Category: Maths Notes PPT

A prism is a five-sided polyhedron with a triangular cross-section. In a prism, there are two identical parallel triangles opposite to each other. Along with the triangles, three rectangular surfaces are inclined to each other. A prism is a transparent solid used for refraction. The two inclined rectangular surfaces through which the light passes are called the refracting surfaces. The angle formed between these two refracting surfaces is called the refracting edge of the prism. The section of the prism that is perpendicular to the refracting edge is called the principal section of the prism. The third rectangular surface at the bottom is the base of the prism. 

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Again, the question of what is a prism can be answered in two ways as the concept of it is used in both mathematics as well as science. In mathematics, a prism is defined as a polyhedron.  In physics (optics), a prism is defined as the transparent optical element that has flat and polished surfaces used for refracting light. There are two formulas of the prism.  

1. The surface area of a prism = (2×BaseArea) +Lateral Surface Area

2. The volume of a prism =Base Area× Height

Properties of Prism 

Now that we know what is a prism, we can know the properties of prism easily. 

1. Among all the properties of the prism the most basic is that the base and top of the prism are parallel and congruent.

2. In a prism, except the base and the top, each face is a parallelogram. These faces are known as Lateral face.

3. The base and the top has one edge common with every lateral face. 

4. The height of the prism is basically the common edge of two adjacent side faces.

Types of Prism

There are different types of prisms. Some of them are: 

1. Rectangular Prism: In a Rectangular Prism, 2 rectangular bases are parallel to each other and 4 rectangular faces. 

2. Triangular Prism: In a Triangular Prism, there are 2 parallel triangular surfaces, 2 rectangular surfaces that are inclined to each other and 1 rectangular base.  

3. Pentagonal Prism: In a Pentagonal Prism, 2 pentagonal surfaces are parallel to each other and 5 rectangular surfaces that are inclined to each other. 

4. Hexagonal Prism: In a Hexagonal Prism, there are 2 hexagonal surfaces parallel to each other and 6 rectangular surfaces that are inclined to each other.  

These were the few different types of prisms. 

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Different types of Prisms

There can be yet two other types of prisms that can also be a right prism and oblique prism.

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Base Area

1. The base area of a rectangular prism formula = base length x base width.

2. The base area of a triangular prism formula = ½ x apothem length x base length. 

3. The base area of a pentagonal prism formula = 5/2 x apothem length x base length.

4. The base area of a hexagonal prism formula = 3 x apothem length x base length.

Surface Area Of A Prism

1. The surface area of a rectangular prism formula = 2 x (base length x base width) + (base width x height) + (height x base length)

2. The surface area of a triangular prism formula = (Apothem length x base length) + 3 x (base length x height)

3. The surface area of a pentagonal prism formula = 5 x(apothem length x base length) +5 x (base length x height)

4. The surface area of a hexagonal prismformula = 6 x (apothem length x base length) + 6 x (base length x height)

Volume of A Prism

1. The volume of a rectangular prism formula = Base width x base length x height

2. The volume of a triangular  formula= ½ x apothem length x base length x height

3. The volume of a pentagonal prism formula = 5/2 x apothem length x base length x height

4. The volume of a hexagonal prism formula = 3 x apothem length x base length x height 

Find the Volume of Prism

The volume of the prism may be calculated using the procedures below:

• First, write down the prism’s dimensions.

• Using the formula V = BH, find the volume of the prism, where “V”, “B”, and “H” are the volume, base area, and height of the prism, respectively.

• After you’ve calculated the volume of the prism, write the unit of volume of the prism at the end (in terms of cubic units).

Solved Examples

Example 1: The base of a right triangle prism where the lengths of the sides arc are 13 cm, 20 cm, and 21 cm. If the height of the prism is 9 cm. Find:

1. The area of the total lateral surface. 

2. Area of the whole surface. 

3. The volume of the prism.

Solution 1: Let the semi-perimeter of the triangular base of the prism be s.

Therefore, S = (13 + 20 + 21)/2 cm. = 27 cm

The area of the prism = √[s(s – a)(s – b)(s – c)]

= √(27(27 – 13)(27 – 20)(27 – 21)) sq. cm.

