Mathematics Kenya
Area of Quadrilaterals and other Polygons
A group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation
Area of Quadrilaterals and other Polygons
Area of Quadrilaterals and other Polygons
Area of quadrilaterals
Area of other polygons (regular and irregular)
When we sum up the lengths of the sides of the polygon we obtain what is called perimeter of a polygon. Therefore, perimeter of a regular polygon is the sum of the lengths of the sides of the polygon.
How to find the perimeter of a Regular Polygon inscribed in a circle?
An inscribed polygon is the one whose vertices lie on the circle. If the lengths of the sides of the polygon are the same we say that the polygon is an inscribed Regular Polygon.
A Regular Polygon with number of sides larger than 2 say n sides can be inscribed in a circle as follows:
For example, if you want to construct an inscribed regular hexagon (6 sides), first draw a circle and locate the center of the circle. Then draw rays that intersect the circle in six points from the center of the circle. Each angle at the center will measure 360°/6 = 60°. Connect the points of intersection on the circle by line segments. The figure formed is an inscribed regular polygon. See the figure below:
Now, to obtain the formula of finding the perimeter of a regular polygon inscribed in a circle with radius r and center O, let AB be the side of the polygon and OC the perpendicular from O to AB as shown in the figure below:
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The angle AOB = 360°/n, since the polygon has n sides.The angle AOC = the angle AOB and the angle BOC = angle AOC +angle BOCTherefore, the angle AOC = ½(360°/n) = 180°/nLet the length of the side of the regular polygon ABbe S.Then,
Therefore the length of a regular polygon with n sides inscribed in a circle is given by
The Perimeter of a Regular Polygon
If we let 2r = d
rom the concept of perimeter that perimeter of a regular polygon is the sum of the lengths of the sides of the polygon , if we have n sides each with length ‘S’ then the sum of the lengths of these sides will be nS. Therefore, Perimeter P of a regular polygon of n sides each with length S is given by:
Example 1
Find the length of one side of eight-sided regular polygon inscribed in a circle with radius 7cm.
Solution
Therefore the length of one side of eight-sided regular polygon with radius of 7cm is 5.358cm
The Formula for Finding the Area of a Regular Polygon
Consider the regular polygon with n sides inscribed in a circle of radius r and center O as shown below:
But since each vertex of the polygon is connected to O, the polygon region is divided into n triangles which are equal.
Now,
Therefore the area of a polygon of n sides inscribed in a circle of radius r is given by:
The Formula to Calculate the Area of a Regular Polygon
Example 2
Find the area of twelve-sided regular polygon inscribed in a circle of radius 14 cm
Solution
Circumference and area of a circle
Circumference of a circle is the distance around it. Circumference of a circle can be estimated by using a regular polygon with many sides inscribed in a circle with radius r.
We know perimeter of the regular polygon is given by:
Here we see that, as n increases the value of nsin 180°/napproaches the value ofπ.When n is very large the perimeter of a regular polygon approaches the circumference of the circle. The value ofnsin 180°/ncan be replaced byπbecause it approaches the value ofπwhen n is very large.
Therefore, circumference of the circle C, is given by C = 2πr
Area of a Circle
In similar way we can generate the formula of calculating the area of a circle by considering area of a regular polygon inscribed in a circle of radius r.
We know that, area of a regular polygon is given by:
Alternatively we can write it as:
Example 3
Find the circumference of a circle of radius 21cm. (takeπ= 3.14).
Solution
Circumference of a circle, C = 2πr
= 2× 3.14 × 21cm = 131.88cm
The Ratio of Areas of Similar Polygons
Let ABC and A’ B’ C’ be two similar triangles:
If we find the ratio of their sides we get;
Generally, if the ratio of the lengths of the corresponding sides of two similar polygons is k, then the ratio of their areas is k2.
Example 4
Problems Related to Ratio for Areas of Similar Polygons
We are given two triangles which are similar. The length of one side is 8cm and the length of the corresponding side is 14cm. if the area of a smaller triangle is 24cm2find the area of the other triangle.
Solution
Therefore the area of the other triangle is 73.5cm2.
Example 5
The ratio of the areas of two similar polygons is 36:48. The length of a side of the smaller polygon is 10cm. find the length of the corresponding side of the other polygon.
Solution
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