Geometry of Plane Shapes
This is a branch of mathematics that is basically concerned with shapes and their properties. The shapes involved here are plane shapes (geometry) which have to do with lines, circles, triangles, rectangles, squares and other polygons which include pentagons, hexagons, octagons etc. other types of shapes are solid geometry (including cubes, pyramids, prisms)
The general properties of planes shapes also called flat surfaces are that we can calculate their areas and perimeter or distance around the shape. We however cannot find their volumes and their total surface areas because their thicknesses (or widths or breadth) are really negligible compared to lengths and widths.
In the following sections of this paper, we look at shapes, pentagon, rectangle and hexagon one at a time.
Pentagon
A pentagon is any five sided figure or shape. When all the sides of the pentagon are equal, it is referred to as regular pentagon. But normally ‘pentagon’ refers to a regular pentagon. The shape of the figure is below.
With respect to the geometry of the figure, each exterior angle of a regular pentagon is 720 and the interior angle sum of the pentagon adds up to 5400. Therefore every interior angle is 1080 (Clarke,1970).
Perimeter of Pentagon
The perimeter or the distance round the pentagon is the sum of all the sides of the pentagon. The perimeter of a regular pentagon is given by, . If the polygon is not regular, the perimeter is given by where a, b, c, d and e are different sides of the pentagon.
Area of a Pentagon
The area of pentagon is given by the formular where r is the radius of the pentagon and s is the side of the pentagon.
Rectangle
A rectangle is a quadrilateral in which the two opposite sides are equal and parallel with a right inside every corner. The exterior and interior angle of a triangle are both 900 (Clarke, 1970). A rectangle has two lines of symmetry -lines which dive it into exactly two halves (Clarke, 1970).
We show a diagram showing a rectangle.
In the diagram l is the length of the rectangle and w is the width or breadth of the rectangle.
The perimeter of the rectangle is given by units.
The area of the rectangle is given by square units.
The length of any diagonal in the rectangle is given units where in this f is the length of the diagonal.
Hexagon
This is a six –sided plane shape. Most hexagons used in geometry are regular polygons. Therefore in this section of the paper, the hexagon shall mean a regular polygon.
The diagram below shows a regular hexagon with side t and radius x
The perimeter of the hexagon is similarly determined as the perimeter of the pentagon by adding all the sides of the hexagon. Sounits.
The area of the hexagon is square units.
Applications of Geometry.
The figures above can be applied as
In tiling designs by making tiles in specific shapes of rectangle, pentagon and hexagon to make the floor or walls beautiful
Some flowers are in form of these shapes for example, morning glory, sea star okra ( are in shapes of pentagon)
Isotropic structures are application of the hexagon.
Given that the geometry we have looked at is plane, its application to stewardship can be realized in the form of responsible planning and management for the proper outlook of the environmental resources. The use of pentagonal and hexagonal go a long way to improve the beauty of the surrounding for example lining a walk way and lobby with multi colored tiles. Truncated, tri-hexagonal and rhombitrihexagonal tiles are an example of this (Gibilisco, 2003). The knowledge of these polygons can also be applied to beautification of football and other games stadia for example the architecture of the birds nest in Beijing (Olympic stadium) and the calabash in South Africa (2010 world cup opening match stadium) are a direct application of the knowledge of polygons.
References
Chakerian, G. D. "A Distorted View of Geometry." Mathematical Plums (R. Honsberger, ed).
Washington, DC: Mathematical Association of America, 1979: 147.
Gibilisco, (2003). Geometry demystified (Online-Aug. edition). New York: McGraw-
Hill.
ISBN 978-0-07-141650-4.
Clarke, L.H (1970): Trigonometry at Ordinary Level. Heinemann Educational Books. London
Parr, H.E (1948): School Mathematics with Answers.
