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Solids

Classifying Solids

Concept

Types of Solids

Solids are usually divided into different types, all with different sets of characteristics. If each face of a solid is a polygon, the solid is called a polyhedron. A finer division can consist of types such as prism, pyramid, cylinder, cone, and sphere.

Concept

Prism

A prism is a three-dimensional object created by connecting a polygon with a translated version of the same polygon, vertex to vertex. All the faces are polygons, so prisms are polyhedra.

Concept Prism.svg

Concept

Pyramid

Pyramids are created by connecting a polygon's vertices with a single point, using line segments. Pyramids are also polyhedra.

Concept Pyramid.svg

Concept

Cylinder

A cylinder is made up of two congruent, parallel circles connected in a way corresponding to how the polygons in a prism are connected. As it contains circles, it is not a polyhedron.

Concept Cylinder.svg

Concept

Cone

Cones look like cylinders, though one of the circles is replaced by a point. Thus, they taper from a circle to a point. They are not polyhedra.

Concept Cone.svg

Concept

Sphere

A sphere is the three-dimensional equivalent of a circle. It consists of every point in space at a certain distance away from the center of the sphere. It is not a polyhedron.

Concept Sphere.svg
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Exercise

Name the solids and tell whether they are polyhedra.

Skills Classify the solids.svg
Show Solution
Solution

Example

First solid

The first solid is made up of a circular base and top that are parallel, connected by a smooth surface. Thus, it is a cylinder, which is not a type of polyhedron.

Example

Second solid

The second solid's faces are six quadrilaterals and two hexagons, meaning six-sided polygons, on the left and right side. The two hexagons are parallel and connected vertex to vertex. This means that the second solid is a prism, which is a type of polyhedron. As the two parallel polygons are hexagons, the solid could be called a hexagonal prism.

Concept

Cross-Section

The intersection between a plane and a solid is a two-dimensional surface called a cross-section. Often, it is possible to get different types of cross-sections from the same solid. For instance, the cross-section of a cylinder could be a rectangle or a circle, among other things.

Concept Cross Section.svg
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Exercise

Name the shapes of the cross-sections.

Skills Identify the cross section shapes.svg
Show Solution
Solution

Example

First cross-section

The cross-section of the first solid has four vertices. As there are no parts of the solid that are curved, the cross-section can't have any curves either. Thus, the cross-section is in the shape of a quadrilateral.

Example

Second cross-section

This cross-section has three vertices instead. Similarly, the cube has no curved parts, meaning the cross-section has no curves. Therefore, the cross-section is a triangle.

Concept

Solid of Revolution

A solid of revolution is a three-dimensional object formed by rotating some two-dimensional object around an axis. The axis is referred to as the axis of revolution. Rotating a rectangle with one side against the axis of revolution creates a cylinder.

Depending on the shape used, many different types of solids can be created.

Rotate

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Exercise

Sketch and describe the solid of revolution corresponding to the following shape and axis.

Show Solution
Solution

Notice that if the top segment of the shape were extended to the left, the shape would be a triangle. Thus, we can think of it as a triangle without its pointy top. This means that the solid of revolution will be the same as though the shape were a triangle, but without the top. Thus, the resulting solid of revolution is a cone without a top.

It's base radius is units, top radius is unit, and its height is units.

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