![](http://www.scienceiq.com/Images/GrayPix.gif)
![](http://www.scienceiq.com/Images/FactsImages/cone.jpg)
Looking directly toward the cut surface, one would see a triangle with two equal angles at the base (an isosceles triangle). But looking directly along the cut surface, one would see a right triangle. This right triangle is the key to the area of a right cone. Imagine that right triangle being turned all the way around a complete circle along its vertical side. The result is the right cone that you started with. Clearly the area of the body of that cone has been described by the hypotenuse of the triangle as it traveled around in a circle, and the area of the base of the cone has been described by the base of the triangle as it traveled around. The total surface area of the right cone is therefore the sum of these two areas. The area of the base is given by the general formula for the area of a circle. The area of the body is given by the length of the hypotenuse of the right triangle multiplied by p and by the radius of the circle.
The total area of the right cone is then given by adding these two areas together to get the general equation A = prh + pr2. As an example of how to use this equation, imagine that you want to make a cloth tent to use as a cabana or changing room at the beach. The main body of the cabana will be made from a large piece of cloth that you already have, but you need to find out how much more material you need to buy in order to make the peak and floor of the cabana. You know that the cabana will be 4 feet wide and 5 feet high. The peak of the cabana should be 6 feet high, so the vertical height of the cone will be 1 foot, and the radius will be 2 feet. The hypotenuse of the angled side of the peak is calculated by h2 = r2 + l2 = 4 + 1 = 5 h = 5. This means that you will have to have at least A = prh + pr2 = (3.14 X 2 X 5) + (3.14 X 4) = 14.13 + 12.56 = 26.69 square feet of extra material to complete the cabana.