[Maths Class Notes] on Areas Related to Circles Pdf for Exam

An Overview on the Areas Related to Circles Class 10

Mathematics involves the study of various interesting concepts, such as geometry, integers, number system, circles etc. The study of areas related to circles is one such engaging mathematical concept. 

Circles are a circular figures without any edges. According to geometry, these round-shaped figures can be of three or four types. 

You can seek areas related to circles NCERT solutions to score good grades.

Let’s start learning to understand area related to circle class 10 NCERT!

What are Areas Related to Circles?

The areas related to circles represent the number of squares within a circle’s space. If a circle’s every square has an area approximately 1cm2, you need to count all the squares to calculate its area. Geometrically, the area enclosing a circle with radius r equals to πr2

Tip: Study all formula of area related to circle to solve problems like a pro!

Exercises on Area Related to Circles Class 10 Solutions

Read the following questions with areas related to circles solutions to score better!

1. What will be the circumference and area of a circle given radius 8 cm? (Sums related to class 10 chapter 12 maths)

Solution: Circumference or perimeter of a circle = 2πr

                                                                               = 2 * 22/7 * 8 = 50.286 cm (approx.)

Area of the circle will be πr2 = 22/7 * 8 * 8 cm2 = 201.143 cm2 (approx.)

2. Suppose, two circles have a radius of 20 cm and 10 cm, respectively. Find the radius of the third circle having a circumference equal to the sum of both circle’s perimeters. (Problem: area related to circle class 10 NCERT)

Solution: Here, we know about the radii of both circles. 

From area related to circle all formula, use perimeter’s formula C = 2* π * r

Radius of 1st circle = 20 cm, and radius of 2nd circle = 10 cm.

Assume, the radius of the 3rd circle to be r. 

Now, perimeter of 1st circle = 2* π* 20 = 40 π

Circumference of 2nd one = 2* π* 10 = 20 π

Given, 3rd circle’s circumference = perimeter of 1st and 2nd circle.

Radius will be 2 * π * r     = (40 + 20) π 

r = 60 π /2 π = 30 cm

3. A car has wheels with a diameter of 70cm each. How many revolutions can each wheel finish in 10 minutes, when the car is running at a speed of 60 km/hour? (Problem – area related to circle class 10 questions with solutions)

Solution: We know that the car wheel’s diameter = 70 cm, and its radius = 35 cm.

Distance travelled in one revolution = wheel’s circumference. 

Therefore, Perimeter = 2πr = 2*π*35 =70 cm

The car’s speed is 60 km/hour = (60 *100000)/60 cm/min = 1,00,000 cm/min. If the distance covered in 10 minutes, then = 1,00,000*10 = 10,00,000 cm

Let, n = no. of complete revolutions, 

If n*distance covered in 1 resolution = distance covered in 10 minutes

Then, n = (10,00,000*7) / (70*22) = 4545.45 (approx.)

So, every wheel will make 4545.45 complete revolutions.

Often, while studying mathematics, pupils face trouble with cumbersome topics. It happens due to lack of proper subject knowledge. For reducing such a crisis, you can take help from areas related to circles class 10 NCERT solutions. Moreover, try seeking area related to circle class 10 extra questions with solutions.

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[Maths Class Notes] on Asymptotes Pdf for Exam

The asymptote of a curve is an important topic in the subject of Mathematics. It is part of analytic geometry. In simple words, asymptotes are in use to convey the behavior and tendencies of curves. When the graph comes close to the vertical asymptote, it curves upward/downward very steeply. This way, even the steep curve almost resembles a straight line. It helps to determine the asymptotes of a function and is an essential step in sketching its graph. We also analyze how to find asymptotes of a curve. The detailed study of asymptotes of functions forms a crucial part of asymptotic analysis. 

Definition of Asymptote

An asymptote of a curve is the line formed by the movement of the curve and the line moving continuously towards zero. This can happen when either the x-axis (horizontal axis) or y-axis (vertical axis) tends to infinity. In other words, an Asymptote is a line that a curve approaches (without a meeting) as it moves towards infinity.

