Polyhydron
What is the meaning of Polyhydron?
A polyhedron is a three-dimensional solid whose faces are polygons.
A prism is a polyhedron that has two parallel, congruent faces called bases. The other faces are parallelograms.
A pyramid is a polyhedron whose base is a polygon and whose faces are triangles with a common vertex.
A cube is a prism whose faces are squares.
This cube has six faces, twelve edges, and eight vertices.
A pyramid is a polyhedron whose base is a polygon and whose faces are triangles with a common vertex.
This triangular pyramid has four faces, six edges, and four vertices.
This square pyramid has five faces, eight edges, and five vertices.
A cylinder is a solid whose bases are circles.
A cone is a solid with one circular base and one vertex.
A sphere is a set of all points in space that are equidistant from a given point called the center.
Polyhedra
Surface Area
The surface area of a three-dimensional figure is the sum of the areas of its faces.
Since the faces of a polyhedron are polygons, the surface area of a polyhedron is the sum of the areas of its polygonal faces.
Let’s look at a prism first.
Each of the faces of the rectangular prism below is a rectangle.
What is its surface area?
The prism has two rectangular faces with dimensions 3 in. x 8 in.
Each has area (3 in)(8 in) = 24 in
Two faces have dimensions 8 in. x 12 in.
Each has area (8 in)(12 in) = 96 in
Two faces have dimensions 3 in. x 12 in.
Each has area (3 in)(12 in) = 36 in
The surface area of the prism is:
Volume is the number of cubic units needed to pack a solid figure without gaps or overlaps. A cubic unit is the volume of a cube measuring 1 unit on each edge.
Q. What is the volume of this rectangular prism?
Use unit cubes to make a model. Each cube of a solid figure is 1 cubic unit.
Develop a Volume Formula:
Q. Kara needs a new aquarium for his fish. The pet store has a fish tank in the shape of a rectangular prism that measures 5 feet long by 2 feet wide by 4 feet high. Kara needs a fish tank that has a volume of at least 35 cubic feet. Will this fish tank be big enough?
Solution;
Read the problem carefully to make sure that you understand what you are trying to find. Show your work!
Remember that volume is the number of cubic units (units3) needed to pack a solid figure without gaps or overlaps.
A formula is a rule that uses symbols to relate two or more quantities.
If the dimensions of a rectangular prism are given as length l, width w, and height h, then use this formula to find the volume V:
Volume = length X width X height
V = l X w X h
= 5 x 2 x 4
= 40 cubic ft
Volume
A general formula for the volume of a prism is
V = Bh,
where B is the area of the base and h is the height.
Since volume is three-dimensional, it is expressed in cubic units,
like cubic inches (in ), cubic centimeters (cm ), and cubic feet (ft ).
Let’s look back at the prism from the previous section.
What is its volume?
The bases are any pair of congruent faces; this prism has three pairs of congruent faces. We’ll let the faces with dimensions 8 in. x 12 in. be the bases since they are on the top and bottom
(and that’s where we like our bases).
Now compute the formula for volume:
