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Name ________________ Geometry Notes Pyramids & Cones Pyramids Our study of pyramids will include only ______________ pyramids. A regular pyramid is a solid that has only one _______ , and that base must be a ___________ The point at the top is called the _________. The ___________ of the regular pyramid is directly above the ___________ of the base and is _______________ to the base. The distance from the vertex to the base is the ___________ (h). The faces other than the base are called ____________ _____________. All lateral faces are ______________. The lateral faces are __________ triangles. The height of a lateral face is called the _________ _____________ of the pyramid. (Notice it’s a “slanted height”!) The slant height is a cursive “ ” on your formula V sheet! Example: In the regular pyramid shown, 2. The vertex is __________ Z W 1. The base is _________ C Y M X 3. The height is ___________. The slant height is ________. 4. Name a lateral face ____________ How many lateral faces are there? ________ 5. Notice that VCM is a _________ triangle. 6. The slant height will always be __________ than the height of the pyramid. 7. Since there are right triangles in a pyramid, we may use the _________________ theorem. 8. If YX = 20 cm, then CM = ______ cm. 9. If CM = 3 in and CV = 4 in, then VM = _______in. 10. If WY = 12 ft, the perimeter of the base of the pyramid is ____________. Lateral Area, Surface Area and Volume of Pyramids Lateral Area of a pyramid (no base!) The _________ of all the areas of its lateral faces (triangular lateral faces!). L. A. formula= p = perimeter of the base = slant height of the pyramid Example A: Find the lateral area of the pyramid. V In the pyramid at right, VC = 12 in and CM = 5 in Z W What is VM?_______ Note that VM is the slant height () Sometimes you may have to use Pythagorean Theorem to find the slant height! C Y Looking at the base of the pyramid M X Since CM = 5 in then MZ=_____ XZ=_____ perimeter of square ZXYW=________. The lateral area is: Surface area of a pyramid S. A. formula= p = perimeter of the base = slant height of the pyramid B = area of base Example: Find the total surface area of the pyramid above. () slant height = _______ (p) perimeter = _______ (B) area of base = Surface Area = Volume of a pyramid Volume formula = B = area of base h = height of pyramid Example: Find the volume of the pyramid above. (B) area of base = _______ (h) height= _______ volume = Surface Area and Volume of Cones A cone is a solid with only one ____________ base. h The cones that we will study will always be right circular cones. The ___________ and the circular _________ will always be _______. Lateral Area of a Cone Lateral area does _______ include the circular __________. L.A. formula= r = radius = slant height of the cone Note: You may have to use Pythagorean Theorem to find the slant height! Example: Find the lateral area of a cone with radius of 6 cm and height 10 cm. (r)radius = () slant height = lateral area = Surface area of a cone S.A. formula= r = radius = slant height of the cone Example: Find the surface area of cone with r= 6 cm and h= 10 cm. (r) radius = () slant height = surface area = Volume of a cone: Volume formula= r = radius h = height of cone Example: Find the volume of a cone with r = 6 cm and h = 10 cm. (r) radius = (h) height = volume = rr Examples 1. A pyramid has a square base with sides of length 220 meters and a height of 145 meters. What is the volume of the pyramid to the nearest meter? 2. A tepee in the shape of a right cone has a slant height of 18 feet and a diameter of 20 feet. How much canvas would be needed to cover the tepee? (hint: the canvas does NOT go on the ground) 3. The base of a square pyramid is 8 cm by 8 cm. If the volume of the pyramid is 128 cm3, what is the height? 4. What is the total capacity (cone and cylinder) of the storage container shown below? 8m 6m 12 m 5. The lengths of the sides of the base of a square pyramid are 4 ft. If the slant height is 9 feet, what is the surface area of the pyramid?