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Key Stage 3 Measures, Shape and Space Dimension Learning Unit: More about Areas and Volumes Learning Objectives: ·understand and use the formulas for volumes of pyramids, circular cones and spheres ·understand and use the formulas for surface areas of right circular cones and spheres Programme Title: Pyramids & Spheres Programme Objectives 1. Recognize the formula for the volume of a pyramid. 2. Recognize the method of calculating the lateral surface area of a pyramid. 3. Compare the volumes and the surface areas of a pyramid and a prism with the same base and height; recognize the difference in volumes is much greater than the difference in surface areas. 4. Recognize the formula for the curved surface area of a right circular cone. 5. Understand the method of calculating the volume and surface area of a frustum. 6. Recognize the formulas for the volume and surface area of a sphere, and apply the formulas in solving simple problems. Programme Content Through a simple experiment, it is demonstrated that the water contained in a prism is three times the water contained in a pyramid with the same base and height. This shows that the volume of a pyramid is one third of the volume of a prism with the same base and height. -1- The programme uses a pyramid with a rectangular base as an example to illustrate the calculation of the volume of a pyramid. By comparing the material used for making a pyramid and a prism with the same base and height, the calculation of the surface area of a pyramid is demonstrated. It also points out that the difference in their volume is much greater than the difference in their surface areas. The programme highlights the special shape of a circular cone when it is spread out flat. The formula for the curved surface area of a circular cone is introduced. It also shows the relation between a frustum and a pyramid. The method of calculating the volume and surface area of a frustum is clearly elaborated with an example. Finally, the programme puts forward a scenario of save the environment to introduce the formulas for the volume and surface area of a sphere and their applications. Worksheet Answers 1. 321.03 m, 160.52 m, 136.9 m; roughly 2 351 000 m 3. 2. 1:2; 1:4; 1:4=(1:2)2 , that is square of the ratio of the corresponding sides; The volumes of the two similar cone are respectively: 1 1 π2.526 and π5212 3 3 Therefore, the ratio of their volumes = 2.526 : 5212 = 1 : 8 = (1:2)3; which is the cube of the ratio of the corresponding sides. 3. roughly 3.9 billion hectares; roughly 0.65 hectares. This is just a bit bigger than the size of a football playground. Therefore, if we do not preserve the resources in the forests, there will be a big problem with oxygen levels. -2- Key Stage 3 ETV Programme 《More about Areas and Volumes》 Worksheet 1. The best-known pyramids are those in Cairo near Giza. For the largest pyramid, a side of its base is approximately 227 m long and its slant height is approximately 211 m in length. Suppose it is a right pyramid with a square base, what is its volume? V A 211 m B M D 227m 227m Solution: C By Pythagoras’ Theorem, the length of a diagonal of the base equals: Therefore, in the vertical right-angled △VMD, the base side: MD = By Pythagoras’ Theorem, the height of the pyramid, VM equals: According to the formula for the volume of a pyramid, the volume of the pyramid equals: -3- 2. Let’s consider the frustum of the programme: For the small cone, base radius is 2.5cm and height is 6cm; and for the large cone, base radius is 5cm and height is 12cm. Therefore, the ratio of the corresponding sides of the two similar cones is: Their curved surface areas are respectively π2.56.5 andπ513. Therefore, the ratio of their curved surface areas is: Hence, for two similar solids, the ratio of their curved surface areas is of the ratio of their corresponding sides. For two similar solids, what is the relation between the ratio of their volumes and the ratio of their corresponding sides? Give your answers with the help of the above two similar cones. 3. It is mentioned in the programme that: the radius of the earth is approximately 6400km, 1/4 of the surface of the earth is land and 30% of its area is occupied by the forests. According to these data, calculate the area, in hectare, of the forests in the earth. (1 square kilometer = 100 hectares) The population of the earth is approximately 6 billion. What is the average area of the forests shared out by each people? In considering the essential oxygen, do you think that the area of the forests is adequate for the supply? -4-