Is a Cube a Polygon

Cube is not a polygon,because cube is a three dimensional shaped figure .But polygon is a two dimensional object. Generally polygon must be flat, plane figure and it’s made up of line segment. Here we are going to study about cube and polygon shape and its example problems.


Shape of cube:

Cube is a regular solid three-dimensional figure it has six square faces .it has 12 edges of equal length and 8 vertices it is otherwise called as regular hexahedron.

Shape of polygon:

It will be triangle, quadrilateral, pentagon, hexagon, octagon, heptagon…etc. All are two dimensional shapes.

Example problems for cube:

Example: 1

Find the volume of the cube with side length 6 meter.

Solution:

We know that volume of the cube is a 3

Given a = 6

Therefore a3 = (6)3

= 6 * 6 * 6

= 216 meter cube

Example: 2

Find the surface area of the cube each side length of a cube is 16 feet.

Solution:

We know that surface area of cube is,

A = 6a2

Here the given  a = 16 feet

Substitute the a value in the above formula we get

A= 6*162

162 = 16*16 = 256

Therefore the area of the cube is

= 6*256

=1536 feet square

The surface area of the given cube is 1536 feet square.

Example problems for polygon:

Example: 1

Find the perimeter of the polygon which is hexagon shape with side length is 9 feet

Solution:

We know that perimeter of the hexagon is =6*a

a represents the side length

Therefore perimeter = 6 *9

= 54

Perimeter of the given hexagon is 54 feet

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Example: 2

Find the perimeter of the regular pentagon with side length is 8 meter

Solution:

We know that pentagon is one of a polygon and formula for finding the perimeter is 5 *a

= 5 * 8

= 40

Perimeter of the pentagon is = 40 meter

Unit of Measurement for Volume

The volume is defined as the space occupied by any object in three dimensions. There are various units for the measurement of the volume of the objects. The most commonly used is the standard international unit of measurement, the cubic meter unit. But there are other units of volume measurement. In this article we will see them in detail.


Unit of measurement for volume:

The various units of volume measurement are related with the standard international unit of volume measurement the cubic meter. With the relations the various units can also be related to one other. The various units of volume measurement and their relations are,

1 cubic meter = 1000 L = 264.2 gallons

1 cubic meter = 35.31 ft3 = 1.308 yd3

1 gallon = 0.1337 ft3 = 3.785 L

1 cubic feet = 7.481 gallons = 0.0283 m3

1 cubic yard = 27 ft3 = 202 gallons = 0.7646 m3 = 764.6 L

1 imperial barrel = 163.7 L = 6.10 m3

The above relations can be used for the conversion between the various units above by relating with each other.

Example problems on unit of measurement for volume:

1. A container volume is measured to be 15000 liters. Convert the volume into m3 and gallons.

Solution:

1 cubic meter = 1000 L

1 liter = `1/1000` m3

15000 liter = `15000*(1/1000)` m3

15000 liter = 15 m3

1000 liter = 264.2 gallons

15000 liters = 15*264.2 gallons

15000 liters = 3963 gallons

2. A water tank can store 20.5 m3 of water. Convert the volume into yd3 and gallons.

Solution:

1 cubic meter = 1.308 yd3

20.5 cubic meters = 20.5 * 1.308 yd3

20.5 cubic meters = 26.8 yd3

1 cubic meter = 264.2 gallons

20.5 cubic meter = 20.5*264.2 gallons

20.5 cubic meter = 5416 gallons

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3. Convert 1.35 Barrel into cubic meters.

Solution:

1 imperial barrel = 6.10 m3

1.35 imperial barrel = 1.35*6.1 m3

1.35 imperial barrel = 8.23 m3
Practice problems on unit of measurement for volume:

1. Convert the volume of 560 gallons into ft3 and liters.

Answer: 74.9 ft3 and 2119.6 L

2. Convert the volume of 2.5 yd3 into liters.

Answer: 1911.5

Median Frequency Table

Median is defined as one of the most important topic in mathematics. Mainly it is used to find the middle values. The values are given in the frequency table. By using the table we can find the median. Both the even numbers and the odd numbers, the median can be find. In this article, we are going to find the calculation of median from the frequency table.

I like to share this Median in Statistics with you all through my article.

Explanation to median frequency table

The explanation to median frequency table are given below the following,

Median for odd values:

Median = `(n + 1)/2`,      if the n value is odd.

Median for even values:

Median = `(n/2)+1` , if the n value is even.