As you can see from the above illustrations, an asymptote of a curve is a line to which the curve converges. There is a peculiar and unique relationship between the curve and its asymptote. They run parallel to each other, but they never meet each other, at any point in infinity. They run very close to each other but are still apart.

The Application of an Asymptote in Real Life 

Asymptotes have several applications, such as:

  • They are in use for significant O notations.

  • They are simple approximations for complex equations.

  • They are useful for graphing rational equations.

  • They are relevant for Algebra: Rational functions and Calculus: Limits of functions.

A value that you grow closer to but never quite achieve is called an asymptote. A horizontal, vertical, or slanted line that a graph approaches but never touches is known as an asymptote in mathematics.

Assume you’re returning to your automobile, which is parked at the far corner of the mall parking lot. You’re fatigued after a long day of shopping. You’re halfway to your car after about a minute of walking, but you notice you’re slowing down. You’ve only walked a quarter of the way to your car after another minute.

 

Your legs are becoming very heavy. You’re only 1/8 of the way thereafter the third minute. The pattern persists. It appears as if you would never get there as your shopping-tired body lags even further. Each additional minute only gets you halfway to the car compared to where you were before.

Well, here’s the kicker: In principle, if this trend continues, you will never get there. Yes, you’re getting closer to your car by the minute, but the distances you’re covering are shrinking. With time, you get incredibly close. You get so near to your car that you can feel it. You get so near that your steps are as little as a pinhead, but you never cross more than half of the remaining distance to the car.

 

In actuality, this circumstance, known as Zeno’s dilemma, is a bit silly: no matter how weary you are, you eventually walk more than half the distance to your car. You just have one step left to travel, so you simply take it and arrive; yet, mathematicians frequently prefer to consider it in terms of theory.

Types of Asymptotes

As you may have noticed in Fig.1 and Fig. 2 above, sometimes a graph or the bend of the curve gets close enough to a line without ever touching it. This line is called an asymptote. Now, it is essential to know that an asymptote can be horizontal, vertical, or oblique/slanted.

Asymptotes are usually straight lines unless stated otherwise. You can even call an asymptote a value that you get closer to but never reach. In maths, as mentioned earlier, asymptotes can be horizontal, vertical, or an oblique/slanted line that a graph approaches, but never touches. Take a look at the illustration depicted in Fig.3 below to have a better understanding of the different types of asymptote (s).

 

As you have seen, there are three types of curves – horizontal, vertical, and oblique.  It is important to note that the directions can also be negative. The curve can take an approach from any side, such as from above or below for a horizontal asymptote. Sometimes, and many times, a curve may even cross over, and move away and back again.

  

In fig.4a, you can find two horizontal asymptotes, in fig.4b, there are two vertical asymptotes, and in fig.4c you can note that there are two oblique asymptotes.  So, these figures explain the character of the curve and the lines (asymptotes) that run parallel to the curve.

How to Find Asymptotes of a Curve 

The asymptote (s) of a curve can be obtained by taking the limit of a value where the function does not get a definition or is not defined. An example would be infty∞ and -infty −∞ or the point where the denominator of a rational function is zero.

Now you know that the curves walk alongside the asymptotes but never overtake them. The method in use to find horizontal asymptote changes- is based on how the degree of the polynomials in the numerator and the denominator of the functions get a comparison. If the polynomials are equal in degree, you can divide the coefficients of the largest degree values.

The vital point to note is that the distance between the curve and the asymptote tends to be zero when it moves from (+) positive infinity to (-) negative infinity.

Essential Characteristics of Asymptotes

In calculus, based on the orientation, curves of the form y = f(x) can be calculated using limits and can be any of the three forms

  • Horizontal Asymptotes – x goes to +infinity or –infinity, the curve approaches some constant value b. In curves in the graph of a function y = ƒ(x), horizontal asymptotes are flat lines parallel to the x-axis that the graph of the function approaches as x moves closer towards +∞ or −∞. 

  • Vertical Asymptote – when x approaches any constant value c, parallel to the y-axis, then the curve goes towards +infinity or – infinity.

  • Oblique Asymptote – when x goes to +infinity or –infinity, then the curve goes towards a line y = mx + b.