= √(27 × 14 × 7 × 6) sq. cm.

= 9 × 7 × 2 sq. cm.

The area of the total lateral surface of the prism = (perimeter of the base) × height

= (486 + 2 × 126) sq. cm. 

The volume of the prism = area of the base × height

= 126 × 9 cu.cm.

= 1134 cu.cm.

Example 2: Find the surface area of the triangular prism if the apothem length = 5 cm, base length = 10 cm, and height = 18 cm. 

Solution 2: We have: 

Apothem length = 5 cm; 

Base length = 10 cm; 

height = 18 cm

The surface area of a triangular prism = ab + 3bh

= (5 cm × 10 cm) + (3 × 10 cm × 18 cm)

= 50 cm² + 540 cm²

= 590 cm² 

Fun Facts

1. The prism helps in the refraction of light. It splits the light into a lot of different colours which is known as a spectrum.

Difference Between Pyramid and Prism

• Both a pyramid and a prism are three-dimensional polyhedron-shaped structures, with the difference in their bases being the most significant.

• A prism, on the other hand, has two bases, whereas a pyramid has just one.

• A pyramid’s sides are always triangular, whereas the sides of a prism are always rectangular.

• The sides of a pyramid are angled to the base, whereas the sides of a prism are perpendicular to the base.

• A pyramid’s sides are always connected together at a point; whereas, a prism’s sides are not always joined together at a point.

• In a pyramid, the point where all of the sides meet is called the apex or vertex, and it is located vertically above the center of the base, but in a prism, there is no such point.

• A pyramid is associated with the field of geometry, whereas a prism is associated with the fields of geometry and optics.

• The existence of an apex distinguishes a pyramid. There is no apex in Prism.

To find the length of an arc of a circle, let us understand the arc length formula. An arc is a component of a circle’s circumference.

Again, if we want an exact answer when working with π, we use π. We substitute a rounded form of π, such as 3.14, if we want to approximate a response. Also, r refers to the radius of the circle, which is the distance from the center to the circumference of a circle.

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What is an Arc?

An arc is a connected subset of a separable curve in Euclidean geometry. Depending on whether they are confined or not, arcs of lines are called segments or rays. Two arcs are determined by every pair of distinct points on a circle. If the two points are not directly opposite each other, an angle at the center of the circle that is less than π radians (180 degrees) will be subtended by one of these arcs, the minor arc, and an angle greater than π radians by the other arc, the major arc.

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What is Arc Length?

Now, let us understand what is arc length. Arc length is the distance along a segment of a curve between two points. Rectification of a curve is often called evaluating the length of an irregular arc section. The advent of infinitesimal calculus led to a general arc angle formula which, in some instances, provides closed-form solutions.

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Finding Arc Length

Now, let us find out how to find the arc length under different conditions. For finding arc length, there are different arc angle formula for different conditions.

• Arc Length Formula Radians

• Arc Length Formula Degrees

• Arc Length Formula Integral Form

 

Where, s: arc length of the circle,

r: radius of the circle,

θ: central angle of a circle.

Conclusion

The arc formula is used to find the length of an arc in the circle. And as seen above, there are different formulas under different conditions. So, while we calculate the arc length we have to focus on the given conditions.

As the name suggests, ‘equi’ means equal, an equilateral triangle is the one in which all sides are equal. The internal angles of any given equilateral triangle are of the same measure, that is, equal to 60 degrees. 

 

Triangles are classified into three sorts based on the length of their sides:

• Scalene triangle: The sides and the angles of the scalene triangle are not equal.

• Isosceles triangle: An isosceles triangle has two equal sides and two equal angles.

• Equilateral triangle: All sides and angles of the equilateral triangle are equal.

Area of Equilateral Triangle

The region enclosed by the three sides of an equilateral triangle is defined as the area of the equilateral triangle. It is expressed in square units. The common units used to express the area of an equilateral triangle are in2, m2, cm2 and yd2

Below the area of the equilateral triangle formula, the altitude of the equilateral triangle formula, the perimeter of the equilateral triangle formula, and the semi-perimeter of an equilateral triangle are discussed.

