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Work in pairs or groups and investigate the history of the development of geometry in the last 1500 years. Describe the various stages of development and how different cultures used geometry to improve their lives.
The works of the following people or cultures should be investigated:
A pyramid is a geometric solid that has a polygon base and the base is joined to a point, called the apex. Two examples of pyramids are shown in the left-most and centre figures in [link] . The right-most figure has an apex which is joined to a circular base and this type of geometric solid is called a cone. Cones are similar to pyramids except that their bases are circles instead of polygons.
Surface Area of a Pyramid
The surface area of a pyramid is calculated by adding the area of each face together.
If a cone has a height of and a base of radius , show that the surface area is .
The cone has two faces: the base and the walls. The base is a circle of radius and the walls can be opened out to a sector of a circle.
This curved surface can be cut into many thin triangles with height close to ( is called the slant height ). The area of these triangles will add up to base height(of a small triangle) which is
can be calculated by using the Theorem of Pythagoras. Therefore:
Volume of a Pyramid: The volume of a pyramid is found by:
where is the area of the base and is the height.
A cone is like a pyramid, so the volume of a cone is given by:
A square pyramid has volume
where is the side length of the square base.
What is the volume of a square pyramid, 3cm high with a side length of 2cm?
The volume of a pyramid is
where is the area of the base and is the height of the pyramid. For a square base this means
where is the length of the side of the square base.
We accept the following formulae for volume and surface area of a sphere (ball).
In order for two polygons to be similar the following must be true:
If
Polygons PQTU and PRSU are similar. Find the value of .
Since the polygons are similar,
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