Polygons
A polygon is a closed figure in two dimensions with a certain number of finite sides. The word polygon refers to the ‘many angles’ found in such shapes. A polygon must have a minimum of three sides and vertices and they are always flat and plane in shape. The line segments of a polygon are known as edges or sides. The points where the line segments of a polygon meet are known as vertices.
For example:
We have a few types of polygons that can be seen in this image.
Coordinate Plane
The coordinate plane is formed by the intersection of horizontal and vertical number lines. The horizontal line represents the x-axis of the plane. The vertical line represents the y-axis of the plane. The point where the x-axis and y-axis meet is known as the origin. The points on the coordinate plane are represented in the form of ordered pairs, and are written as, (x, y), where x tells us the distance from the origin along the x-axis, and y tells us the distance from the origin along the y-axis.
For example, (2, 3)
The first number, 2, is the x-coordinate and it marks the horizontal distance from the origin along the x-axis. The second number, 3, is the y-coordinate, and it denotes the vertical distance from the origin along the y – axis.
Polygon in the Coordinate Plane
A polygon is drawn in a coordinate plane by plotting the points and connecting them. There are a few steps to follow:
Step 1: Plot each point on the coordinate plane
Step 2: Label each point of the coordinate plane
Step 3: Join the adjacent points to form a polygon
For example: Draw a polygon with the following coordinates: A(2, 3), B(7, 3), C(7, -2), D(2, -2).
Solved Examples
Example 1:
Draw a polygon with the following points as vertices The points are: A(-3, 5), B(3, 3), C(6, 5), D(4, -1), E(-1, -2), F(1, 1).
Solution:
The vertices are A(-3, 5), B(3, 3), C(6, 5), D(4, -1), E(-1, -2), and F(1, 1).
First, plot the vertices on the coordinate plane and then join the adjacent points.
The resulting polygon is an irregular polygon.
Example 2:
Find the area of the polygon formed by the coordinates (1, 3), (1, -1), and (6, -1).
Solution:
First, plot the vertices on the coordinate plane and then join the adjacent points.
The polygon that is formed is a right angle triangle.
To find the area of a triangle, calculate the length of its base BC and height AB.
The coordinates of the points are A(1, 3) and B(1, -1).
The coordinate points lie in different quadrants and have the same x-coordinates.
So, the distance between A and B is the sum of the absolute values of the y-coordinates.
|-1| + |3| = 1 + 3 = 4 units.
The coordinates of the points are B(1, -1) and C(6, -1).
The coordinate points lie in the same quadrant and have the same y-coordinates. So, the distance between B and C is the difference in the absolute value of the x-coordinates.
|6| – |1| = 6 – 1 = 5 units.
The area of a triangle can be calculated by using its area formula.
A = \(\frac{1}{2}bh\) [Formula for the area of a triangle]
= \(\frac{1}{2}\times 5\times 4\) [Substitute 5 for b and 4 for h]
= 10 [Simplify]
Hence, the area of the polygon formed is 10 square units.
Example 3:
Sam was plotting some random points A(-2, 4), B(-2, -2), C(6, -2), D(6, 4) on a coordinate plane. Draw the polygon on a plane and find the perimeter of the polygon.
Solution:
First, plot the vertices on the coordinate plane and then join the adjacent points. The resulting polygon is a rectangle.
To find the perimeter of a rectangle, calculate the length (BC) and width (AB).
The coordinates of the points are A(-2, 4) and B(-2, -2).
The coordinate points lie in different quadrants and have the same x-coordinates. So, the distance between A and B is the sum of the absolute values of the y-coordinates.
|-2| + |4| = 2 + 4 = 6 units.
The coordinates of the points are B(-2, -2) and C(6, -2).
The coordinate points lie in different quadrants and have the same y-coordinates. So, the distance between B and C is the sum of the absolute values of the x-coordinates.
|6| + |-2| = 6 + 2 = 8 units.
The perimeter of a rectangular shape can be calculated by using the formula for the perimeter.
P = 2(l + w) [Formula of the perimeter of a rectangular shape]
= 2(6 + 8) [Substitute 6 for l and 8 for w]
=28 [Simplify]
Hence, the perimeter of the shape is 28 units.
A polygon is a closed figure in two dimensions with a certain number of finite sides.
The minimum number of sides and vertices of a polygon is 3. The polygon formed by 3 sides is known as a triangle.
The polygon with all sides and interior angles equal is known as a regular polygon.
The perimeter of a polygon is the sum of the lengths of all its sides.