 

Area of the Equilateral Triangle Formula

The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional plane. To recall, an equilateral triangle can be defined as a triangle in which all the sides are equal and the measure of all the internal angles is 60°. So, an equilateral triangle’s area can be calculated if the length of any one side of the triangle is known.

 

The area occupied between the sides of an equilateral triangle in a plane is calculated using the equilateral triangle area formula.

 

The formula for calculating the area of a triangle with a known base and height is:

 

Area = 1/2 × base × height

 

The following formula can be used to compute the area of an equilateral triangle:

 

Area = √3/4 × (side)2 square units

  

Perimeter of the Equilateral Triangle Formula

The perimeter of a triangle is equal to the sum of the length of its three sides, whether they are equal or not.

An equilateral triangle’s perimeter is the sum of its three sides. 

 

P = 3a is the basic formula for calculating the perimeter of an equilateral triangle, where ‘a’ denotes one of the triangle’s sides. The sum becomes a + a + a = 3a since all three sides of an equilateral triangle are equal.

 

 Height = √3a/ 2

 

 Semi perimeter = (a + a + a)/2 = 3a/2

 

Formulas and Calculations for an Equilateral Triangle:

• Perimeter of Equilateral Triangle: P = 3a

• Semiperimeter of Equilateral Triangle Formula: s = 3a/2

• Area of Equilateral Triangle Formula: K = (1/4) * √3 * a2

• The altitude of Equilateral Triangle Formula: h = (1/2) * √3 * a

• Angles of Equilateral Triangle: A = B = C = 60 degrees

• Sides of Equilateral Triangle: a equals b equals c.

 

1. Given the side of the triangle, find the perimeter, semiperimeter, area, and altitude.

• a is known here; find P, s, K, h.

• P equals 3a

• s = 3a/2

• K = (1/4) * √3 * a2

• h = (1/2) * √3 * a

 

2. Given the perimeter of the triangle , find the side, semiperimeter, area, altitude.

• Perimeter(P) is known; find a, s, K, and h.

• a = P/3

• s = 3a/2

• K = (1/4) * √3 * a2

• h = (1/2) * √3 * a

 

3. Given the semi perimeter of a triangle, find the side, perimeter, area, and altitude.

• Semiperimeter (s) is known; find a, P, K, and h.

• a = 2s/3

• P = 3a

• K = (1/4) * √3 * a2

• h = (1/2) * √3 * a

 

4. Given the area of the triangle find the side, perimeter, semiperimeter, and altitude.

• K is known; find a, P, s and h.

• a = √

• (4/√3)∗K

• (4/√3)∗K equals 2 * √

• K/√3

• K/√3

• P = 3a

• s = 3a / 2

• h = (1/2) * √3 * a

 

5. Given the altitude/height find the side, perimeter, semiperimeter, and area

• Altitude (h) is known; find a, P, s, and K.

• a = (2/√3) * h

• P = 3a

• s = 3a/2

• K = (1/4) * √3 * a2    

 

Solved Examples

1. Apply the equilateral triangle area formula and find the area of an equilateral triangle whose each side is 12 in.

Solution:

 

Side = 12 in

 

Applying the equilateral triangle area formula,

 

Area = √3/4 × (Side)2

 

= √3/4 × (12)2

 

= 36√3 in2

 

Answer: Area of an equilateral triangle area 36√3 in2

 

2. Calculate the perimeter and semi perimeter of an equilateral triangle with a side measurement of 12 units.

Solution:

The perimeter  = 3a 

 

Semi-perimeter = 3a/ 2

 

Given, side a = 12 units

 

 Now, the perimeter of an equilateral triangle is equal to:

 

3 × 12 = 36 units

 

  And, Semi-perimeter of an equilateral triangle is equal to:

 

36/2 = 18 units.

 

3. Suppose you have an equilateral triangle with a side of 5 cm. What will be the perimeter of the given equilateral triangle?

