Calculator Techniques for Polygons in Plane Geometry
What Are Polygons?
Polygons are closed plane surfaces with n number of sides and n number of vertices. There are two types of polygons, a convex and a concave polygon. A convex polygon is a polygon in which no side, when extended will pass inside the polygon, otherwise, the polygon is called concave.
There are also subcategories for polygons. Equilateral polygons are polygons whose sides are equal, while polygons whose angles are equal are called equiangular polygons. Then, a polygon which is both equilateral and equiangular is called a regular polygon. Polygons have diagonals which are line segments joining two nonadjacent sides of the polygon.
Polygons are classified based on the number of sides. Here is a list of different polygons from twosided polygon to the thousandsided polygon.
Number of Sides
 Name of the Polygon
 Number of Sides
 Name of the Polygon


1
 No name
 16
 Hexadecagon

2
 Digon
 17
 Heptadecagon

3
 Triangle
 18
 Octadecagon

4
 Quadrilateral (Tetragon)
 19
 Enneadecagon

5
 Pentagon
 20
 Icosagon

6
 Hexagon
 30
 Triacontagon

7
 Heptagon
 40
 Tetracontagon

8
 Octagon
 50
 Pentacontagon

9
 Nonagon
 60
 Hexacontagon

10
 Decagon
 70
 Heptacontagon

11
 Undecagon
 80
 Octacontagon

12
 Dodecagon
 90
 Enneacontagon

13
 Tridecagon
 100
 Hectogon

14
 Tetradecagon
 1000
 Chilliagon

15
 Pentadecagon
 10000
 Myriagon

Calculator Techniques for Polygons
Calculator techniques for problems related to polygons are more on algebra and trigonometry. Memorization of formulas is what is needed. Here are the contents of the article.
 Sum of interior angles of a polygon
 Number of sides of a polygon
 Area of polygons given the length of one side
 Number of diagonals of a polygon
 Number of sides of a polygon given the number of diagonals
 Polygons inscribed in a circle
 Polygons circumscribed a circle
 Fivepointed star inscribed in a circle
 Variation of problems
Problem 1: Sum of Interior Angles of a Polygon
The sum of the interior angles of a polygon is 1,440. Find the number of sides.
Calculator Technique
a. The formula for solving the sum of the interior angles is:
Sum of Interior Angles = (n  2) (180) 1440 = (n  2) (180) 1440 / 180 = n  2 8 = n  2 n = 8 + 2 n = 10 sides
Final Answer
The polygon has ten sides. Therefore, it is a decagon.
Problem 2: Number of Sides of Equiangular Polygons
How many sides are in an equiangular polygon if each of its interior angles is 165°?
Calculator Technique
a. Let n be the number of sides. Use the same formula and solve for the number of sides
(n  2) (180) = Sum of Interior Angles (n  2) (180) = 165n n = 24 sides
Final Answer
The equiangular polygon has twentyfour sides. Therefore, it is a Tetra Icosagon.
Problem 3: Finding the Area of a Polygon Given a Side
One side of a regular octagon is three units. Find the area of the octagon.
Calculator Technique
a. Divide the octagon into eight equal parts. Calculate the interior and exterior angles of the octagon.
Interior Angle = 360 / n Interior Angle = 360 / 8 Interior Angle = 45 degrees 180 = 2 (Exterior Angle) + 45 Exterior Angle = 67.5 degrees
b. Use sine law in solving for the remaining sides.
a / sin (alpha) = b / sin (beta) x / sin 67.5 = 3 / sin 45 x = 3.92 units
c. Solve for the area of the triangle. Multiply it by the number of sides which is 8.
Area = 1/2 (b) (h) Area = 1/2 (3) (3.92 sin 67.5) Area = 5.43 square units Total Area = 8 (Area) Total Area = 8 (5.43) Total Area = 43.44 square units
Final Answer
The total area of the octagon is 44.34 square units.
Problem 4: Diagonals of a Polygon Given the Number of Sides
How many diagonals have a dodecagon?
Calculator Technique
a. The formula for solving the diagonals of a polygon is:
Diagonals = n (n  3) / 2 Diagonals = 12 (12  3) / 2 Diagonals = 54
Final Answer
The total number of diagonals in a dodecagon is 54.
Problem 5: Number of Sides of a Polygon Given the Number of Diagonals
The number of diagonals of a polygon is 252. How many sides are there?
Calculator Technique
a. The formula for solving the diagonals of a polygon is:
Diagonals = n (n  3) / 2 252 = n (n  3) / 2 n = 24 sides
Final Answer
The total number of sides is 24.
Problem 6: Number of Sides of a Polygon
How many sides has a polygon is the sum of its exterior angles equals the sum of its interior angles?
Calculator Technique
a. The formula for solving the sum of the interior angles is:
Sum of Interior Angles = (n  2) (180) Sum of Exterior Angles = 360 degrees (n  2) (180) = 360 degrees n = 4 sides
Final Answer
The total number of sides of a polygon whose sum of its exterior angles equals the sum of its interior angles is 4.
Problem 7: Area of Big and Small Polygons
A regular pentagon has sides of 25 cm. An inner pentagon with sides of 15 cm is inside and concentric to the larger pentagon. What is the area inside the larger pentagon and outside the smaller pentagon?
Calculator Technique
a. Solve for the value of h for both the small and larger triangle formed.
Larger Triangle: tan 36 = 12.5 / h h = 17.20 centimeters Smaller Triangle: tan 36 = 7.5 / h h = 10.32 centimeters
b. Solve for the area of the big and small pentagon.
Area of bigger triangle = 1/2 (25) (17.20) Area of bigger triangle = 215 square centimeters Area of bigger pentagon = 5 (215) Area of bigger pentagon = 1075 square centimeters Area of smaller triangle = 1/2 (15) (10.32) Area of smaller triangle = 77.40 square centimeters Area of smaller pentagon = 5 (77.40) Area of smaller pentagon = 387 square centimeters
c. Solve for the area difference between the larger pentagon and smaller pentagon.
Area difference = Area of bigger pentagon  Area of smaller Area difference = 1075  387 Area difference = 688 square centimeters
Final Answer
The area inside the larger pentagon and outside the smaller pentagon is 688 square centimeters.
Problem 8: FivePointed Star Inscribed in a Circle
Find the sum of the interior angles of the vertices of a fivepointed star inscribed in a circle.
Calculator Technique
a. Solve for the value of the interior angles using the formula:
Interior angle = 360 / n Interior angle = 360 / 5 Interior angle = 72 degrees
b. Solve for the sum of interior angles of the vertices.
Sum of interior angles = 5 (36 degrees) Sum of interior angles = 180 degrees
Final Answer
The sum of the interior angles of a fivepointed star is 180 degrees.
Problem 9: Area of a Circle Inscribed in a Hexagon
The side of the hexagon is h. Find the area of the biggest circle inscribed in a hexagon.
Calculator Technique
a. Solve for the radius of the circle which is same as the height of the triangle shown.
radius = h sin(60) radius = 0.866 (h)
b. Calculate the area of the circle.
Area = (pi) (r)^2 Area = (pi) (0.866 (h))^2 Area = 2.356 h^2
Final Answer
The area of the largest circle inscribed in a hexagon is 2.356h^{2}.
Problem 10: Equilateral Triangle Inscribed in a Hexagon
Given a variable 'h' as the side of a hexagon, find the area of the equilateral triangle that can be inscribed in a hexagon.
Calculator Technique
a. Set your calculator in Complex mode. Then input the equation below. The variable 'x' is the length of the side of the triangle.
x =  1 < 120  1  (h) x = 1.732 (h)
b. Solve for the area of the equilateral triangle. The formula in solving for the area of equilateral triangles is:
Area = (x)^2 √(3) / 4 Area = (1.732h)^2 √(3) / 4 Area = 1.3 h^2
Final Answer
The area of the equilateral triangle inscribed in a hexagon is 1.3h^{2}.
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