How do you find the volume and surface area of a cone?

Category: 3D Shapes »

Course: Geometry »

 
Relevant MM Lessons:
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Cones (of Dunshire?)

It is only somewhat likely that your teacher will want you to know the cone formulas for volume and surface area. We don't use them as often as we do cylinders, and many standardized tests actually give you the formulas.We will also look at the volume formulas for oblique (slanted) cones, but not surface area as there is no plainly expressible algebra formula.

Right Cone - Volume

It is probably not worth understanding where the cone volume formula comes from, but it is related to cylinder volume the same way pyramids are related to prisms.$$V = \frac{1}{3} \,\, \pi r^2 h$$where $r$ is the radius of the base, and $h$ is the height of the cone measured perpendicular to the ground, not the slant height $l$. Compare that with the volume of a cylinder, which was $\pi r^2 h$ (no $1/3$ coefficient).

Right Cone - Surface Area

The total surface area is the sum of the circular base area and the lateral surface area. The area of the bottom is $\pi r^2$ since it is the area of a circle with radius $\pi$. The lateral surface area is less intuitive, but simple enough. It is the product of $\pi$, the radius, and the slant height ($\pi r l$).Therefore we have$$S = \pi r^2 + \pi r l$$

Oblique Cone - Volume

Oblique cones are slanted so that they are not symmetric - think of a cone that has a center of mass off to the side. Note the slanted center axis that would be perpendicular to the ground in a right cone:The volume formula is identical for oblique cones as it is for right cones:$$V = \frac{1}{3} \,\, \pi r^2 h$$Make sure you use the actual height $h$ which measures how high off the floor the top of the cone is. Neither the slant height nor the axis length is used for volume in any way, and the slant height is only used for the lateral surface area.

Oblique Cylinder - Surface Area

As mentioned, there is no algebraic or formulaic way to get the lateral surface area of a slanted cone. There may be one I haven't come across, but given how difficult it is to get it for cylinders which are a better behaved shape (slanted cylinder surface area involves perimeter of an ellipse), it's certainly not going to see the light of day in any typical geometry or algebra course.

Tips

  • The cone volume formula is similar to that for cylinders but with a $1/3$ in front
  • The lateral surface area is hard to derive but just three things multiplied together
  • Total surface area is the sum of the lateral area and the circular base - depending on the problem you may want only the lateral area
  • Unless you are specifically told otherwise, I wouldn't memorize any of these formulas for cones, especially the surface area, as teachers often provide the formulas