What are Asymptotes and How can I Find Them?

The equation for an asymptote is x = a, y = a, or y = axe + b because it is a horizontal, vertical, or slanting line. The
rules for finding all forms of asymptotes of a function y = f are as follows (x).

A horizontal asymptote has the form y = k, where x or x – is a positive or negative number. lim x f(x) and lim x – f(x) are the values of one or both of the limits (x). Click here to learn how to discover the horizontal asymptote using tricks and shortcuts.

A vertical asymptote has the form x = k, where y or y – is a positive or negative number. A slant asymptote has the form y = mx + b, where m is less than zero. An oblique asymptote is another term for a slant asymptote. It is commonly found in rational functions, and mx + b is the quotient obtained by dividing the numerator by the denominator of the rational function. In the next sections, we’ll look at the process of locating each of these asymptotes in greater depth.

Finding a Rational Function’s Horizontal Asymptotes

  • The degrees of the polynomials in the numerator and denominator of the function are used to find the horizontal asymptote.

  • Divide the coefficients of the leading phrases if both polynomials have the same degree. It will give you the value of the asymptote. 

  • If the numerator’s degree is smaller than the denominator, the asymptote is found at y = 0. (which is the x-axis).

  • There is no horizontal asymptote if the numerator’s degree is bigger than the denominator. 

Finding a Rational Function’s Vertical Asymptotes

To locate the vertical asymptote of a rational function, reduce it to its simplest form, set the denominator to zero, then solve for x values.

Examples of Asymptotes

In the question, you will have to follow some steps to recognise the different types of asymptotes. 

1. Find the domain and all asymptotes of the following function: 

Y= x² +3x +1

4x² – 9 

Solution = 4x² – 9 = 0 (take denominator as zero) 

x²= 9/4 = 3/2 

Y = x²/ 4x²  

= 1/4    

Domain x ≠ 3/2 or -3/2, Vertical asymptote is x = 3/2, -3/2, Horizontal asymptote is y = 1/4, and Oblique/Slant asymptote = none 

2. Find horizontal asymptote for f(x) = x/x²+3.

Solution= f(x) = x/x²+3. As you can see, the degree of numerator is less than the denominator, hence, horizontal asymptote is at y= 0 

Fun Facts About Asymptotes 

1. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at y= 0. 

2. If the degree of the denominator is less than the degree of the numerator by one, we get oblique asymptote

3. If the degree of the numerator is equal to the degree of the denominator, horizontal asymptote at a ratio of leading coefficients.

[Maths Class Notes] on Beta Function Pdf for Exam

In Mathematics, the two most popular functions are Beta and Gamma Function. Beta is a two-variable function, while Gamma is a single variable function. And the relation between the Beta Function and the Gamma Function will help solve many Physics and Mathematics problems. The Beta Function is a one-of-a-kind function, often known as the first type of Euler’s integrals. β is the notation used to represent it. The Beta Function is represented by (p, q), where p and q are both real values.

It clarifies the relationship between the inputs and outputs. The Beta Function tightly associates each input value with one output value. Many Mathematical processes rely heavily on the Beta Function.

Functions are a very important part of Mathematics. A function acts as the link between a set of input and output values, such that if you pass a certain input value through a given function, it will always yield one specific output. Therefore, a function is a special correlation between two data sets. Now, we can have some special types of functions. These functions can act as solutions for integral and differential equations. One such set of functions is Euler’s Integral Functions. This group consists of two types, namely Gamma and Beta Function. In this article, we are going to discuss the Beta Function, its definition, properties, the Beta Function formula, and some problems based on this topic.

 

Mathematical Functions can be represented in different ways, such as – in the form of an algorithm or formula that shows how to calculate an output for a given value or in the form of a graph or an image.

 

There is a term known as special functions. These are the specific Mathematical functions having special notations as well as established names because of their importance in various branches of Physics and Mathematics.

Few special functions appear as integrals of differential equation solutions at times. Some commonly studied special functions are step function, absolute value function, floor function, triangle wave function, error function, Bessel’s function, Riemannian zeta function, Euler integral function, and more.

