Determine the surface area of the following square-based pyramid. Determine the volume of the following triangular prism. ) + 9.

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Abigayle Dennis
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1 MPM 1D Name: Unit: Measurement Date: Calculating and of Three Dimensional Figures Use the Formula Sheet attached to help you to answer each of the following questions. Three problems are worked out for you below so that you can see what is expected in terms of showing work. (Yes work MUST be shown.) Note that volume will have cubed units and surface area will have squared units. Unless otherwise state, you may round your final solution to the nearest 10 th of a unit (so to one decimal place.) Determine the volume of the following triangular prism. Determine the surface area of the following square-based pyramid. Slant height = s s= 13.5 cm.3 m 3.4 m 4.8 m 9.3 cm = b V = blh V = (.3)(4.8)(3.4) V = m 3 The volume of the triangular prism is about 18.8 m 3. A total = 4A triangle + A base A total = 4 ( bs ) + b A total = 4 ( (9.3)(13.5) ) A total = 4(6.775) A total = cm These equations come from the equation sheet. The total area of the square-based pyramid is about cm. If you are not given s = slant height for a pyramid or a cone, you will have to use the Pythagorean Theorem to calculate it first. Example: Calculate the surface area of the cone. Assume h = 1 cm and r = 9 cm. A total = πrs + πr So, we need to determine the value of s, the slant height. The side, s, in the diagram is the hypotenuse of a right triangle. The Pythagorean Theorem says a + b = c. With the labels on the diagram we would rewrite this as h + r = s. So, = s and s = 5. Thus, s = 15 cm. Now, A total = πrs + πr becomes A total = π(9)(15) + π(9). A total = A total = cm Note that the value 3.14 was used for π here. If you use the π key on the calculator your answer will be a bit larger.

2 MPM 1D Name: Practice With and Calculations Date: For Three Dimensional Objects Name each type of 3-D object shown below. Determine the surface area and volume of each one. Name: Name: h = 3 mm l = 8 mm w = 8 mm Answer: 40 cm Answer: 144 cm 3 Name: Determine the slant height of this figure, then determine its surface area and volume. h = 4 cm and r = 7 cm Answer: 4 cm Answer: 19 cm 3 Name: r = 5 m Calculate Slant Height, s, here: Answer: 5 cm Answer: cm Answer: cm 3 Answer: 314 m Answer: m 3

3 Name: h = 4 cm and r = 7 cm Name: 3 mm Calculate Slant Height, s, here: 8 mm 8 mm s Answer: 5 mm Answer: cm Answer: cm 3 Note that the radius and the height of this figure are the same as the radius and the height of the cone on this page. Describe the relationship that exists between the volume of this figure and that of the cone with the same dimensions. Answer: 144 mm Answer: 64 mm 3 Note that the base and height of this figure are the same as those for the rectangular prism on this page. Describe the relationship that exists between the volume of this figure and that of the rectangular prism with the same dimensions. Word Problems 1) Tennis balls are stacked four high in a rectangular prism package as shown below. The diameter of one ball is 6.5 cm. a) Calculate the volume of the rectangular prism package. 6.5 cm (Answer: cm 3 ) b) Determine the minimum amount of material to make the required box. (Hint: surface area of the box = the amount of material needed to construct it) (Answer: cm ) c) Determine the amount of empty space in the rectangular prism package. (Answer: 53.3 cm 3 ) d) List the assumptions you had to make when working out the math.

4 ) A cone-shaped glass holds a volume of 500 ml of water which is equivalent to 500 cm 3 of water. If the height of the glass is 10 cm, determine the radius of the glass. (Hint: You will need to rearrange the cone volume equation to solve for the missing value, r.) (Answer: 6.9 cm) 3) A propane tank is in the shape of a cylinder with a hemisphere (a half sphere) at both ends. The tank has a radius of 0.4 m, which is also the radius of the hemispheres. The cylinder alone is m in length. Calculate the volume of the tank, to the nearest tenth of a cubic metre. (Answer: 1.3 m 3 ) r=0.4 m.8 m 4) Aqua Aquariums sell aquariums shaped like rectangular prisms in two different sizes: Large and Small. Each aquarium has glass sides, a glass bottom and NO top. The diagrams below are not drawn to scale but the measurements are accurate. You will likely need to answer the following questions on a separate sheet of paper a) Calculate the volume of each aquarium. [V large = 19000cm 3 ] [V small = 4000 cm 3 ] Large b) Calculate the total outside surface area of each aquarium (Remember there are no tops.) [A large = cm ; A small = 4400 cm ] 60 cm c) Determine the total cost of the materials to build each aquarium if the cost is $0.00 per cm of surface area. d) Complete this sentence: The cost of the materials needed to build the Large aquarium is times the cost of the materials to build the Small aquarium. e) The selling price of the small aquarium is $4. The selling price of the large aquarium is $115. Do the selling prices seem appropriate according to your calculations? (Look at ALL of the calculations before answering.) Justify your answer. Small 40 cm 80 cm 30 cm 0 cm 40 cm

5 5. A box of crackers has a volume of 5000 cm 3. If its length is 5 cm and its width is 8 cm, what is its height? (Answer: 5 cm) 6. The radius of a cone is tripled. Does this triple the surface area of the cone? Justify your answer. 7. Sophia has constructed a cone-shaped funnel from paper. The funnel has a volume of 6 cm 3 and a radius of 4 cm. What is the height of the paper cup? (Answer: 4 cm) 8. A sphere has a surface area of 47.5 cm. Find its radius. (Answer: 1.9 cm)

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth.

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth. Volume of Cylinders As with prisms, the area of the base of a cylinder tells the number of cubic units in one layer. The height tells how many layers there are in the cylinder. The volume V of a cylinder