Profile of a solid

The idea

When we see a solid object we see its profile from a specific viewpoint, and the outline of this profile can vary greatly with the viewpoint.

One cone, many looks

When we see a solid object, we really see its profile: its outline from the place where we stand. Think of a cone, like a party hat.

Which outline can a cone have?

• Only a triangle
• Only a circle
• A triangle or a circle, depending on where we look from

Turn the solids

Drag the sliders, or drag the solid itself. The dark shape is the outline you would see. Find the outlines in the list below.

Same solid, different outlines

When we see a solid object we see its profile from a specific viewpoint, and the outline of this profile can vary greatly with the viewpoint.

A cone seen from the side has a triangle outline. Seen from straight above or below it has a circle. A cylinder gives a rectangle from the side and a circle from the end.

So one outline never tells the whole story of a solid.

Notes

A solid has a profile from each viewpoint. The outline of the profile can be very different from different viewpoints: a cone is a triangle from the side and a circle from below.

Check yourself

A coin lies flat on a table. You look straight down at it. What outline do you see?

Which solid can show both a circle and a rectangle as its outline?

Which of these can never be the outline of a cube?

Asha looks at this cone from the side and says, "A cone is a triangle." What is wrong with that?

• A thin rectangle. That is the outline from the side, at table level. From straight above you look at the flat face.
• A circle — correct. Yes! From above you see the round face of the coin.
• A triangle. A coin has no pointed part, so no view of it is a triangle.
• Cone. A cone gives a triangle and a circle, but never a rectangle.
• Cube. A cube gives squares, rectangles and hexagons, but never a circle.
• Cylinder — correct. Yes! From the end it is a circle and from the side a rectangle.
• Sphere. A sphere gives only a circle from every viewpoint.
• A square. Looking straight at a face of the cube, the outline is a square.
• A circle — correct. Yes! A cube has straight edges and flat faces, so its outline is made of straight lines.
• A rectangle. Turn the cube half-way (45°) about a vertical edge and the outline is a rectangle.
• A hexagon. Look at a corner of the cube head-on and the outline is a hexagon.
• A cone has no outline. Every solid has an outline from a given viewpoint.
• A cone is a triangle from every viewpoint. Not so. The outline changes with the viewpoint.
• The outline depends on the viewpoint: from below the same cone shows a circle — correct. Yes! A solid has many profiles, one for each viewpoint.