Cone cross-sections are obtained when we cut a cone with a plane. We can obtain different cross-sections depending on the orientation of the plane. It is possible to obtain circular, elliptical, parabolic, and hyperbolic cross-sections.

Here, we will learn about each of the cross-sections of a cone using diagrams.

## Circular cross-section

When a cone is cut by a plane that is parallel to the bases, a circular cross-section is formed.

Circles have the following characteristics.

- Circles have a central point, called the
**center**. - The
**radius**is the constant distance from the center to any point on the circle. - The circles have an
**eccentricity**of e=0.

## Elliptical cross-section

When a cone is cut by a plane that has an inclination with respect to the bases, an elliptical cross-section is formed. The angle of inclination of the plane must be less than the angle of the lateral sides.

Ellipses have the following characteristics:

- The longest diameter of the ellipse is called the
**major axis**. - The shortest diameter is called the
**minor axis**. - The
**center**of the ellipse is the intersection of the two axes. - The sum of the distances from any point on the ellipse to the two
**foci**is constant.

## Parabolic cross-section

When a plane intersects a cone with an inclination parallel to the lateral sides of the cone, a parabolic cross-section is formed.

Parabolas have the following characteristics:

- The
**vertex**is the lowest or highest point on the parabola. The vertex is the point where the curve changes direction. - The
**focus**is on the inside of the parabola. - The
**directrix**is on the outside of the parabola. - The
**axis of symmetry**is the line that connects the vertex and the focus and divides the parabola into two equal parts.

## Hyperbolic cross-section

When a plane cuts a cone at a higher angle to the base of the cone, the cross-section formed is hyperbolic. The angle must be greater than the angle of the lateral sides.

Hyperbolas have the following characteristics:

- They are composed of two branches.
- The two
**vertices**are located one on each branch. These points are located where each branch changes direction. - The
**asymptotes**are two straight lines that the curve approaches but never touches. - The
**center**is the intersection of the two asymptotes. - The two
**foci**are the fixed points, which define the shape of each branch.

## See also

Interested in learning more about cross-sections of geometric figures? Take a look at these pages:

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