Area of Triangular Prism Formula

Introduction to formula of Area of Triangular prism:

Triangular prism:

In geometry the concept of triangular prism is one of the most important concept.A triangular prism is a polyhedron which is composed of two triangular bases and three rectangular sides. It is a pentahedron where two faces are parallel when it is in surface normal of the other three are in the same plane. In triangular prism cross-sections parallel to the base faces are the same triangle. So the faces are parallelograms. When the base faces of a triangular prism are equilateral triangles, and the other three faces are squares, then it becomes semi regular. A 3-sided bi-pyramid can be made by dual of triangular prism. It has five faces in which three are rectangles’ and two are triangles. It is with 6 vertices and 9 edges.

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There are different types of prism-

a) triangular prism,
b) square prism,
c)rectangular prism,
d)hexagonal prism,
e)pentagonal prism, and
f)octagonal prism.

Key features of a triangular prism are -

Faces: 5;
Corners: 6;
Edges:9
Different formula of Triangular prism:

Area of Base (A) = ½ * a * b
Perimeter of Base (P) = s1 + s2 + s3
Surface Area of Prism = ab + (s1 + s2 + s3)h = ab + Ph
Volume of Prism = ½ * a * b * h = Ah
[where
a = altitude, b = base, h = height and s1, s2, s3 are sides]
Area of Triangular Prism

Area of Triangular prism:

Pro: Calculate the surface area and area of base of triangular prism when altitude 12 cm, base of triangular 13 cm, height 14 cm and the sides are 10 cm, 12cm  and 13 cm.


Sol:

Step 1:To calculate the surface area of triangular prism we can use the formula,

Surface Area of Prism = ab + (s1 + s2 + s3)h = ab + Ph

[Where,a = altitude, b = base, h = height and s1, s2, s3 are sides]

Step 2:Substituting the value we will get,

Surface Area of Prism=12*13+(10+12+13)*14

=646 cm2

Step 3: Area of base= ½ * a * b

[Where, a = altitude, b = base]

=½*12*13

=78 cm2

Practice Problems on Area of a Triangular Prism:

Calculate the surface area of triangular prism when altitude 5, base of triangular 8, height 12 and the sides are 8, 6 and 8.( Surface Area of Prism=304, Area of base=20)

1. Calculate the surface area of triangular prism when altitude 4, base of triangular 3, height 5 and the sides are 1, 2and 3.( Surface Area of Prism=42, Area of base=6)

Two Equivalent Fractions

Introduction to two equivalent fractions:

Fraction is a number which has two parts. One part is in upper and the other part is lower. The upper part is known as the numerator and the lower part is known as the denominator.  It is possible to find equivalent fraction for the given fraction. Two equivalent fractions means finding two equivalent fractions for the given fraction.

Example Problems – Two Equivalent Fractions:

1) Find two equivalent fractions for the fraction `(1)/(2)` .

Solution:Given , `(1)/(2)`

To find the two equivalent fraction, multiply and divide the given fraction by 2.

The given fraction would be  `(1 * 2)/(2 * 2)`    =  `(2)/(4)`

which is one of the equivalent fractions.

To find the second equivalent fraction, multiply and divide the given fraction by 3.

The given fraction would be  `(1 * 3)/(2 * 3)`  =   `(3)/(6)`

which is second  equivalent fractions.

The two equivalent fraction of the fraction `(1)/(2)`  is `(2)/(4)`  and `(3)/(6)` .

2) Find two equivalent fractions for the fraction `(2)/(4)` .

Solution:Given , `(2)/(4)`

To find the two equivalent fraction, multiply and divide the given fraction by 2.

The given fraction would be  2 * `(2)/(4)`  * 2  =  `(4)/(8)`

which is one of the equivalent fractions.

To find the second equivalent fraction, multiply and divide the given fraction by 3.

The given fraction would be  2 * `(3)/(4)`  * 3  =   `(6)/(12)`

which is second  equivalent fractions.

The two equivalent fraction of the fraction `(2)/(4)`  is `(4)/(8)`  and `(6)/(12)`



More Examples on Equivalent Fractions:

3)Find two equivalent fractions for the fraction `(1)/(3)`.

Solution:Given , `(1)/(3)`

To find the two equivalent fraction, multiply and divide the given fraction by 2.

The given fraction would be  1 * `(2)/(3)`  * 2 = `(2)/(6)`

which is one of the equivalent fractions.

To find the second equivalent fraction, multiply and divide the given fraction by 3.

The given fraction would be  1 *`(3)/(3)`  * 3  = `(3)/(9)`

which is second  equivalent fractions.

The two equivalent fraction of the fraction `(1)/(3)` is `(2)/(6)` and `(3)/(9)` .

4) Find two equivalent fractions for the fraction `(1)/(4)`

Solution:Given , `(1)/(4)` .

To find the two equivalent fraction, multiply and divide the given fraction by 2.

The given fraction would be  1 * `(2)/(4)`  * 2 = `(2)/(8)`

which is one of the equivalent fractions.

To find the second equivalent fraction, multiply and divide the given fraction by 3.

The given fraction would be  1 * `(3)/(4)`  * 3 = `(3)/(12)`

which is second  equivalent fractions.

The two equivalent fraction of the fraction `(1)/(4)`  is `(2)/(8)`  and `(3)/(12)`

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Introduction to absolute value rules

 Introduction to absolute value rules

                The numerical value of a real number without considering it sign is called absolute value. For example we can consider the value |-5| absolute value of this is 5. We have to follow some rules for writing the absolute values. Which is called absolute value rules. Here we didn’t consider the value of the sign. If it is positive or negative the absolute value is positive. To find the absolute value of the  complex number is square root of the sum of the real and imaginary parts of the number. If we found like that we will get a real number there is no imaginary part in that.Absolute value rules are used to write the absolute values in the standard form.

