Polygons

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Polygons

Polygons are the part of the plane bounded by a dashed line enclosed. So, basically a polygon is made from straight lines.

Polygons are classified  by the number of sides that they have.

Three sided polygons

A three sided polygon is a triangle.

The internal angles of all triangles add up to 180°.

Triangles can be classified based on the number of sides and also based on the angles.

Based on their angles

Based on their side lengths

Don’t forget:

Each of this triangles can also be either Equilateral, Isosceles or Scalene.

Area and the Perimeter of triangle

polygons 1

The area of a triangle is the total region that is defined by the three sides of the triangle.The area of a triangle is equal to half the product of a side with the height above it.

$\displaystyle A=\frac{b\cdot h}{2}$

The perimeter of a triangle is the distance around the shape and to find it we sum the three side lengths of the triangle.

P = a + b + c

Example 1: Find the area of the triangle where is given the base 5 cm and the height 3 cm.

polygons 2

Solution: Based on the principal formula $\displaystyle S=\frac{b\cdot h}{2}$ we obtain

$\displaystyle A=\frac{5\cdot 3}{2}$

$\displaystyle A=\frac{15}{2}=7.5c{{m}^{2}}$

A = 7,5 cm²

Example 2: Find the perimeter of the triangle where are given the three side lengths 5 cm, 8 cm and 10 cm.

P = the sum of all the lengths of the triangle

P = 5 + 8 + 10 = 23 cm

Four Sided Polygons

A four sided polygon is a quadrilaterals.

The internal angles of all quadrilaterals add up to 360°.

The group of quadrilaterals include.

Square – is a four sided shape that has equal sides and each angle is 90°.

Rectangle – is a four sided shape where every angle is 90°.

Parallelogram – is a four sided shape where opposite sides are equal and parallel.

Rhombus – is a four sided shape that has equal sides.

Trapezium – is a four sided shape where two sides are parallel,side lengths and angles are not equal.

Isosceles Trapezium – is a four sided shape where  2 sides are parallel and base angles are equal,non parallel sides are equal lengths.

Kite – is a four sided shape with two pairs of adjacent (touching), congruent (equal-length) sides.

Irregular Quadrilaterals – is a four sided shape where no side are equal in length and no internal angles are the same.

There are two main types of polygons: ‘’Regular and Irregular’’

A regular Polygon is a polygon where all the sides have the same lengths and all angles between sides are equal.

An Irregular Polygon is a polygon which has unequal side lengths and unequal angles between sides.

Two other types of Polygons : “Concave or Convex”

convex polygon is defined as a polygon with all its interior angles less than 180°.

concave polygon is defined as a polygon if any internal angle is greater than 180°.

polygons 4

Parallelogram

Parallelogram – is a four sided shape where opposite sides are parallel.

Properties:

1. Opposite sides are congruent.

2. Opposite angles are congruent.

3. Consecutive angles are supplementary.

4. The diagonals of a parallelogram bisect each other.

5. Each diagonal of a parallelogram separates it into two congruent triangles .

6. If one angle is right ,then all the angles are right.

The Area and Perimeter of Parallelogram

The area of parallelogram is measurement of the surface of a shape.

The area of a parallelogram is found by multiplying the base with the height.

$latex \displaystyle A=b\cdot h$

The perimeter of a parallelogram is the total length of the sides.

Perimeter is found by adding the lengths of all sides. In this case, the perimeter of parallelogram is:

 P = 2(B + S)

Example: Find the perimeter and area of the parallelogram, whose base is 18 cm, the length is 8 cm and the height is 4 cm.

Solution: As we know, the perimeter of parallelogram is:

P = 2(a + b) so,

$\displaystyle P=2\left( 18+8 \right)$

$\displaystyle P=2\cdot 26=52cm$

Now, let’s find the area of parallelogram:

$\displaystyle A=b\cdot h$

$\displaystyle A=18\cdot 4=72c{{m}^{2}}$

Rectangle

A Rectangle is a four sided-shape where all the angles are right-angled (90o).

Properties

1. It’s a parallelogram with four right angles.

2. Its diagonals bisect each other.

3. The opposites side of  a rectangle are equal.

4. The opposites side of a rectangle are parallel.

The area and the perimeter of Rectangle

polygons

The area of a Rectangle is the plan enclosed by the sides of the rectangle and it’s found by multiplying the length with the width.

