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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.
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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
π2.526 and π5212
3
3
Therefore, the ratio of their volumes = 2.526 : 5212 = 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.
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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:
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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
π2.56.5 andπ513. 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?
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