There are so many theorems on areas and deductions from that main theorem. Any planar closed figure has an area. If the figure is a noted one like triangle, rectangle, circle, polygon etc. we have specific formula so that we can find out areas by applying formula.
For Geometric closed figures like quadrilateral, irregular polygons we use intersecting lines to divide into triangles and add all areas of the triangle, so that all the triangles taken for this purpose are different segments of the polygon.
In this article, we study about how to calculate area for regular polygons.
Area of a polygon:
A polygon is a closed curve consisting of line segments in a plan which are called sides of the polygon. There are no restrictions to the number of sides of a polygon in a plane. Starting from 3, it can have any no of sides up to infinity.
Regular polygons are polygons which have equal sides and equal angles. Regular polygons are special types of polygons with certain extra properties.
For an irregular polygon, no standard formula but can be considered as the sum of all triangles formed by joining the centroid of the polygon to each vertex. If there are n sides, n triangles will be formed. The full area of the irregular polygon will be the sum of area of all triangles.
Corollary to the above for a regular polygon:
For a regular polygon of n sides we know there exists a centre and from centre lines are drawn to each vertex, the polygon will be a combination of n congruent triangles. In other words, for a regular polygon the lines drawn from centre divides the polyton into n number of congruent triangles. If we calculate one triangle area, then total area is calculated by multiplying by number of sides.
So Area of the polygon = n(area of each triangle) = n(1/2*side*height from centre of polygon to any side)
= 1/2 (n*side)*apothem (Since apothem is the altitude of the side from the centre)
= 1/2 * Perimeter*apothem (since n multiplied by no of sides give perimeter)
Learning problems solved for regular polygons - area theorem
Learning area theorems:
Problem1: Find area of a regular hexagon with side 6.
Solution:
Area of hexagon of 6 equal sides = 1/2 (perimeter)(apotnem)
We know that a hexagon when divided into 6 triangles, each triangle is equilateral as central angle is 60.
Area of an equilateral triangle =√ 3 (a^2)/4 = 1.732*6*6/4 = 15.588
Hence area of hexagon = 6*15.588 = 93.528
Applications of regular area theorems in squares and triangles:
This theorem applies for equilateral triangle and square also.
For triangle, centroid is the centre and we know the apothem for the triangle formed with two vertices and one side is 1/3 h where h is the height of the triangle. So area of the triangle = 3*a*1/2*1/3h = a2h/2 = √3/2a/2*a = √3a^2/4 = Area of the equilateral triangle
For square, the intersection of diagonals is the centroid and the apothem for each triangle is a/2.
So area of square by applying this theorem = 4a*a/2*1/2 =a^2
Problem 2: Find area of a regular pentagon with side 7.
Solution: Here we know each angle for a regular pentagon = 540/5 =108.
Hence line joining each vertex to the centre forms an angle of 108/2 = 54 degree with the side.
So 5 triangles are formed with base as 7 and angles two equalling 54 degrees.
Now we can find apothem using this. Apothem / 1/2 side = tan 54: or apothem = 7tan 54/2 =7(1.3763)/2
= 4.8173
Area of pentagon = 5*7*4.8173/2 = 168.61/2 =84.303
Thus given side and no of sides, the area of a regular polygon we can find out by using area theorems for polygons.
CONCLUSION:
In this article, we learnt about the area theorems for polygons and worked out problems on that. An interesting thing in a regular polygon is if we know the side we can find out the area of the polygon.
For Geometric closed figures like quadrilateral, irregular polygons we use intersecting lines to divide into triangles and add all areas of the triangle, so that all the triangles taken for this purpose are different segments of the polygon.
In this article, we study about how to calculate area for regular polygons.
Area of a polygon:
A polygon is a closed curve consisting of line segments in a plan which are called sides of the polygon. There are no restrictions to the number of sides of a polygon in a plane. Starting from 3, it can have any no of sides up to infinity.
Regular polygons are polygons which have equal sides and equal angles. Regular polygons are special types of polygons with certain extra properties.
For an irregular polygon, no standard formula but can be considered as the sum of all triangles formed by joining the centroid of the polygon to each vertex. If there are n sides, n triangles will be formed. The full area of the irregular polygon will be the sum of area of all triangles.
Corollary to the above for a regular polygon:
For a regular polygon of n sides we know there exists a centre and from centre lines are drawn to each vertex, the polygon will be a combination of n congruent triangles. In other words, for a regular polygon the lines drawn from centre divides the polyton into n number of congruent triangles. If we calculate one triangle area, then total area is calculated by multiplying by number of sides.
So Area of the polygon = n(area of each triangle) = n(1/2*side*height from centre of polygon to any side)
= 1/2 (n*side)*apothem (Since apothem is the altitude of the side from the centre)
= 1/2 * Perimeter*apothem (since n multiplied by no of sides give perimeter)
Learning problems solved for regular polygons - area theorem
Learning area theorems:
Problem1: Find area of a regular hexagon with side 6.
Solution:
Area of hexagon of 6 equal sides = 1/2 (perimeter)(apotnem)
We know that a hexagon when divided into 6 triangles, each triangle is equilateral as central angle is 60.
Area of an equilateral triangle =√ 3 (a^2)/4 = 1.732*6*6/4 = 15.588
Hence area of hexagon = 6*15.588 = 93.528
Applications of regular area theorems in squares and triangles:
This theorem applies for equilateral triangle and square also.
For triangle, centroid is the centre and we know the apothem for the triangle formed with two vertices and one side is 1/3 h where h is the height of the triangle. So area of the triangle = 3*a*1/2*1/3h = a2h/2 = √3/2a/2*a = √3a^2/4 = Area of the equilateral triangle
For square, the intersection of diagonals is the centroid and the apothem for each triangle is a/2.
So area of square by applying this theorem = 4a*a/2*1/2 =a^2
Problem 2: Find area of a regular pentagon with side 7.
Solution: Here we know each angle for a regular pentagon = 540/5 =108.
Hence line joining each vertex to the centre forms an angle of 108/2 = 54 degree with the side.
So 5 triangles are formed with base as 7 and angles two equalling 54 degrees.
Now we can find apothem using this. Apothem / 1/2 side = tan 54: or apothem = 7tan 54/2 =7(1.3763)/2
= 4.8173
Area of pentagon = 5*7*4.8173/2 = 168.61/2 =84.303
Thus given side and no of sides, the area of a regular polygon we can find out by using area theorems for polygons.
CONCLUSION:
In this article, we learnt about the area theorems for polygons and worked out problems on that. An interesting thing in a regular polygon is if we know the side we can find out the area of the polygon.
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