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.

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
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.

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.

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.

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.

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

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.

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.

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.

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.

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..