Example problems to median frequency table

Problem 1: Find median for the following frequency table,
Values 2 3 4 5 6
Frequency 3
4 5
6
7

Solution:

Step 1: Given:

Values = 2, 3, 4, 5, 6

Frequency = 3, 4, 5, 6, 7

Step 2: Find:

Values = 2 + 3 + 4 + 5 + 6

= 20 ( Its a even function)

Step 3: Formula:

Median = `(n/2)+1` , if the n value is even.

Step 4: Solve:

Median = `(n/2)+1`

= `(20/2) + 1`

= 10 + 1

= 11

Therefore, the median is in the position of 11.

Step 5: To find position:

Add values and the frequencies, we get,
Values 2 3 4 5 6
Frequency 3 4 5 6 7
Position 2 + 3 = 5 5 + 4 = 9 5 + 9 = 14

Since the frequency is at 11 position, it will be between the 9 and the 14 position, So, 4 is the median value.

Result: Median = 4

Problem 2: Find median for the following frequency table,
Values 1 3 4 5 6
Frequency 2
4 6
8
10

Solution:

Step 1: Given:

Values = 1, 3, 4, 5, 6

Frequency = 2, 4, 6, 8, 10

Step 2: Find:

Values = 1 + 3 + 4 + 5 + 6

= 19 ( Its a odd function)

Step 3: Formula:

Median = `(n + 1)/2`,  if the n value is odd.

Step 4: Solve:

Median = `(n+1)/2`

= `(19 + 1)/2`

= `20/2`

= 10

Therefore, the median is in the position of 10.

Step 5: To find position:

Add values and the frequencies, we get,
Values 1 3 4 5 6
Frequency 2 4 6 8 10
Position 2 + 1 =3 4 + 3 =7 7 + 6 =13

Since the frequency is at 10 position, it will be between the 7 and the 13 position, So, 4 is the median value.

Result: Median = 4


My forthcoming post is on Divide Polynomials and Regular Convex Polygon will give you more understanding about Algebra.

Practice problems to median frequency table

Problem 1: Find median for the following frequency table,
Values 3
5 7 9 10
Frequency 5
6
8
8
9

Answer: 7

Problem 2: Find median for the following frequency table,
Values 3
6
9
12
15
Frequency 2
4
6
8
10

Answer: 9

Study Definition of Subset

Online gives the definition of subset as the elements of subset are contained by another set. By studying the definition of subset we can understand that subsets are part of another set which is used a symbol `sube`. Online gives a clear definition of subsets which helps to study problems easily.

Explanation to study definition of subset:

The definition of subset is as follows.

A set X is called as a subset of a set Y if some of or all the elements of X are existing in the set Y which can be denoted as X `sube` Y. We can also write the set as the set y is a superset of set X and denoted as Y `supe` X. A empty set is also taken as a subset for any kind of set.

Representation to study definition of subset:

The elements in X are existing in the set Y called X is the subset of Y.

Example problems to study definition of subset:

Example: 1

Write the subset relation for the following sets.

A = {m, n, o} , B = {o, p} and C = {m, n, o, p, q, r}

Solution:

Given sets are,

A = {m, n, o}

B = {o, p}

C = {m, n, o, p, q, r}

A and B:

The elements in A is not in B as well as the elements in B are not in A.

A and C:

The elements in A are in C. So, A is called as subset for the set C = {m, n, o, p, q, r} which can be represented as A `sube` C.

B and C:

The elements in B are in C. So, B is called as subset for the set C = {m, n, o, p, q, r} which can be represented as B `sube` C.

Example: 2

Say whether the set P = {12, 14, 15} is a subset for a set Q = {11, 12, 13, 14, 15}.

Solution:

Given sets are,

P = {12, 14, 15}

Q = {11, 12, 13, 14, 15}

The elements in P are in Q. So, P is called as subset for the set Q = {11, 12, 13, 14, 15} which can be represented as P` sube ` Q.

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Practice problems to study definition of subset:

Problem: 1

Write the subset relation for the sets C = {9, 7} and D = {9, 8, 7, 6}

Answer: C `sube` D

Problem: 2

Write the subset relation for the sets S = { } and P = {4, 5, 10}

Answer: S `sube` P

Preparation for Subset

In mathematics subsets are the terms used in set theory. The preparation depends on the elements in the sets. For the preparation for subset we have the derive a set by having elements of another set. The subset preparation uses the symbol `sube` .

For example, C `sube` A denotes C is subset of the set A and A `supe` C denotes A is a superset of C.

Understanding subset of a set is always challenging for me but thanks to all math help websites to help me out. 

Explanation to preparation for subset:

The preparation for subset is as follows.
Every set has a subset which is derived from the set. This subset may have all elements from the given set and it may be an empty set.

For example, set V = {l, p m , n}. Some of the possible subset for the given set V are { } , {l, p, m , n} , {l}, {l , p} , { m, n} , {p, m, n} etc. In this the set V is called as superset.

