendicular to SC. Consider two right-angled triangles ASO ACO, in which the AO side is common and equal to the radius of both cones.

We denote the radius of the AO through X, and the segment of the OC through Y.

Then, by the Pythagorean theorem:

R ^ 2 = AS ^ 2 – (SC – Y) ^ 2.

R ^ 2 = AC ^ 2 – Y ^ 2.

Let’s equate both equalities.

AS ^ 2 – (SC – Y) ^ 2 = AC ^ 2 – Y ^ 2.

400 – 441 + 42 * Y – Y2 = 169 – Y ^ 2.

42 * Y = 210.

Y = 210/42 = 5 cm.

R ^ 2 = AC ^ 2 – Y ^ 2.

R ^ 2 = 169 – 25 = 144.

R = AO = 12 cm.

Find the volume of the rotation figure.

V = V1 + V2 = (n * R2 * SO) / 3 + (n * R2 * CO) / 3 = n * R2 * (SO + CO) / 3 = n * 144 * 21/3 = 1008 * n cm3 …

Let’s find the area of ​​the lateral surface of the figure of revolution.

S = S1 + S2 = n * R * AS + n * R * AC = n * R * (AS + AC) = n * 12 * 33 = 396 * n cm2.

Answer: V = 3024 * n cm3, S = 396 * n cm2.