## Space Figures and Basic Solids

# Space figures and basic solids

Space figures

Cross-section

Volume

Surface area

Cube

Cylinder

Sphere

Cone

Pyramid

Tetrahedron

Prism

### Space Figure

A space figure or three-dimensional figure is a figure that has depth in addition to width and height. Everyday objects such as a tennis ball, a box, a bicycle, and a redwood tree are all examples of space figures. Some common simple space figures include cubes, spheres, cylinders, prisms, cones, and pyramids. A space figure having all flat faces is called a polyhedron. A cube and a pyramid are both polyhedrons; a sphere, cylinder, and cone are not.

### Cross-Section

A cross-section of a space figure is the shape of a particular two-dimensional "slice" of a space figure.

Example:

The circle on the right is a cross-section of the cylinder on the left.

The triangle on the right is a cross-section of the cube on the left.

### Volume

Volume is a measure of how much space a space figure takes up. Volume is used to measure a space figure just as area is used to measure a plane figure. The volume of a cube is the cube of the length of one of its sides. The volume of a box is the product of its length, width, and height.

Example:

What is the volume of a cube with side-length 6 cm?

The volume of a cube is the cube of its side-length, which is 6^{3} = 216 cubic cm.

Example:

What is the volume of a box whose length is 4cm, width is 5 cm, and height is 6 cm?

The volume of a box is the product of its length, width, and height, which is 4 × 5 × 6 = 120 cubic cm.

### Surface Area

The surface area of a space figure is the total area of all the faces of the figure.

Example:

What is the surface area of a box whose length is 8, width is 3, and height is 4? This box has 6 faces: two rectangular faces are 8 by 4, two rectangular faces are 4 by 3, and two rectangular faces are 8 by 3. Adding the areas of all these faces, we get the surface area of the box:

8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3 =

32 + 32 + 12 + 12 +24 + 24=

136.

### Cube

A cube is a three-dimensional figure having six matching square sides. If *L* is the length of one of its sides, the volume of the cube is *L*^{3}* = L *× *L* × *L*. A cube has six square-shaped sides. The surface area of a cube is six times the area of one of these sides.

Example:

The space figure pictured below is a cube. The grayed lines are edges hidden from view.

Example:

What is the volume and surface are of a cube having a side-length of 2.1 cm?

Its volume would be 2.1 × 2.1 × 2.1 = 9.261 cubic centimeters.

Its surface area would be 6 × 2.1 × 2.1 = 26.46 square centimeters.

### Cylinder

A cylinder is a space figure having two congruent circular bases that are parallel. If *L* is the length of a cylinder, and *r* is the radius of one of the bases of a cylinder, then the volume of the cylinder is *L* × *pi* × *r*^{2}*, and the surface area is 2 × r* × *pi* ×* L* + 2 × *pi* × *r*^{2}*.*

Example:

The figure pictured below is a cylinder. The grayed lines are edges hidden from view.

### Sphere

A sphere is a space figure having all of its points the same distance from its center. The distance from the center to the surface of the sphere is called its radius. Any cross-section of a sphere is a circle.

If *r* is the radius of a sphere, the volume *V* of the sphere is given by the formula *V* = 4/3 × *pi* ×*r*^{3}*. *The surface area S of the sphere is given by the formula

*S*= 4 ×

*pi*×

*r*

^{2}

*.*

Example:

The space figure pictured below is a sphere.

Example:

To the nearest tenth, what is the volume and surface area of a sphere having a radius of 4cm?

Using an estimate of 3.14 for *pi*,

the volume would be 4/3 × 3.14 × 4^{3} = 4/3 × 3.14 × 4 × 4 × 4 = 268 cubic centimeters.

Using an estimate of 3.14 for *pi*, the surface area would be 4 × 3.14 × 4^{2} = 4 × 3.14 × 4 × 4 = 201 square centimeters.

### Cone

A cone is a space figure having a circular base and a single vertex.

If *r* is the radius of the circular base, and *h* is the height of the cone, then the volume of the cone is 1/3 × *pi* × *r*^{2}* × h*.

Example:

What is the volume in cubic cm of a cone whose base has a radius of 3 cm, and whose height is 6 cm, to the nearest tenth?

We will use an estimate of 3.14 for *pi*.

The volume is 1/3 × *pi* × 3^{2} × 6 = *pi *×18 = 56.52, which equals 56.5 cubic cm when rounded to the nearest tenth.

Example:

The pictures below are two different views of a cone.

### Pyramid

A pyramid is a space figure with a square base and 4 triangle-shaped sides.

Example:

The picture below is a pyramid. The grayed lines are edges hidden from view.

### Tetrahedron

A tetrahedron is a 4-sided space figure. Each face of a tetrahedron is a triangle.

Example:

The picture below is a tetrahedron. The grayed lines are edges hidden from view.

### Prism

A prism is a space figure with two congruent, parallel bases that are polygons.

Examples:

The figure below is a pentagonal prism (the bases are pentagons). The grayed lines are edges hidden from view.

The figure below is a triangular prism (the bases are triangles). The grayed lines are edges hidden from view.

The figure below is a hexagonal prism (the bases are hexagons). The grayed lines are edges hidden from view..