Welcome to Geometry for beginners. This article discusses the surface area and volume of the cones. Probably the most common visual depiction of people with a cone is the ice cream cone; but my personal favorite comes from carnivals and state fairs — a cinnamon salad, roasted with pecans or cashews, in a red and white striped cone. Both ice cream and smoked nuts give us excellent examples of surface area and volume applications.
As with all three-dimensional shapes, the application of surface area is a package or container. For our visual images, the surface area will be represented by the ice cream cone itself, which contains ice cream and red and white cone-shaped paper, which contains nuts. The application of the volume would be the ice cream itself and the nuts that fall into the paper cone. For sellers at exhibitions, fairs and carnivals, both of these concepts are extremely important. Suppliers cannot afford to exit containers or products that go inside. Poor planning can be costly in terms of lost sales. These examples, of course, are not the only applications of cones, but they are among the best tasting.
If you have already read articles about prisms and pyramids, you know that they are similar to each other and have similar formulas. The same is true for cylinders and cones. The difference is that one base (cone) and 2 bases (cylinder).
The formula for the surface area of the cones: SA = B + LA, where sa refers to the surface area, b refers to AREA base, and LA refers to the lateral area.
This formula is exactly the same as the original formula for the pyramid. ATTENTION! This formula will be very different and can be difficult to memorize. In other situations, I advised only to remember this original formula, and then substitute the corresponding previously developed formula of the polygon. However, at this time everything is different. The form that we receive when we open our cone is NOT one of the polygons when they have learned before, and we will need a new terminology.
For a cone, the base is a circle, so the first change to the original formula looks like SA = pi r ^ 2 + LA. It is this side area that will give us problems.
The picture makes a vertical cut in the cone, like an Indian Ti Pei, and then opens it and lays out the open form. The shape will look like a big wedge of pizza, but it will not be all pizza. Now, using the same "limiting" process that we used in calculating the area of circles, we mentally cut this wedge into many parts and fit them together, alternating a point up and down. We will again use the "taking limit" of this process. The end result of this process is a rectangle whose length is equal to half the circumference of the main circle - 1/2 (2 pi r) or pi r, and its height - inclined height s.
Slope height is a new terminology that we must study. While the height of the cone is the perpendicular distance straight to the ground, the height of the slope is the height of the SIDE of the cone. This is the height of the material (skin) from the mosaic, measured from top to bottom. This is the length or height of the sloping side of the cone.
Having made the final replacement, SA = B + LA becomes SA = (pi r ^ 2) + (pi r) s, where r is the radius of the lower circle, s is the inclined height of the side of the cone.
Phew! Now you understand why I said that you need to remember this final formula. Fortunately, the volume formula is not that complicated.
The formula for the volume of cones: V = (1/3) B h, where B is the zone of the base, h is the perpendicular height of the cone, 1/3 comes from the fact that, as in the case of pyramids and prisms, 3 cones are required to fill a cylinder with the same base and height. Thus, V = (1/3) B h becomes V = (1/3) (pi r ^ 2) h.
Summarize:
(1) The formula for the surface area of the cone SA = B + SA or SA = pi r ^ 2 + pi r s; and area is always measured in square units.
(2) The formula for the cone volume is V = (1/3) B h or V = (1/3) pi r ^ 2 h; and volume is always measured in cubic units.
