How do I find the surface area of a cone with just the radius and the height?

Question

I have to find the surface area of a cone with a radius of 45.7 mm and a height of 39.7 mm there’s also a hint that tells me to use the Pythagorean theorem to calculate the slant height..

Answer 1

So the lateral surface area of a cone is pi*r*L, where L is the slant height (the distance from any point on the outer circumference of the base to the vertex at the top of the cone). This formula gives the surface area of everything but the base. The base surface area is of course pi*r², so the total surface area of a cone is:

pi*r*L + pi*r²

We already know r, so now we need to find L. If you imagine a cross section of the cone, you get a triangle. The base is 2r, the height is h, and the length of either of the two sides is L. If we only look at half of this, we get a right triangle. The base of this triangle is r, the height is still h, and the hypotenuse is L.

From that point, we use the Pythagorean Theorem to find L.

L² = r² + h²

L = sqrt( r² + h² )

At this point, plug n play. We already know r and h, so evaluating L gives us L = 60.53577...

Plug everything back into our total surface area formula, and we get:

pi*r*L + pi*r² = 15252.4 mm²

Answer 2

If you think about a cross section of the cone, that cross section would be an isosceles triangle:

You can split the isosceles triangle down the middle to make a right triangle where the hypotenuse is the slant height and the height of the cone is one leg and the radius of the base is the other leg. Does that help?

Answer 3

The slant height, z = sqrt(r² + h²) Pythagorean theorem

The surface area of a cone is the sum of πr² and πz²