Definition of Beta Function

We would first like to define the Beta Function before we proceed with the properties and problems. A Beta Function is a special kind of function which we classify as the first kind of Euler’s integrals. The function has real number domains. We express this function as B(x,y) where x and y are real and greater than 0. The Beta Function is also symmetric, which means B(x, y) = B(y ,x). The notation used for the Beta Function is “β”. The Beta Function in calculus forms an association between the input and output sets in integral equations and many more Mathematical operations.

The Beta Function is a one-of-a-kind function, often known as the first type of Euler’s integrals. “β” is the notation used to represent it. The Beta Function is represented by (p, q), where p and q are both real values.

It clarifies the relationship between the inputs and outputs. The Beta Function tightly associates each input value with one output value. Many Mathematical processes rely heavily on the Beta Function.

Different forms of special functions have become vital tools for scientists and engineers in many fields of applied Mathematics. The classical Beta Function β(α, β) is undoubtedly one of the most fundamental special functions due to its essential significance in a variety of fields such as Mathematics, Physics, statistics, and engineering. Various writers have produced numerous fascinating and valuable extensions of various special functions like the Gamma and Beta Functions, the Gauss hypergeometric function, and so on in the last few decades.

Many complex integrals in Calculus can be simplified to formulations involving the Beta Function. Because of its close relationship to the Gamma Function, which is an extension of the factorial function, the Beta Function is essential in calculus. Because of the following feature, the Gamma Function is related to the Factorial Function.

 Γ (n+1)=nΓn

The only snag is that n must be a positive (+) integer.

 

Using the Beta Function to Integrate

When evaluating integrals in terms of the Gamma Function, the Beta Function comes in handy. We demonstrate the evaluation of various distinct forms of integrals that would otherwise be inaccessible to us in this article.

Beta Function Formula

The Beta Function formula is as follows:

Here, p and q are greater than 0 and real numbers.

 

The Beta Function plays a very important role in calculus as it has a very close relationship with the Gamma Function. The Gamma Function itself is a general expression of the factorial function in Mathematics. The application of the beta-Gamma Function lies in the simplification of many complex integral functions into simple integrals containing the Beta Function.

 

Relationship Between Beta and Gamma Functions

The beta-Gamma Function relationship is as follows:

B(p,q)=(Tp.Tq)/T(p+q)

 

Here, the Gamma Function formula is:

The Beta Function can also find expression as the factorial formula given below:

B(p,q)=(p−1)!(q−1)!(p−1)!(q−1)!/(p+q−1)!

 

Here, p! = p. (p-1). (p-2)… 3. 2. 1

 

These relationships formed by the beta-Gamma Function are extremely crucial in solving integrals and Beta Function problems.

 

Beta Function Properties

The following are some useful Beta Function properties that one should keep in mind:

  • The Beta Function is symmetric which means the order of its parameters does not change the outcome of the operation. In other words, B(p,q)=B(q,p).

  • B(p, q+1) = B(p, q). q/(p+q)q/(p+q).

  • B(p+1, q) = B(p, q). p/(p+q)p/(p+q).

  • B (p, q). B (p+q, 1-q) = π/ p sin (πq).

Incomplete Beta Function

The incomplete Beta Function is basically the formula expressed in a generalized form. We show it by the following relation:

B (z: a, b) =

 

The notation for the same is B(a,b). When we put z = 1, we obtain our normal Beta Function. Therefore, B(1:a,b) = B(a, b).

 

The incomplete Beta Function finds application in Physics, calculus, Mathematical analysis and many other domains.

 

Beta Function Applications

The Beta Function finds implementation in many areas of science and Mathematics. For instance, in string theory, which is a part of complex Physics, the function computes and represents the scattering amplitudes of the Regge trajectories. The bet
a-Gamma Function duo also has numerous applications in calculus.

 

Now, the basic concepts are clear, we will look at Beta Function examples and Beta Function problems with solutions.