Solution) We know that the formula of the perimeter of an equilateral triangle is 3a.

 

Here, a = 5 cm

 

Therefore, Perimeter = 3 * 5 cm = 15 cm.

Trigonometry is an interesting domain of higher mathematics that students love to study and solve problems. The ratio of the sides of a right-angled triangle expressed in the form of trigonometric ratios and their relation with the angles is quite fascinating to learn. For this, you will have to study the different trigonometric ratios and their identities. These identities will explain what the Sin squared x formula stands for. In this section, you will find out what this expression represents and how it can be used.

What do You Mean by the Formula of Sin Square X?

Trigonometry is the fundamental branch of higher mathematics that is concerned about the ratios of the sides of the triangles and their representation in the form of symbols such as sine, cosine, secant, cosecant, tangent, and cotangent. These symbols represent a particular ratio of the sides of a right-angled triangle. One such trigonometric ratio is Sine or Sin. An angle is written after it to complete the representation of a particular value of a ratio of two sides. When it is squared, the Sine becomes:

Sin X = perpendicular/hypotenuse = p/h

Sin²X = (p/h)²

Hence, the formula of Sin square X is represented in this way. When this square is doubled then the representation becomes a little different.

Sin²X = (p/h)²

When doubled, 2 Sin²X = 2(p/h)²

As you can see, 2 is multiplied simply with the square of this ratio to get a value. Here, the value of X can be anything. A change in the value of X will also change the value of Sin X.

Let us consider another example here. If the angle is doubled then it will become 2X. What will happen to the Sin squared 2X formula? Let us check.

If the angle is 2X then the ratio will become Sin 2X. When it is squared, the result will be different and represented in the following way.

(Sin 2X)² = Sin²2X

Check and understand the difference between the square of the sine of an angle and the same of the angle doubled. Learn how to represent these expressions properly so that you can avoid confusion while solving trigonometric problems in the exercise and in the exams.

Learn the Difference Between the Sin Squared X Formula and Others

It is of utmost importance to learn how to represent the Sin squared X formula when you are studying the trigonometric ratios and the relevant identities. When you will proceed to the next level in an advanced class, these representations will be required. Understand the meaning of every expression first and then learn the differences.

Conclusion

Follow the explanation given above and find out what the formula of Sin square x means. Then proceed to the other explanations as well to grab hold of the concepts. These concepts will be required in mathematics and science subjects in the future.

Yield is a percentage word for the earnings created and realised on an investment over a certain period. The percentage is calculated using the amount invested, the current market value, or the investment security’s face value. Interest or dividends obtained from owning certain securities for a specified period are included in yield. However, financial gains are not taken into account. Yields are characterised as known or expected depending on their nature and valuation (whether constant or fluctuating).

Percentage Yield Formula

The ratio of actual yield to theoretical yield is the percentage yield formula. Yield is a measure of the number of moles of a product generated in a chemical reaction in proportion to the amount of reactant consumed, generally given as a percentage. The percentage yield is the difference between the actual amount of product produced and the maximum calculated yield.

The experimental yield divided by the theoretical yield multiplied by 100 is the percentage yield formula. The percent yield is 100 percent if the actual and theoretical yields are equal. Because the actual yield is typically lower than the theoretical value, the percent yield is usually less than 100 per cent.

Percentage yield =[frac{Actual, Yield}{Theoretical, Yield}] x 100%

Theoretical Yield Formula

The yield of a reaction is defined as the quantity of a product produced from the reaction. The theoretical yield is the quantity of product anticipated by stoichiometry, whereas the actual yield is the amount achieved in reality. The proportion of the theoretical quantity is represented as a reaction yield.

For calculating theoretical yield formula is as follows:

Theoretical yield = [frac{Actual, Yield}{Percentage, Yield}] x 100

Annual Percentage Yield (APY)

The annual percentage return is readily calculated by the APY formula. The yearly interest rate and the number of compounding periods are used to calculate it. The formula for APY is as follows:

APY = [(1+ frac{r}{n})^{n} – 1]

Where, 

r = Annual Interest Rate

n = Number of Compounding Periods Each Year

Percentage Atom Economy Formula

Barry Trost of Stanford University (US) developed the Atom Economy idea, for which he earned the Presidential Green Chemistry Challenge Award in 1998. It’s a way of expressing how well a reaction uses the reactant atoms.

