A cone is a three-dimensional closed figure that has a circular base connected to a vertex (or apex) point outside the plane of the base.
You are watching: Perpendicular cross section of a cone
If the segments joining the center of the circle base and vertex point is perpendicular to the base, the cone is a right circular cone. If the segment joining the center of the circle base and vertex point in not perpendicular to the base, the cone is called an oblique circular cone.
h = height; r = radius; s = slant height
The formula for the volume of a cone is very similar to the formula for the volume of a pyramid. The volume of a cone is one-third the product of the base area, πr2, times the height of the cone. Note: A cone is not a pyramid since its base is circular (not a polygon).
Since the base of a cone is a circle, you can see how replacing the B value in the volume of the pyramid with the area of a circle gives us the volume formula for a cone.
Justification of formula by "pour and measure": (For this discussion, our cone will be a right circular cone.) We can conduct an experiment to demonstrate that the volume of a cone is actually equal to one-third the volume of a cylinder with the same base and height. We will fill a right circular cone with water. When the water is poured into a cylinder with the same base and height as the cone, the water fills one-third of the cylinder.
By measurement, it can be concluded that the height (depth) of the water in the cylinder is one-third the height of the cylinder. Since the formula for the volume of the cylinder is V = π r2h, it follows that the volume of the cone can be represented by
Justification of formula by "comparison to pyramid": To use Cavalieri"s Principle, we must have solids whose bases have equal areas and whose cross sections parallel to the bases have equal areas. Can we find a way to have bases of equal area on a right circular cone and a regular square pyramid? The area of the circular base of the cone is πr2. The area of the base of the square pyramid is s2. If these two areas are equal, we must have πr2 = s2. Solving for s, tells us that the side of the square base must have a length of . Now, the circular base and the square base have the same area. If we can establish that cross sections parallel to the bases yield the same areas, we will be able to employ Cavalieri.
Let"s say that our cross section is drawn k units down from the top of both solids. By similar triangles, we know the proportion x / r = k / h, and x = (r)•(k / h). In the cone, with a radius x = (r)•(k / h), the area of the circular cross section isπx2 = π<(r)•(k / h)>2. In the pyramid, we know the proportion y / = k / h, which gives length y = • (k / h). The area of the cross section in the pyramid = <• (k / h) >2. Now, π<(r)•(k / h)>2 = π • r2 • (k / h)2. And, <• (k / h) >2 = π • r2 • (k / h)2. Since the cross sectional areas are also equal, we can employ Cavalieri"s principle and state that the volume of the cone equals the volume of the pyramid. We know that the volume of the pyramid is
. Since the height is the same in both solids, B must equal πr2 for the cone, making the formula for the volume of a cone
The surface area of a closed right cone is a combination of the lateral area and the area of the base.
When cut along the slant side and laid flat, the surface of a cone becomes one circular base and the sector of a circle (lateral surface), as seen in the net at the right.
The length of the arc in the sector is the same as the circumference of the small circular base.
Remember that the area of a sector is a portion of the area of a complete circle. With this in mind, we can use proportions to find the area of a sector.
These calculations refer to the "sector" section of the cone"s net.
The arc length of the sector equals the circumference of the base circle.
The radius of the base circle is r, while the radius of the sector is s.
The base area = area of a circle = πr2. The lateral area (sector) = sπr.
Total Surface Area of a Closed Cone SA = sπr + πr2 SA = surface area r = radius of the base s = slant height of the cone