Polygon Formula and Sum Of Squares Formula and Surface Area Formula

Polygon Formula

Polygons can be found everywhere in our surroundings as well as in geometrical math. Many objects are in the shapes of polygons. In this article, students will learn what are polygons as well as various polygon formula. Some examples will help to understand the concept and formula. Let us begin the concept!

Polygon Formula

What is Polygon?

A polygon is any two-dimensional or 2D shape formed with the straight lines. Triangles, quadrilaterals, pentagons, and hexagons are related shapes. The name tells us that how many sides the shape has. For example, a triangle is having three sides, and a quadrilateral has four sides.

Therefore, any shape that can be drawn by connecting three straight lines is a triangle, and any shape that can be drawn by connecting four straight lines is called a quadrilateral. All of these shapes are polygons. If the shape had curves it will not be a polygon.

A special class of polygon exists in mathematics, it happens for polygons whose sides are all the same length and whose angles are all the same. This is called regular polygons. A stop sign as the traffic signal is an example of a regular polygon with eight sides.

All the sides are the same and no matter how you lay it down, it will look the same. You wouldn’t be able to tell which way was up because all the sides are the same and all the angles are the same.

A pentagon with all sides of equal length and equal angles, then it is called a regular pentagon.

Polygon is a word taken from the Greek language, where poly means many and gonna means angles. So we can say that in a plane, a closed figure with many angles is called a polygon. There are many properties in a polygon such as sides, diagonals, area, angles, etc.

Some popular Polygons are:

1. Triangle – 3 sides
2. Quadrilateral – 4 sides
3. Pentagon – 5 sides
4. Hexagon – 6 sides
5. Heptagon – 7 sides
6. Octagon – 8 sides
7. Nonagon – 9 sides
8. Decagon – 10 sides

Various Polygon Formula

1. Sum total of all internal angles:

(T=(n−2)×180

Where,

T	Sum of internal angles
n	Number of sides

2. Each Interior Angle:

IA = (n−2)×180n

Where,

IA	Each internal angle
N	Number of sides

3. Each Exterior Angle:

EA = 360n

Where,

EA	Each exterior angle
N	Number of sides

4. Perimeter:

P=n×s

Where,

P	Perimeter
n	Number of sides

5. Area of the polygon:

A=s2tan(180n)

Where,

A	Area of polygon
n	Number of sides

Solved Examples

Q.1: A polygon is an octagon and its side length is 5 cm. Calculate its perimeter and value of one interior angle.

Solution:

Given in the problem:

The polygon is an octagon. Hence, n = 8

Length of one side,

s = 5 cm

The perimeter of the octagon

P = n × s

P = 8× 5

=40 cm.

Now, to compute interior angle,

IA = (n−2)×180n

= (8−2)×1808

= 10808

IA=135degrees

So, each interior angle will be 135degrees

Sum of Squares Formula

The sum of squares formula is used to calculate the sum of two or more squares in a given expression. To describe how well a model represents the data being modeled, this formula is used. Also, the sum of squares will give the measure of deviation from the mean value of the data. Therefore, it is calculated as the subtraction of the total summation of the squares and the mean.

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Sum of Squares Formula

Concept of the sum of squares

The sum of squares is a very useful tool used by statisticians and scientists. It is used to evaluate the overall variance of a data set from its mean value. A large sum of squares denotes the large value of variance. It means that individual readings fluctuate widely around its mean value.

This information is very useful in many situations. For example, computing the variance in blood pressure readings over some period of time. This is needed for instability in the cardiovascular system requiring medical attention.

For financial advisors, a large variance in daily stock values indicates the market instability and higher risks for investors. When we are taking the square root of the sum of squares, we get the standard deviation i.e. an even more useful number.

The formula of Sum of Squares:

The calculation of sample variance is typically stated as a fraction. The numerator of this fraction involves a sum of squared deviations from its mean value. In the statistics domain, the formula for this total sum of squares is:

Σ(xi–x¯)2

Where,

x¯	the sample mean
x_i	ith data from the sample.
Σ	Sum
(xi–x¯)	The difference of data with the mean value.