The Rules Applied to Absolute Values

                Normally the absolute value is the distance between the number and the origin. It connects the absolute value of the complex number and magnitude of a vector.

Definition

                                          |a| =  `{(a if a>=0),(-a if a<0 span="span">

 It is the piecewise definition for absolute value.

Rule 1:

                A nonnegative that mean a positive  it must be multiplied times itself to equal a given number.

The square root of x can be written as square root (x) or  x^½. Totally we are having seven absolute value rules.

For example:

                Square root of (16) = 4  and  4²  = 16

Here there square root(x) never refers a negative value. Because (-4) * ( -4) is also +16 we can’t say

Square root (x) is negative if it is negative mean then it is imaginary.

Rule 2:

                |- a| = |a|

                |-a| = Square root ((-a) * ( -a) ) = a =|a|

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More Rules on Absolute Value

 Rule 3:

                |a| >0

                |-a| = |+a | =a  (|a| is always greater than 0)

Rule 4:

                Products of any two absolute value will also give a absolute value       

                                          |a b| = |a||b|

Rule 5:

                The result of dividing one absolute value by another is  equal.

                                |a/b| = |a| / |b|

Rule 6:

                Exponents the absolute values are equal.

                                                |aⁿ| = |a| ⁿ

Rule 7:

                Triangle Inequality:

                                |a +b | ≤ |a| + |b|

                 Alternative triangular property:

                                 |a – b| ≥ |a| – |b|

If we are going to write any absolute value we have to consider all the absolute value rules.

Introduction to natural logarithm base

Introduction to natural logarithm base:

The natural logarithm of a given function is called as the logarithm to the base of e, where e is known as unreasonable constant.

                                 e = 2.7182818.

 Representation of natural logarithm base can be shown as ln(x) and loge(x).

Let us take a number x and the natural logarithm base of that number is given as the power in which e put to be raised.

                                   eln(x) = x    if x > 0

                                   ln(ex) = x

In the natural logarithm multiplication can be formed into a addition which is shown below

                                  ln (xy) = ln (x) + ln (y).

Natural Logarithm Base:

 The base value of natural logarithm can be given as follows,

  ` b = n^ (1/(log_b(n)))`

 The base 2 logarithm is simply denoted as lg(x) or lb(x).

 The base 10 logarithm is commonly specified as log(x).

 When the base is 10 the logarithm is also known as common logarithm(log10 or log).

 When the base is 2  the logarithm is also known as binary logarithm(log2 or ln).

Formulae for finding natural logarithm base are as follows

                   logb x = loga x /loga b(change of base formula).

                   loge ab = loga a+ loga b(multiplication to addition formula).

                   loge a/b = loga a- loga b(division to subtraction formula).

                   logeab = b loge a(power formula).

Example Problems for Natural Logarithm Base:

Example 1:

Give the equation in exponential form for the following natural logarithm base ln 11 = 67.

Solution :

  The following definition will explain you to find the equation.

  If ln x = y ,then x=e^y    

  So 11 = e⁶⁷

Hence the equation can be written as e⁶⁷ -11 = 0.

Example 2:

Evaluate the following natural logarithm base ln e⁶ without the calculator.

Solution :

  we know that loge e = 1.

  So ln e⁶ =6*1 = 6..

So the answer is6.and ln e^x =x

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Introduction to surface area of rectangle

Introduction to surface area of rectangle

DEFINITION:

• The rectangle is a 4 sided polygon
• Opposite sides of the rectangle will be equal
• Diagonals of the rectangle will also be equal
• Length and width are the four sides of the rectangle
• All the angles of the rectangle will be equal to 90 degree.

Surface area of rectangle:

Surface Area of the rectangle can be found by using the area formula
Surface Area of rectangle is equal to length * width
Here by knowing the length and width of the rectangle, we can calculate the area.
Similarly if we know the area and any one side whether length or width we can find the other side.
The unit of measuring the area is square units.

Problems on Surface Area of Rectangle:

Surface area of rectangle:

 Surface Area of rectangle is equal to length * width

                                          

• Example 1:

Find the surface area of a rectangular field 15 m length and 8 m width.

Solution:

The surface area of a rectangle is equal to its length multiplied by its width. 

Surface area = Length * width

                        = 15 * 8

Surface area = 90 m²  

• Example 2:

Find the surface area of a rectangular field 8 m length and 12 m width.

Solution:

The surface area of a rectangle is equal to its length multiplied by its width. 

Surface area = Length * width

                        = 8 * 12

Surface area = 96 m²

• Example 3:

Find the surface area of a rectangular field 5.2 cm length and 3.7 cm width.

Solution:

The surface area of a rectangle is equal to its length multiplied by its width. 

Surface area = Length * width

                        = 5.2 * 3.7

Surface area = 19.24 cm²

Examples on Surface Area of Rectangle:

• Example 4:

Find the surface area of a rectangular field 7.3 cm length and 4.6 cm width.

Solution:

The surface area of a rectangle is equal to its length multiplied by its width. 

Surface area = Length * width

                        = 7.3 * 4.6

Surface area = 33.58 cm²

• Example 5:

Find the surface area of a rectangular field 3/2 in length and 8/3 in width.

Solution:

The surface area of a rectangle is equal to its length multiplied by its width. 

Surface area = Length * width

                        = (3/2) * (8/3)

Surface area = 4 in²