# Finding the Volume and Surface Area of a Right Circular Cone

## Volume and Surface Area of a Right Circular Cone

The volume of a cone is found by taking \(\frac{1}{3}\) of pi times the radius squared times the height. \(V = \frac{1}{3}πr^2h\). The surface area of a cone is found by finding the area of the base, a circle, pi times radius squared, and adding the lateral area or the area around the side pi times radius times the square root of radius squared plus height squared. \(S = πr^2 + πr \sqrt{r^2 + h^2}\). Let’s look at an example.

Find the volume and surface area of the cone in terms of pi and to the nearest whole number. We’ll start with volume. First we’re going to write our volume formula. Volume is 1/3 times the area of the base pi times radius squared times the height of the cone. \(V = \frac{1}{3} πr^2h\).

Now we need to substitute our values. Volume is 1/3 times pi times the radius, 5 inches, squared times the height of the cone which is giving us 12 inches. \(V = \frac{1}{3} π (5in)^2(12in)\). Now we’ll simplify that according to PEMDAS. 5 inches squared is 25 inches squared. Now that all we have left is multiplication we can multiply in any order we want. We could first take \(\frac{1}{3}\) of 12 and we would get 4. \(\frac{1}{3}\) of 12 inches is 4 inches or 12 inches divided by 3 is 4 inches. 4 inches times 25 inches squared is a hundred inches cubed. Then times pi that’s a hundred pi inches cubed so there’s our answer in terms of pi.

Now for the nearest whole number, we multiply times pi and we get that our volume is 314 inches cubed. There are two answers for volume of that cone. Now to find surface area. We’ll start with our formula surface area is the area of the base PI R squared plus pi times radius times the square root of radius squared plus height squared. \(S = πr^2 + πr \sqrt{r^2 + h^2}\).

Then we need to substitute our radius and our height. Surface area is pi times the radius five inches squared plus pi times the radius again five inches times the square root of the radius squared five inches squared plus the height squared twelve inches squared. \(S = π(5in)^2 + π(5in) \sqrt{(5 in)^2 + (12 in)^2}\). Now we need to simplify.

I’ll start here with five inches squared. 25 inches squared times pi. 25 pi inches squared. Now to get to after our plus sign. Five inches times pi is just five pi inches for now and then we’re going to simplify under our radical. Five inches squared 25 inches squared plus 12 inches squared is 144 inches squared. We still need to simplify under our radical so the rest of this isn’t going to change. 25 pi inches plus 5 PI inches times 25 inches squared plus 144 inches squared is 169 inches squared.

Then we need to take the square root of that. The surface area is 25 pi inches plus 5 PI inches times the square root of 169 inches squared is 13 inches. Now we can multiply 5 pi inches times 13 inches. Surface area is 25 pi inches plus 5 times 13. 5 times 10 is 50. 5 times 3 is 15. 65 times pi 65 pi inches times inches squared.

Now finally we can add. 25 pi inches plus 65 pi inches 20 and 60 is 85 and 5 is 10 so 90 PI inches squared. That’s our answer in terms of pi and then to find it to the nearest whole number we need to multiply times PI. 90 times pi is 283 inches squared.