A plane intersects a cone…?

When a cone is cut by a plane that is parallel to its base and that does not pass through its vertex, the plane figure formed by the intersection is a circle.

Think about all the other ways a plane can intersect a cone. Make a sketch of each different shape that can be formed. For each shape, explain in words or with a sketch how the plane must cut the cone to form the shape.

Okay, so I know that if it intersects perpendicular to the base, it’ll form a triangle-ish shape. If it intersects diagonally, wouldn’t it make an oval? And how many other ways can it intersect?

Top 3 Answers

A few days ago

Mike

Favorite Answer

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A few days ago

Anonymous

You’ve got the idea. What you’re calling a “triangularish shape” is actually two crossed lines. What you’re calling an oval is called an ellipse, but the idea is the same.

There are two other ways it can intersect. Think about the plane being parallel to one of the sides of the cone. the plane will intersect only one side of the cone – the top or the bottom – and the intersection will be a U-shaped curve. This is called a parabola.

Now tip that plane a little bit so it intersects the other side of the cone. Now you’ll have two U-shapes. This is called a hyperbola. (If you’re British, that would be ‘an hyperbola’.)

So: circle, ellipse, parabola, hyperbola, and two crossed lines.

5 years ago

you SHOULD know { if it cuts the cone } that perpendicular to axis is a circle , parallel to axis is hyperbola , parallel to the cone is a parabola , any other is an ellipse if it cuts the cone { a point is a degenerate ellipse and a line (pair) is a degenerate hyperbola }

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A plane intersects a cone…?

A plane intersects a cone…?

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