[Maths Class Notes] on Bisection Method Pdf for Exam

In Mathematics, the bisection method is used to find the root of a polynomial function. For further processing, it bisects the interval and then selects a sub-interval in which the root must lie and the solution is iteratively reached by narrowing down the values after guessing, which encloses the actual solution. In this method, the interval distance between the initial values is treated as a line segment. It then successively divides the interval in half and replaces one endpoint with the midpoint so that the root is bracketed. This method is based on The Intermediate Value Theorem and is very simple robust and easy to implement. There are various names given to this method such as “ the interval halving method”, “the binary search method”, “the dichotomy method”, and “Bolzano’s Method”.

Finding Root by Bisection Method

As stated above, the Bisection method program is a root-finding method that we often come across while dealing with numerical analysis. It is also known as the Bisection Method Numerical Analysis. It is based on Bolzano’s theorem for continuous functions. So let us understand what Bolzano’s theorem says.

Theorem (Bolzano): If on an interval a,b and f(a)·f(b) < 0, a function f(x) is found to be continuous, then there exists a value c such that c ∈ (a, b) or which f(c) = 0. The graph given below shows a continuous function.

The bisection method problems can be solved by using the bisection method formula to find the value c of the function f(x) that crosses the x-axis. In this case, the value c is an approximate value of the root of the function f(x). In this bisection method program, the value of the tolerance we set for the algorithm determines the value of c where it gets to the real root. One such bisection method is explained below.

Bisection Method Procedure

To solve bisection method problems, given below is the step-by-step explanation of the working of the bisection method algorithm for a given function f(x):

Step 1:

Choose two values, a and b such that f(a) > 0 and f(b) < 0 .

Step 2:

Calculate a midpoint c as the arithmetic mean between a and b such that c = (a + b) / 2. This is called interval halving.

Step 3:

Evaluate the function f for the value of c.

Step 4:

The root of the function is found only if the value of f(c) = 0.

Step 5:

If (c) ≠ 0, then we need to check the sign:

  1. we replace a with c if f(c) has the same sign as f(a) and we keep the same value for b

  2. we replace b with c if f(c) has the same sign as f(b), and we keep the same value for a

To get the right value with the new value of a or b, we go back to step 2 And recalculate c. 

Advantages of Bisection Method

  • Guaranteed convergence. The bracketing approach is known as the bisection method, and it is always convergent.

  • Errors can be managed. Increasing the number of iterations in the bisection method always results in a more accurate root.

  • Doesn’t demand complicated calculations. There are no complicated calculations required when using the bisection method. To use the bisection method, we only need to take the average of two values.

  • Error bound is guaranteed. There is a guaranteed error bound in this technique, and it reduces with each repetition. Each cycle reduces the error bound by 12 per cent.

  • The bisection method is simple and straightforward to programme on a computer.

  • In the case of several roots, the bisection procedure is quick.

Disadvantages of Bisection Method

  • Although the Bisection method’s convergence is guaranteed, it is often slow.

  • Choosing a guess that is close to the root may necessitate numerous iterations to converge.

  • Some equations’ roots cannot be found. Because there are no bracketing values, like f(x) = x².

  • Its rate of convergence is linear.

  • It is incapable of determining complex roots.

  • If the guess interval contains discontinuities, it cannot be used.

  • It cannot be applied over an interval where the function returns values of the same sign.

Bisection Method Problems

The best way of understanding how the algorithm works are by looking at a bisection method example and solving it by using the bisection method formula.

Example 1:  Find the root of f(x) = 10 − x².

Solution:

The calculation of the value is described below in the table:

i

a

b

c

f(a)

f(b)

f(c)

0

−2

5

1.5

6

−15

7.75

1

1.5

5

3.25

7.75

−15

−0.5625

2

1.5

3.25

2.375

7.75

−0.5625

4.359375

3

2.375

3.25

2.8125

4.359375

−0.5625

2.0898438

4

2.8125

3.25

3.03125

2.0898438

−0.5625

0.8115234

5

3.03125

3.25

3.140625

0.8115234

−0.5625

0.1364746

6

3.140625

3.25

3.1953125

0.1364746

−0.5625

−0.210022

7

3.140625

3.1953125

3.1679688

0.1364746

−0.210022

−0.036026

8

3.140625

3.1679688

3.1542969

0.1364746

−0.036026

0.0504112

9

3.1542969

3.1679688

3.1611328

0.0504112

−0.036026

0.0072393

At initialization (i = 0), we choose a = −2 and b = 5. After evaluation of the function in both points, we find that f(a) is positive while f(b) is negative. This means that between these points, the plot of the function will cross the x-axis at a particular point, which is the root we need to find.