While this formula works for calculations, still there is an equivalent and shortcut formula. It does not require the calculation of the sample mean. This shortcut formula for the sum of squares is given below:

Σ(x2i)–(Σxi)2n

Where,

n	Sample data size
xi	ith data from the sample.
Σ	Sum

Solved Examples

Q.1: Compute the sum of squares of the following data using the mean calculation.

X : 74.01 , 74.77 , 73.94 , 73.61, 73.40.

Solution:

Given sample data is an ungrouped type of data. First, we have to calculate the mean of the given data.

Formula for mean value is:

X¯=SumofalldatavaluesSampledatasize

Where σxissumofthedataandnisthenumberofsampledata.

X¯=74.01+74.77+73.94+73.61+73.405

= 369.735

= 73.95

Now, we will apply the formula for computing sum of squares using mean value, as below:

Σ(xi–x¯)2

= (74.01 – 73.95)² + (74.77 – 73.95)² + (73.94 – 73.95)² + (73.61 – 73.95)² + (73.40 – 73.95)²

= (0.06)² + (0.82)² + (-0.01)²  + (-0.34)²  + (-0.55)²

= 1.0942

Therefore the sum of squares is 1.0942.

Surface Area Formula

A three-dimensional shape is a solid shape with height or depth. For example, the sphere, cuboid, sphere, etc. are three-dimensional. The surface area of a three-dimensional shape is the sum total of all of the surface areas of each of the sides. Children will like to think of the shape as a birthday present and the surface area as the wrapping gift paper. If we carefully took the wrapping paper to cover up the gift item, then we have to add the area of all the sides. The total value will be the surface area of the shape. In this article, we will explore the surface area formula of various objects with different shapes. Let us learn the interesting concept!

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Surface Area Formula

What is Surface Area?

When we are finding the surface area of a 3-D shape, think of it as unfolding the shape, or flattening it out, and then finding the area of each side. When we add all of these areas up, we have the surface area. In order to find the area of a 3-D shape, we must know how to find the area of the basic shapes that make up the sides of the 3-D shape.

For a solid object, the space covered from all sides is termed as the surface area of the object.  By measuring surface area we can measure the area of material required to cover the 3-D object completely. Since it is computation of the area, therefore its unit is a square meter or square centimeter or likewise. Computation of the surface area depends upon the shape and size.

Various Surface Area Formulae:

The volume of different objects with different sizes and shapes will be calculated as follows:

1. Surface Area of a cuboid:

S =2 × (LB + BH + HL)

Where,

S	Surface Area of Cuboid
L	Length of Cuboid
B	Breadth of Cuboid
H	Height of Cuboid

2. Surface Area of a cube:

S = 6 × A²

Where,

S	Surface Area of Cube
A	Side of Cube

3. Surface Area of a Cylinder is:

S=2π×R×(R+H)

Where,

S	Surface Area of Cylinder
R	The radius of Circular Base
H	Height of Cylinder

4. Surface Area of a Sphere is:

S=4π×R2

Where,

S	Surface Area of Sphere
R	Radius of Sphere

5. Surface Area of a Right circular cone:

S=π×r(l+r)

Where,

S	Surface Area of Cone
R	The radius of Circular Base
L	Slant Height of Cone

6. Surface Area of a Hemisphere:

S=3π×R2

Where,

S	Surface Area of Sphere
R	Radius of Sphere

Solved Examples

Example-1: The dimensions of a rectangular box are given as 5m, 3m and 2m. This tank has to be covered from all sides by cloth. Compute the cost for covering it, if rate of cloth is Rs. 25 per square meter.

Solution:

As given the dimensions of the box in cuboid shape is,

L= 5m

B= 3m

H= 2m

Now, we have to compute the surface area by using formula as:

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Putting values,

S=2×(5×3+3×2+2×5)

S=2×(15+6+10)

S=62SquareMeter

So, cost to cover it will be,

Cost=S×Rs.25

Cost=62×25

Cost = Rs. 1550

Hence the cost to cover is Rs. 1600.