10.1 Prisms and Pyramids

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1 AreasandVolumesofprismsandpyramids nb 0. Prisms and Pyramids We have already learned to calculate the areas of plane figures. In this chapter we will be calculating the surface areas and volumes of special solids. Rather then beginning with postulates about areas and volumes as we did for plane figures we will present the formulas as theorems straightaway. The first two solids we will study are prisms and pyramids. A prism is shown in the figure below. The two shaded faces of the prism are its bases. Notice that the bases of the prism are congruent polygons which lie in parallel planes. An altitude of a prism is a segment joining the two bases and perpendicular to both. The length of an altitude is the height, h, of the prism. As we did with plane figures we will use the term altitude to refer to the segment and its height. base lateral edges altitude base lateral face The faces of a prism that are not its bases are called lateral faces. Adjacent lateral faces intersect in parallel segments called lateral edges. The lateral faces of a prism are parallelograms. If they are rectangles, the prism is a right prism. A right triangular prism is shown below.

2 AreasandVolumesofprismsandpyramids nb 2 Lateral edges are altitudes Right Triangular Prism Notice that the prism is named a right (because the altitudes are perpendicular to the bases) triangular (because the bases are triangles) prism. We will continue the practice of naming prisms in this way throughout the text. If a prism is not a right prism (i.e. if the lateral faces are not rectangles then the prism is an oblique prism. Notice in this case that the lateral edges are not altitudes. Lateral edge is not an altitude Oblique Pentagonal Prism

3 AreasandVolumesofprismsandpyramids nb 3 The surface area of a solid is measured in square units. 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 sume of the areas of all its faces (the lateral area plus the areas of its two bases). Using B to denote the area of a base, we have the following formula for the total area of a prism. T.A. = L.A + 2B If a prism is a right prism, the next theorem gives us an easy way to find the lateral area. Theorem: The lateral area of a right prism equals the perimeter of a base times the height of the prism. (L.A. = ph) The formula for lateral areas applies to any right prism. Since the area of each face is the base times the height of each face you will be multiplying the same height times the length of each base. The sum of the bases is the perimeter of the base of the prism, hence the formula. Prisms have volume as well as area. A rectangular solid with square faces is a cube. Since each edge of the blue cube in the figure below is one unit long, the cube is said to have a volume of cubic unit. The rectangular solid below has three layers of cubes, each layer containing (4 2) 3, or 24 cubic units. 3 4 Voulme = Base area height = (4 2) 3 =24 cubic units

4 AreasandVolumesofprismsandpyramids nb 4 We can find the volume of any right prism using the same sort of reasoning. Theorem: The volume of a right prism equals the area of a base times the height of the prism (V = Bh). Volume is measured in cubic units. Some common units for measuring volume are cubic feet Hft 3 ), cubic yards Hyd 3 ), cubic centimeters Hcm 3 ), and cubic meters Hm 3 ). Let's look at some examples before we move on to pyramids. Example : Find (a) the lateral area, (b) the total area, and (c) the volume of the right trapezoidal prism shown below. 2 cm 5 c m 4cm 5 cm 6cm 0 cm Right Trapezoidal Prism Solution: a. First find the perimeter of the base. p = = 28 cm Now use the formula for lateral area L.A. = ph = 28 0 = 280 cm 2

5 AreasandVolumesofprismsandpyramids nb 5 b. First find the area of the base B = ÅÅÅÅ 4 (2 + 6) = 36 cm2 2 Now use the formula for total area. T.A = L.A. + 2B = = 352 cm 2 c. V = Bh = 36 0 = 360 cm 3 Example 2: Find the volume of the solid shown below. 2 in. 2 in. 0 cm. 20 in. 20 i n. 40 in. Solution: Think of this one as two problems. First, find the volume of the rectangular prism as if it were solid (no holes). Then find the volume of the two holes and subtract them from the total volume. V of solid prism = = 6,000 in 3 Volume of one hole = = 2400 in 3 Volume = 6,000-2 (2400) = 6, =, 200 in 3 Now, let's take a look at pyramids,

6 AreasandVolumesofprismsandpyramids nb 6 The diagram below shows a regular hexagonal pyramid. h O Regular Hexagonal Pyramid The point at the top of the pyramid is the vertex of the pyramid and the hexagon is the base. The segment from the vertex perpendicular to the base is the altitude and its length is the height, h, of the pyramid. The six triangular faces with the vertex in common are lateral faces. The faces intersect in segments called lateral edges. Most of the pyramids we will study will be regular pyramids. These are pyramids in which the base is a regular polygon, all lateral edges are congruent, and all lateral faces are congruent isosceles triangles. The height of a lateral face is called the slant height, l, of the pyramid, as marked on the figure. Many times, when finding the volume and areas of pyramids you will need to use the Pythagorean Theorem to find missing lengths. Look at the example below.

7 AreasandVolumesofprismsandpyramids nb 7 Example 3: A regular square pyramid has base edges 0 and lateral edges 3. Find (a) its slant height, and (b) its height. V h 3 D C O M A 0 B Solution: (a) In right DVMC, can be found using the Pythagoran Theorem. = "################ = è!!!!!!!! 44 = 2 (b) In right DVOM, h = "################ = è!!!!!!!! 9 Notice that the lateral faces in the figure above are congruent isosceles triangles. The lateral area of the pyramid can be found by finding the area of one face and multiplying by the number of faces. Or, since the slant height of the pyramid represents the height of each lateral face, you can add the bases of the triangles and multiply by ÅÅÅÅ the slant height. The sum of the 2 bases is the perimeter of the base of the pyramid which gives us the following formula. Theorem: The lateral area of a regular pyramid equals half the perimeter of the base times the slant height. (L.A. = ÅÅÅÅ 2 p )

8 AreasandVolumesofprismsandpyramids nb 8 Look at the prism and pyramid below. They have equal bases and equal heights. h V = Bh h V = ÅÅÅÅÅ 3 Bh Since the volume of the prism is Bh, the volume of the pyramid must be less than Bh. In fact, there is a direct mathematical relationship. The volume of the pyramid is exactly ÅÅÅÅ the volume of the prism. This fact is stated in the theorem 3 below. Theorem: The volume of a pyramid equals one third the area of the base times the height of the pyramid. (V = ÅÅÅÅ 3 Bh)

9 AreasandVolumesofprismsandpyramids nb 9 Example 4: Suppose the regular hexagonal pyramid shown above has base edges 6 and height 2. Find its volume. Solution: Look at the figure below. To find the volume we must first find the area of the base. Recall that the area of a regular polygon is given by the formula A = ÅÅÅÅ 2 ap where a is the apothem and p is the perimeter è!!! 3 B = ÅÅÅÅ 2 3 è!!! 3 ÿ 36 = 54 è!!! 3 Then V = ÅÅÅÅ è!!! ÿ 54 3 ÿ 2 = 26 è!!! 3 3