The value of f(c) after 9 iterations is less than our defined tolerance (0.0072393 < 0.01). This means that the value that approximates best the root of the function f is the last value of c = 3.1611328.

By solving our quadratic equations in a classic way, we can check our result:

10 − x2 =0

x² = 10

x = √10

x = ±3.16227766

The article discusses all the important points related to the bisection method such as its advantages, disadvantages and solved problems etc. Step by step bisection method procedure is also given in the article.

[Maths Class Notes] on Calculus Maths App Pdf for Exam

All About Calculus Application 

Maths is a subject of practical information. Concepts of Mathematics are used widely in practical terms to understand its nature. As it is used widely, hence requires deep learning of the concept. To understand it even better, an app to learn calculus is introduced which guides students further conveniently and easily.

Calculus is a branch of Mathematics which deals in the study of rates of change. Before calculus, all Maths was static. Earlier it could only calculate the objects that were still. Seeing this, it is practically not possible as the universe is never constant. All objects from stars in space to cells in the human body are never at rest. These all are constantly moving. Calculus helps in determining how particles, stars, and matter move and change in real life.

Calculus is not only confined to Mathematics, but it is also used widely in other fields as well

Some of the known fields where calculus concepts are used are:

  1. Physics

  2. Engineering

  3. Economics

  4. Statistics

  5. Medicine

Calculus is used in other disparate areas as well including:

  1. Space travel

  2. Determining how medications interact with the human body

  3. How to build safer structures

Using calculus, scientists, astronomers, mathematicians, and chemists could chart the orbit of planets and stars and also the path of electrons and protons at the atomic level.

Calculus apps like provide the best solution to learn calculus. Here Calculus formulas like integral formula, limits, and derivative formula are explained in an easy manner. Calculus help app provides assistance to understand calculus in a fun and exciting way and also ensures that the context is easy to understand. 

Types of Calculus

Basic calculus is the study of differentiation and integration. In basic calculus, there are two branches of calculus namely –

  1. Differential calculus

  2. Integral calculus

Differential calculus studies the rate of change of quantity and also examines the rate of change of slopes and curves. This part is concerned with continuous change and its application. Here are many topics to shield on in differential calculus. These are as follows:

  1. Limits: It is a degree of closeness to any value or the approaching term.

  2. Derivatives: It is an instantaneous rate of change of quantity with respect to the other.

  3. Continuity and differentiability: Any function is always continuous if it is differentiable at any point.

  4. Chain rule: To find the derivative of the composition of a function, the chain rule is applied.

  5. Quotient rule: To find the differentiation of a function, the quotient rule is used.

Integral calculus finds the quantity where the rate of change is known. It basically focuses on concepts as slopes of tangent lines and velocities and concerns with space under the curve. It resembles the reverse of differentiation. It is the study of integrals and their property. Below listed are other topics covered in integral calculus:

  1. Integration: It is simply defined as the reciprocal of differentiation.

  2. Definite Integral: In a definite integral, the upper limit and lower limit of the independent variable of a function is specified.

  3. Indefinite integral: Here, it is not confined in a specified boundary and hence the integration value is always accompanied by a constant value.

Application of Calculus 

Calculus is not only confined to Mathematics subjects, but it has many practical applications in the outside world. Here are some of the concepts that use calculus including:

  1. Motion

  2. Heat

  3. Electricity

  4. Harmonics

  5. Astronomy

  6. Acoustics

  7. Photography

  8. AI

  9. Robotics

  10. Video Games

  11. Movies

  12. Predict birth and death rate

  13. Study of gravity

  14. Planetary motion

  15. Bridge engineering

Talking about economics, Calculus is even used in economics to determine the price elasticity of demand. Calculus allows us to determine points on changing supply and demand curves.

