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Geometry (Greek: geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have become very complex and abstract and are barely recognizable as the descendants of early geometry.
Work in pairs or groups and investigate the history of the foundation of geometry. Describe the various stages of development and how the following cultures used geometry to improve their lives. This list should serve as a guideline and provide the minimum requirement, there are many other people who contributed to the foundation of geometry.
In this section we study how to calculate the surface areas and volumes of right prisms and cylinders. A right prism is a polygon that has been stretched out into a tube so that the height of the tube is perpendicular to the base. A square prism has a base that is a square and a triangular prism has a base that is a triangle.
It is relatively simple to calculate the surface areas and volumes of prisms.
The term surface area refers to the total area of the exposed or outside surfaces of a prism. This is easier to understand if you imagine the prism as a solid object.
If you examine the prisms in [link] , you will see that each face of a prism is a simple polygon. For example, the triangular prism has two faces that are triangles and three faces that are rectangles. Therefore, in order to calculate the surface area of a prism you simply have to calculate the area of each face and add it up. In the case of a cylinder the top and bottom faces are circles, while the curved surface flattens into a rectangle.
Surface Area of Prisms
Calculate the area of each face and add the areas together to get the surface area. To do this you need to determine the correct shape of each and every face of the prism and then for each one determine the surface area. The sum of the surface areas of all the faces will give you the total surface area of the prism.
In pairs, study the following prisms and the adjacent image showing the various surfaces that make up the prism. Explain to your partner, how each relates to the other.
The volume of a right prism is calculated by multiplying the area of the base by the height. So, for a square prism of side length $a$ and height $h$ the volume is $a\times a\times h={a}^{2}h$ .
Volume of Prisms
Calculate the area of the base and multiply by the height to get the volume of a prism.
Now, what happens to the surface area if one dimension is multiplied by a constant? For example, how does the surface area change when the height of a rectangular prism is divided by 2?
The size of a prism is specified by the length of its sides. The prism in the diagram has sides of lengths $L$ , $b$ and $h$ .
The volume of a prism is given by: $V=L\times b\times h$
The surface area of the prism is given by: $A=2\times (L\times b+L\times h+b\times h)$
If all the sides of the prism get rescaled, the new sides will be:
The new volume will then be given by:
The new surface area of the prism will be given by:
When the length of one of the sides is multiplied by a constant the effect is to multiply the original volume by that constant, as for the example in [link] .
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