Area = length x width

$\displaystyle A=a\cdot b$

The perimeter of a Rectangle is the total lengths of all sides.

The perimeter of the rectangle is 2 times the(length of one side + the other side )

P = 2(a + b)

Example: Find the perimeter and area of the rectangle, whose the base is 10 cm and the length is 8 cm.

Solution: The perimeter of rectangle is:

P = 2(a + b)

$\displaystyle P=2\left( 10+8 \right)$

$\displaystyle P=2\cdot 18=36cm$

The area of rectangle is:

$\displaystyle A=a\cdot b$

$\displaystyle A=10\cdot 8=80c{{m}^{2}}$

Square

Square is a four sided shape which have all the sides of equal length and also the angles are equal, all 90°.

The area and the perimeter of a rectangle

The area of a square is the plane surrounded by the sides.

To find the area of the square we multiply side times side or we just square the side since all sides are equal.

$\displaystyle A=a\cdot a={{a}^{2}}$

The perimeter of a square is the total lengths of all sides. Since all sides are equal, we can say the perimeter of a square is 4 times the side of square.

$\displaystyle P=4\cdot a$

Properties

1. All the interior angles are right angles (90°)

2. All the sides of the square are equal.

3. The opposite sides of the square are parallel to each other.

4. The diagonal bisects each other at 90°

5. The two diagonals of the square are equal to each other.

6. The diagonal of the square divides it into two similar isosceles triangles.

Example: Find the area and perimeter of the square, whose the base is 7 cm.

Solution: The perimeter of square is P = 4a, so we have:

$\displaystyle P=4\cdot 7$

P = 28 cm

The area of square is:

$\displaystyle A={{a}^{2}}$

$\displaystyle A={{7}^{2}}=49c{{m}^{2}}$

Trapezium

Trapezium is a four sided shape where two sides are parallel, side lengths and angles are not equal.

The parallel sides of a Trapezium are called bases and the non parallel sides are called legs.

Types of Trapezium

Isosceles Trapezium “The legs or the not parallel sides are equal.”

Scalene Trapezium “A trapezium with all the sides and angles of different measures’’

Right Trapezium  “A right trapezium has at least two right angles”

Properties

1. In a trapezium at least two opposite sides are parallel.

2. The diagonals intersect each other .

3. The sides which are not parallel in a trapezium are not equal except in the Isosceles trapezium.

4. The line that joins the mid-points of the non-parallel sides is always parallel to the bases or parallel sides which is equal to half the sum of the parallel sides.

The Area and Perimeter of trapezium

The parallel sides on a Trapezium are called the bases. The one that is longer is called the big base ( B) and the other the small base (b) of the trapezium.

The area of the trapezium is equal to half the product of the sum of the bases with height.
The formula is:
$\displaystyle A=\frac{h\left( B+b \right)}{2}$

The perimeter of a square is the total lengths of all sides. Since all sides are equal, we can say the perimeter of a square is 4 times the side of square.

$\displaystyle P=4\cdot a$

Find the are and perimeter in cm of the trapezium shown below.

The perimeter of trapezium is P = a + b +c

P = 5 + 10 + 4 + 5.
P = 24 cm.
The area of the trapezium is: $\displaystyle A=\frac{h\left( B+b \right)}{2}$.
Firstly we have to find the height of the trapezium because that’s the only data that´s missing from the formula.

We try to observe the figure first to see the data we have. Based on the data we see that our trapezium is an isosceles trapezium. That’s why the height is perpendicular with the base and appoints the same length on both side, so both are 3 cm.

Now, to find the height of the trapezium we use Pythagoras Theorem because we have a right angle triangle where two sides are known.

h= (5)– (3)2
h= 25 – 9
h= 16
h = 4

Now we have all the data so we use the formula

$\displaystyle A=\frac{(10+4)\cdot 4}{2}$

$\displaystyle A=\frac{14\cdot 4}{2}=\frac{56}{2}=28c{{m}^{2}}$

The perimeter of a square is the total lengths of all sides. Since all sides are equal, we can say the perimeter of a square is 4 times the side of square.

$\displaystyle P=4\cdot a$

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