Introduction
A cone narrows from a circular base to a point, so its horizontal cross sections do not all have the same area. To find its volume by slicing, we need to describe how the radius of a cross section changes with its position. The straight sides of a cone make that relationship especially simple: similar triangles show that the radius changes in direct proportion to the distance from the tip.
Once the radius is expressed as a function of height, the area of each circular slice becomes a function of height as well. Integrating those areas over the cone’s height gives its volume. This calculation also explains why a cone’s volume is one-third the volume of a cylinder with the same base and height.
Setting Up the Cone and Its Slices
Consider a right circular cone with base radius \(R\) and height \(H\), where \(R>0\) and \(H>0\). A right circular cone has a circular base and an axis perpendicular to the base that joins the base’s center to the vertex. Place the vertex at height \(y=0\) and the base at height \(y=H\). A horizontal slice at height \(y\) is a circle; call its radius \(r(y)\).
The slice principle says that a solid’s volume is the definite integral of its cross-sectional area over the interval being sliced. Here the slices are perpendicular to the cone’s axis, and the area at height \(y\) is the area of a circle with radius \(r(y)\). Thus, once \(r(y)\) is known, the volume is obtained by integrating \(\pi r(y)^2\) from \(y=0\) to \(y=H\).
Finding the Radius by Similar Triangles
Take a vertical cross section through the cone’s axis. It forms a triangle whose full height is \(H\) and whose half-base is \(R\). At height \(y\), the smaller triangle from the vertex to the slice has height \(y\) and half-base \(r(y)\). These two triangles are similar: they share the angle at the vertex, and both have a right angle.
Corresponding sides of similar triangles have equal ratios. The radius therefore scales in the same proportion as the distance measured from the vertex:
For \(0<y\le H\), solving for \(r(y)\) gives \(r(y)=Ry/H\). At \(y=0\), the slice is the vertex itself and has radius zero, which agrees with the same formula. The slice area is consequently:
The square is essential: circle area depends on the square of the radius, not the radius alone. The area starts at zero at the tip and grows quadratically until it reaches \(\pi R^2\) at the base.
Deriving the Cone Volume Formula
The area \(A(y)\) is continuous on the full interval from the vertex to the base, so it can be integrated there. Applying the slice principle and substituting the area found from similar triangles gives the volume.
This derivation shows precisely where the factor of one-third comes from: integrating \(y^2\) from zero to \(H\) gives \(H^3/3\). The result requires a right circular cone, with circular slices perpendicular to its axis and a radius that changes linearly from zero at the vertex to \(R\) at the base.
The conclusion is not based on assuming that all the slices have the same area. They do not: the radius, and therefore the slice area, changes continuously along the cone. The integral accounts for each area at its correct height.
Worked Examples
Worked Example: A Cone with a Vertex-Based Coordinate
A right circular cone has base radius \(3\) cm and height \(8\) cm. Find its volume by integrating the circular slice areas, with \(y=0\) at the vertex.
Step 1: Find the slice radius. Similar triangles give \(r(y)/y=3/8\), so \(r(y)=3y/8\) cm for \(0\le y\le8\). The slice area is therefore \(A(y)=\pi(3y/8)^2=9\pi y^2/64\) square centimeters.
Step 2: Integrate over the full height.
Check. The base-area-times-height expression is \(\pi(3)^2(8)=72\pi\ \text{cm}^3\). One-third of this is \(24\pi\ \text{cm}^3\), matching the integral. The units also agree: an area in square centimeters integrated over a length in centimeters gives cubic centimeters.
Worked Example: Measuring Height Down from the Base
A right circular cone has base radius \(5\) m and height \(12\) m. Let \(x\) measure distance from the base upward toward the vertex. Find its volume using \(x\) as the coordinate.
Step 1: Write the radius in terms of \(x\). At \(x=0\), the radius is \(5\) m; at \(x=12\), it is zero. By similar triangles, \(r(x)=5(1-x/12)\) m for \(0\le x\le12\). The area of a slice is \(25\pi(1-x/12)^2\) square meters.
Step 2: Expand and integrate.
Check. Using the cone formula gives \(\frac13\pi(5)^2(12)=\frac13(300\pi)=100\pi\ \text{m}^3\), as required. Measuring from the base changes the radius function and the order in which slice areas are encountered, but not the volume.
Worked Example: The Volume of the Tip Section
A cone has base radius \(6\) cm and height \(10\) cm. Find the volume from its vertex up to a horizontal slice \(7\) cm above the vertex.
Step 1: Model the slices in the section. With \(y=0\) at the vertex, the radius is \(r(y)=6y/10=3y/5\) cm. The required portion extends only from \(y=0\) to \(y=7\), so its slice area is \(A(y)=\pi(3y/5)^2=9\pi y^2/25\) square centimeters.
Step 2: Integrate over the stated bounds.
Check. The top slice has radius \(r(7)=3(7)/5=21/5\) cm. The section is itself a cone of height \(7\) cm and base radius \(21/5\) cm. Its cone-formula volume is \(\frac13\pi(21/5)^2(7)=\frac13\pi(441/25)(7)=1029\pi/25\ \text{cm}^3\), matching the integral. The check uses the radius at the section’s own top, not the original cone’s full base radius.
A Common Modeling Error
A frequent mistake is to treat every slice as if it had the full base radius \(R\). For a cone with radius \(3\) cm and height \(8\) cm, that incorrect model would set \(A(y)=9\pi\) for every height and calculate:
This is the volume of a cylinder with the cone’s base and height, not the cone’s volume. The error ignores that the slice radius is zero at the vertex and grows gradually toward the base. The correction is to find the changing radius first: \(r(y)=3y/8\). Then the slice area is \(9\pi y^2/64\), and its integral is \(24\pi\ \text{cm}^3\).
Why the Slice Model Is Useful
The derivation gives more than a formula to memorize. It connects the geometry of the solid to the function being integrated: similar triangles determine the radius, the circle-area formula determines each slice area, and the definite integral combines those areas through the cone’s height. The coordinate choice affects how the radius is written, but consistent bounds and a correctly modeled radius lead to the same volume.
The same setup is also useful when only part of a cone is included. In that case, the integral bounds must match the portion being measured, as in the tip-section example. Using the full height would calculate the whole cone instead. In every case, check that the radius at the endpoints agrees with the geometry and that the final units are cubic units.
Check Your Understanding
Use similar triangles and slice areas to answer these questions.
- A cone has base radius \(4\) cm and height \(9\) cm. With \(y=0\) at the vertex, what is the radius function \(r(y)\)?
- For the cone in Question 1, write an integral for its volume using circular slice areas. What are the bounds?
- A cone has base radius \(5\) m and height \(6\) m. What is the radius of its slice \(3\) m from the vertex?
- Why does setting every slice radius equal to the base radius produce a cylinder’s volume rather than a cone’s?
- A horizontal slice is \(4\) cm from the vertex of a cone with base radius \(3\) cm and height \(6\) cm. What is the radius of that slice?