Example problems to preparation for subset:

Example: 1

Prepare the subsets of a set C = {2, 3}

Solution:
Given: C = {2, 3}

Subsets has the elements from the given set C = {2, 3} such as { }, {2}, {3}, {2, 3}

Example: 2

Which of the following is true for the sets A = {1, 2, 3, 4} B = {3, 4}?

a) B `supe` A

b) A `sube` B

c) A = B

d) B` sube` A

Solution:
Given A = {1, 2, 3, 4} B = {3, 4}
        B has the elements 3, 4 which is in A and has the elements 1, 2, 3, 4 where 1, 2 are not in B.So, B is a subset of A and A is a super set of B. (B `sube` A)
Answer: d

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Practice problems to preparation fo subset:

Problem: 1

Which of the following is true for the sets X = {a, c, b, m, l, i} Y = {a, b, l}?

a) Y `supe` X

b) Y `sube` X

c) Y = X

d) X `sube` Y

Answer: b
Example: 2

Which of the following is true for the sets P = {p, m} Q = {p, q, r, s}?

a) P `supe` Q

b) P `sube` Q

c) P = Q

d) Q `sube` P

Answer: b

Height of Trapezoid

Geometry deals with shapes, structures, lines, planes and angle’s. Geometry learning is also known as architectural learning. Basic shapes of geometry are square, triangle, rectangle, parallelogram, trapezoid etc. Trapezoid is one of the basic shapes in geometry. Trapezoid is a quadrilateral which has 4 sides. The total internal angle of the trapezoid is 360 degree. In trapezoid, one pair of opposite sides is parallel.

Formula for finding the height of the Trapezoid

The formula for finding the area of pyramid is given as,

Area of pyramid = h (b1 + b2)/2

Where,

h = height of the pyramid,

b1, b2 = bases of the trapezoid.

From the given area formula we can find the height of the trapezoid when the area of trapezoid is given,

A = h (b1 + b2)/2

2A = h (b1 + b2)

h = 2A/b1+ b2



Problems on height of pyramid:

Example 1:

Find the area of the trapezoid, whose bases are 10 cm and 12 cm, height, is 6 cm.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

= 6 (10 + 12) / 2

= 6 (22) / 2

= 3 * 22

= 66 cm2.

The answer is 66cm2.



Example 2:

Find the height of the trapezoid, whose bases are 8 cm and 12 cm, area, is 120 cm2.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

120 = h (8 + 12)/2

120 * 2 = h (20)

240 =   h*20

Divide 20 on both sides,

240/20 = 20*h/20

12 = h

The height is 12 cm.

I am planning to write more post on Statistics Quartiles and Calculate Geometric Mean. Keep checking my blog.

Example 3:

Find the height of the trapezoid, whose bases are 6 cm and 4 cm, area, is 100 cm2.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

100 = h (6 + 4)/2

100 * 2 = h (10)

200 =   h*10

Divide 10 on both sides,

200/10 = 10*h/10

20 = h

The height is 20 cm.

Least Common Multiple of 3 and 6

In mathematics, the least common multiple of two rational numbers a and b is the smallest positive rational number that is an integer multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then LCM(a, b) is defined to be zero. (Source: From Wikipedia).

Least common multiple of two numbers can be found by their multiples. Here we are going to learn how to find the least common multiple of two or more numbers.

Least common multiple of 3 and 6

The least common multiple of 3 and 6 can be found by finding the multiples 3 and 6.

The list of multiples of 3 and 6 are given below

The multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60.

The multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

Here 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 are the common factors, among those 6 is the lowest common number.

So, 6 is the lowest common multiple of 3 and 6.

Example problems for least common multiple

Example 1

Find the least common multiple of 3 and 16

Solution

The least common multiple of 3 and 16 can be found by finding the multiples 3 and 16.

The list of multiples of 3 and 16 are given below

The multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60.

The multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160.

Here 48 is the lowest common number. So, 48 is the lowest common multiple of 3 and 16.

Example 2

Find the least common multiple of 13 and 6

Solution

The least common multiple of 13 and 6 can be found by finding the multiples 13 and 6.

The list of multiples of 13 and 6 are given below

The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143.

The multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 78, 84.

Here 78 is the lowest common number. So, 78 is the lowest common multiple of 13 and 6.

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Example 3

Find the least common multiple of 13 and 16

Solution

The least common multiple of 13 and 16 can be found by finding the multiples 13 and 16.

The list of multiples of 13 and 16 are given below

The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208.

The multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208.

Here 208 is the lowest common number. So, 208 is the lowest common multiple of 13 and 16.