Space figures
Crosssection
Volume
Surface area
Cube
Cylinder
Sphere
Cone
Pyramid
Tetrahedron
Prism



A space figure or threedimensional 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 crosssection of a space figure is the shape of a particular twodimensional "slice" of a space figure.
Example:
The circle on the right is a crosssection of the cylinder on the left.
The triangle on the right is a crosssection 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 sidelength 6 cm?
The volume of a cube is the cube of its sidelength, 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 threedimensional 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 squareshaped 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 sidelength 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 crosssection 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.
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 triangleshaped sides.
Example:
The picture below is a pyramid. The grayed lines are edges hidden from view.
A tetrahedron is a 4sided 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..
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