Advanced Calculus

Advanced calculus includes topics such as advanced series and power series and it covers the basic calculus topics such as differentiation, derivatives, and so on. The important areas to be covered are vector spaces, matrices, linear transformation. It helps us to understand the knowledge on few concepts such as:

  1. Vector fields

  2. Multilinear algebra

  3. Continuous differentiability

  4. Integration of forms

  5. Quadratic forms

  6. Tangent space

  7. Normal space via gradients

  8. Critical point analysis 

[Maths Class Notes] on Centroid of a Trapezoid Pdf for Exam

In this article, students will be able to learn about the topic of the centroid of a trapezoid. We will also look at the centroid of the trapezoid formula. But before we learn how to find the centroid of a trapezoid, students need to focus on the basics and start from the beginning.

The first thing that one needs to learn is the definition of a trapezoid. A trapezoid can be defined as a quadrilateral in which there are two parallel sides. A trapezoid is also known as a trapezium. So, if you see trapezium written in some other book, then don’t be confused. It means the same thing as a trapezoid.

A trapezoid can also be defined as a four-sided figure that is closed. It also covers some areas and has its perimeter. We will learn the formula for both area and perimeter of a trapezoid at a later point in this article.

It should be noted that a trapezoid is a two-dimensional figure and not a three-dimensional figure. The sides that are parallel to one another are known as the bases of the trapezoid. On the other hand, the sides that are not parallel to each other are known as lateral sides or legs. The distance between the two parallel sides is also known as the altitude.

Some readers might find it interesting to learn that there is also a disagreement over the exact definition of a trapezoid. There are different schools of mathematics that take up different definitions.

According to one of those schools of mathematics, a trapezoid can only have one pair of parallel sides. Another school of mathematics dictates that a trapezoid can have more than one pair of parallel sides.

This means that if we consider the first school of thought to be true, then a parallelogram is not a trapezoid. But according to the second school of thought, a parallelogram is a trapezoid. There are also different types of trapezoids. And those different types of trapezoids are:

A right trapezoid contains a pair of right angles. We have also attached an image of a right trapezoid below.

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In an isosceles trapezoid, the non-parallel sides of the legs of the trapezoid are equal in length. An image depicting an isosceles trapezoid is attached below.

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A scalene trapezoid is a figure in which neither the sides nor the angles of the trapezium are equal. For your better understanding, an image of a scalene trapezoid is attached below.

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The Formula for Area and Perimeter of a Trapezoid

Now, let’s look at the formula for calculating the area and perimeter of a trapezoid. According to experts, the area of a trapezoid can be calculated by taking the average of the two bases and multiplying the answer with the value for the altitude. This means that the formula for the area of a trapezoid can also be depicted by:

Area = ½(a + b) x h

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Moving on to the formula for the perimeter of a trapezoid, it can be described as the simple sum of all the sides. This means that if a trapezoid has four sides like a, b, c, and d, then the formula for the perimeter of a trapezoid can be represented by:

Perimeter = a + b + c + d.

The Properties of a Trapezoid

There are various important properties of a trapezoid. We have discussed those properties in the list that is mentioned below.

  • The diagonals and base angles of an isosceles trapezoid are equal in length.

  • If a median is drawn on a trapezoid, then the median will be parallel to the bases. And the length will also be the average of the length of the bases.

  • The intersection point of the diagonals is collinear to the midpoints of the two opposite sides.

  • If there is a trapezoid that has sides, including a, b, c, and d, and diagonals p and q, then the following equation stands true.

p2 + q2 = c2 + d2 + 2ab

In the next section, we will look at the centroid of a trapezoid formula.

The Formula for Centroid of a Trapezoid

In this section, we will look at the trapezoid centroid and the centroid formula for the trapezoid. As you must already know, a trapezoid is a quadrilateral that has two sides parallel. The centroid, as the name indicates, lies at the centre of a trapezoid. This means that for any trapezoid that has parallel sides a and b, the trapezoid centroid formula is:

X = {b + 2a / 3 (a + b)} x h

In this formula, h is the height of the trapezoid. Also, a and b are the lengths of the parallel sides.

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