The amount of starting materials that end up as useful products is measured by the atom economy of a process.

The use of reactions with a high atom economy is important both for sustainable development and economic reasons.

The percentage atom economy formula is given as,

Atom economy =[frac{Mass,of, atom , in, desired, Product}{Mass,of, atom , in,Reactants}] x 100%

Solved Examples

Example.1. During a chemical reaction, 0.9 g of product is made. The maximum calculated yield is 1.8g. Calculate the percentage yield of this reaction by using percent yield formula chemistry?

Solution:

Substitute the values in the corresponding  percentage yield formula,

Percentage yield =[frac{Actual, Yield}{Theoretical, Yield}] x 100%

Percentage yield =[frac{0.9}{1.8}] x 100%

Percentage yield = 0.5 x 100%

Percentage yield =50%

Example.2. Determine the theoretical yield of the formation of geranyl formate from 465 g of geraniol. A chemist making geranyl formate uses 465 g of starting material and collects 419g of purified product. Percentage yield is given as 92.1%

Solution: 

The actual yield is 419 g which is the quantity of the desired product.

Percentage yield is 92.1%

Therefore, by using the theoretical yield formula chemistry is as,

Theoretical yield = [frac{Actual, Yield}{Percentage, Yield}] x 100

Theoretical yield = [frac{419}{92.1}] x 100

Theoretical yield = 4.545 x 100

Theoretical yield =454 g

Example.3. Find the APY on $1000 at the compound interest rate of 9%, compounded monthly.

Solution:

Using the APY formula

APY = [(1+ frac{r}{n})^{n} – 1]

APY = ( 1 + 0.09/12)12 – 1

APY = [(1+ frac{0.09}{12})^{12} – 1]

APY = 0.0938

 Hence the APY is 9.38%.

In trigonometry mathematics, every function has an inverse and arctan is the inverse of the tangent function. Arctan is also referred to as the tan^-1. Arctan x is used to find the angle. The tangent on the other hand is described as the ratio of the opposite side to the adjacent side of a particular angle of a right-angled triangle. Arctan formula is used to identify an angle.

What is the Arctan Formula?

A fundamental arctan formula is written as:

Other arctan formulas are as given below:

• arctan(x) = 2arctan (x/1+√1+x²)

• arctan(x) = ∫^x₀ 1/z²+1dz;|x|≤1

• ∫arctan(z) dz = z arctan(z) – 1/2 ln(1+z²) + C  

Arctangent formulas for π are as given below:

• π/4 = 4 arctan(1/5) – arctan(1/239)

• π/4 = arctan(1/2) + arctan(1/3)

• π/4 = 2 arctan(1/2) – arctan(1/7)

• π/4 = 2 arctan(1/3) + arctan(1/7)

• π/4 = 8 arctan(1/10) – 4 arctan(1/515) – arctan(1/239)

• π/4 = 3 arctan(1/4) + arctan(1/20) + arctan(1/1985)

• π/4 = 24 arctan(1/8) + 8 arctan(1/57) + 4 arctan(1/239)

Solved Examples Using Arctan Formula

The arctan formula can be thoroughly understood for use and application referring to solved examples below.

Example:

In the right-angled triangle PQR, the base of which measures 17 cm and the height is 9cm. Determine the base angle.

Solution:

To calculate: base angle

How: Using arctan formula

θ = arctan(opposite ÷ adjacent)

θ = arctan(9 ÷ 17)

= arctan(0.52)

θ = 27.47 degrees or 27⁰

Answer: The angle is 27⁰

 

Example:

Find out the value of θ, given that the base of the triangle ABC is 24 ft and the height is 11 ft

Solution:

arctanθ = opposite / adjacent

arctanθ = 11 ÷ 24 =0.24

arctanθ = 24.60

θ = 24⁰