Find the volume of the surface of the figure generated by rotating a 30-60-90 triangle about its longest leg if the hypotenuse is 8 in.

Question

I’m so confused on where to start with this

Answer 1

Hi Jess!

OK, any cylinder has a height h and a radius r. Your cylinder, like any other cylinder, has a volume equal to pi*r*r*h. pi*r*r is the area of the circle that is the cylinder's base.

The 'longest leg' of your triangle is the leg that is longer. It isn't the hypoteneuse; that isn't a leg, right? So now, let's think about how to find which leg is the longer.

In any triangle, the angle opposite to a side is larger if the side is longer. More generally, the ninety-degree right angle in this 30-60-90 triangle is the largest angle, and it is indeed opposite to the longest side. Further, the smaller the angle, the shorter the opposite side, so this implies:

Because 90 > 60 > 30, then

(hypoteneuse length) > (length of leg opposite to 60) > (length of leg opposite to 30)

OK, so now we can use trig to find the lengths of the legs. The mnemonic term SOHCAHTOA can be used to remember that:

*Sine of an angle is opposite leg over hypoteneuse,

*Cosine of an angle is adjacent leg over hypoteneuse, and

*Tangent of an angle is opposite leg over adjacent leg.

You know your hypoteneuse length. You also know that the leg opposite to the 60 is the longest leg, so you can write:

height = h = (hypoteneuse)*sin 60

and

radius = r = (hypoteneuse) * cos 60

Substitute these into your formula for volume, plug in numbers, and you're done!

Cheers,

Joe