PRISMS Lateral Area and Total Area of a Prism (11th grade)

home 1. PRISMS The two shaded faces of the prism shown are its bases. The bases are congruent polygons lying in parallel planes. home If they are rectangles, the prism is a right prism. Otherwise the prism is an oblique prism. home The faces of a prism that are not its bases are called lateral faces. The lateral faces of a prism are parallelograms. Right triangular prism Right rectangular prism (Rectangular solid) Oblique pentagonal prism Lateral edge is not an altitude Lateral edges are altitudes home The diagrams below show that a prism is also classified by the shape of its bases. The lateral area (L.A.) of a prism is the sum of the areas of its lateral faces. The total area (T.A.) is the sum of the areas of all its faces. home Lateral Area and Total Area of a Prism L.A. = ah+bh+ch+dh+eh = (a+b+c+d+e)h = perimeter • h = ph The right pentagonal prism can be used to illustrate the development of the lateral area formula: home Lateral Area and Total Area of a Prism home The Method applies to any right prism. The lateral area equals the perimeter of a base times the height of the prism. (L.A. = ph) home Lateral Area and Total Area of a Prism The larger rectangular solid has 3 layers of cubes, each layer containing (4•2) cubes. Hence its volume is (4•2)•3, or 24 cubic units. home Volume = Base area x height Prisms have volume as well as area. A rectangular solid with square faces is a cube. Since each edge of the blue cube shown is 1 unit long, the cube is said to have a volume of 1 cubic unit. home The same sort of reasoning is used to find the volume of any right prism. The volume of a right prism equals the area of a base times the height of the prism. (V = Bh) home The Volume of a Right Prism home Example 1 A right trapezoidal prism is shown. Find the lateral area, total area, volume. home home A right triangular prism is shown. The volume is 315. Find the total area. Example 2 home home The container shown has the shape of a rectangular solid. When a rock is submerged, the water level rises 0.5 cm. Find the volume of the rock. Example 3 home home A right prism has square bases with edges that are three times as long as the lateral edges. The prism's total area is 750 m2. Find the volume. Example 4 home The volume of an oblique prism equals the area of a base times the height of the prism or right cross-section area times the length of the lateral edge. The Volume of an Oblique Prism home home Volume = Base Area x Altitude = Right Cross-section Area x Length of lateral edge The Volume of an Oblique Prism home Example 5 home home In the parallelepiped ABCD, right cross-section area is a square with sides 1 cm. If =60° and cm, find the volume of the prism. Example 6 home home A diagonal of a right rectangular prism joins two vertices not in the same face. The length of the diagonals is Note home home If the diagonals of a cube are cm long, what is the volume? Example 7