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A cone is a three-dimensional closed number that has actually a one base connected to a crest (or apex) suggest outside the airplane of the base.

You are watching: Perpendicular cross section of a cone

comparable Cross part (parallel come base)
ideal Circular Cone

If the segment joining the center of the circle base and also vertex suggest is perpendicular come the base, the cone is a appropriate circular cone. If the segment involvement the center of the circle base and also vertex allude in no perpendicular come the base, the cone is dubbed an slope circular cone.

h = height; r = radius; s = slant height

The formula for the volume of a cone is very comparable to the formula because that the volume that a pyramid. The volume the a cone is one-third the product of the basic area, πr2, time the elevation of the cone. Note: A cone is no a pyramid due to the fact that its basic is circular (not a polygon).
Since the base of a cone is a circle, you deserve to see just how replacing the B worth in the volume the the pyramid with the area of a circle offers us the volume formula because that a cone.

Justification of formula by "pour and measure": (For this discussion, our cone will certainly be a best circular cone.) We have the right to conduct an experiment to demonstrate that the volume the a cone is actually equal to one-third the volume of a cylinder through the same base and height. We will certainly fill a appropriate circular cone v water. Once the water is poured into a cylinder v the very same base and also height as the cone, the water filling one-third that the cylinder.
The basic of the cone is a circle, with an area π r2.
The basic of the cylinder is likewise a circle through an area of π r2.
The elevation of the cone and the cylinder is h.
The volume of the cylinder is V = π r2 h.
Since the water native the cone filling one-third that the cylinder, the volume of the cone is one-third the volume of the cylinder:

By measurement, it deserve to be concluded the the height (depth) the the water in the cylinder is one-third the height of the cylinder. Since the formula for the volume the the cylinder is V = π r2h, it adheres to that the volume the the cone can be represented by


Justification of formula by "comparison come pyramid": To usage Cavalieri"s Principle, we must have solids who bases have actually equal areas and whose overcome sections parallel to the bases have equal areas. Can we find a means to have bases of equal area top top a right circular cone and also a continuous square pyramid? The area the the circular basic of the cone is πr2. The area that the basic of the square pyramid is s2. If this two locations are equal, us must have actually πr2 = s2. resolving for s, tells us that the next of the square basic must have a length of . Now, the one base and the square base have actually the very same area. If we can create that cross sections parallel to the bases productivity the very same areas, we will be able to employ Cavalieri.


Let"s say the our cross section is drawn k systems down indigenous the optimal of both solids. By comparable triangles, we understand the ratio x / r = k / h, and also x = (r)•(k / h). In the cone, with a radius x = (r)•(k / h), the area that the circular cross ar isπx2 = π<(r)•(k / h)>2. In the pyramid, we understand the ratio y / = k / h, which gives length y = • (k / h). The area that the cross ar in the pyramid = <• (k / h) >2. Now, π<(r)•(k / h)>2 = πr2 • (k / h)2. And, <• (k / h) >2 = πr2 • (k / h)2. Due to the fact that the cross sectional areas are likewise equal, we deserve to employ Cavalieri"s principle and state the the volume of the cone equates to the volume the the pyramid. We know that the volume the the pyramid is

. Because the elevation is the exact same in both solids, B have to equal πr2 because that the cone, do the formula for the volume of a cone

The surface area of a closed right cone is a mix of the lateral area and also the area the the base.
When cut along the slant side and also laid flat, the surface of a cone i do not care one circular base and the sector of a one (lateral surface), as seen in the network at the right.

The length of the arc in the sector is the same as the circumference of the little circular base.

Remember the the area the a sector is a section of the area of a complete circle. V this in mind, we can use proportions to find the area that a sector.

These calculations refer to the "sector" section of the cone"s net.
The arc length of the sector amounts to the one of the basic circle.
The radius of the basic circle is r, when the radius that the sector is s.

The basic area = area of a circle = πr2. The lateral area (sector) = sπr.

See more: How Many Grams Are In 1/8 Of An Ounce, 1/8 Of An Ounce

Note: The area of the ar is fifty percent the product of the slant height and also the circumference of the base. sπr = ½ s • 2πr

Total surface ar Area the a closeup of the door Cone SA = sπr + πr2 SA = surface area r = radius that the basic s = slant elevation of the cone

In a best circular cone, the slant height, s, can be discovered using the Pythagorean Theorem.