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mathsai · 1 year

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A Comprehensive Guide to Areas Related to Circles:

Understanding the properties and calculations related to circles is fundamental in mathematics. From the ancient times to modern-day applications, circles have played a crucial role in various fields such as geometry, physics, engineering, and more. In this article, we will delve into the world of areas related to circles, exploring the concepts, methods, and real-world applications, while providing step-by-step examples to facilitate understanding.

Basic Concepts

Before we delve into the methods of calculating areas related to circles, let's review some fundamental concepts:

Radius and Diameter:

The radius is the distance from the center of a circle to any point on its edge, while the diameter is twice the radius.

Circumference:

The circumference of a circle is the distance around its edge. It is given by the formula: (C = 2\pi r), where (r) is the radius.

Area of a Circle:

The area of a circle is the space enclosed by its edge. It is calculated using the formula: (A = \pi r^2), where (r) is the radius.

Methods for Calculating Areas Related to Circles

There are several scenarios where circles interact with other geometric shapes, leading to the calculation of areas related to circles. Let's explore two common methods for solving these problems: sector area and segment area calculations.

Method 1: Sector Area Calculation

A sector of a circle is a region enclosed by two radii and the corresponding arc. To calculate the area of a sector, follow these steps:

Step 1:

Identify the radius (r) and the angle (θ) (in radians) subtended by the sector at the center of the circle.

Step 2:

Use the formula for the area of a sector: (A_{\text{sector}} = \frac{1}{2} r^2 θ).

Example: Calculating the Area of a Sector

Consider a circle with a radius of 8 cm, and a sector subtending an angle of (60^\circ) at the center. Let's calculate the area of the sector.

Step 1: Given (r = 8) cm and (θ = 60^\circ), convert (θ) to radians: (θ = \frac{60}{180} \pi = \frac{\pi}{3}).

Step 2: Use the formula (A_{\text{sector}} = \frac{1}{2} r^2 θ): [A_{\text{sector}} = \frac{1}{2} \times 8^2 \times \frac{\pi}{3} = \frac{64}{3} \pi \approx 67.03 \, \text{cm}^2].

Methods for Calculating Areas Related to Circles

A segment of a circle is the region enclosed by a chord and the arc it subtends. To calculate the area of a segment, use these steps:

Step 1:

Identify the radius (r), the angle (θ) (in radians), and the length of the chord.

Step 2:

Calculate the area of the corresponding sector using (A_{\text{sector}} = \frac{1}{2} r^2 θ).

Step 3:

Use the formula (A_{\text{segment}} = A_{\text{sector}} - \text{Area of Triangle}), where the triangle is formed by the two radii and the chord.

Example: Calculating the Area of a Segment

Consider a circle with a radius of 10 cm, and a chord of length 12 cm, subtending an angle of (120^\circ) at the center. Let's calculate the area of the segment.

Step 1:

Given (r = 10) cm, (θ = 120^\circ), and chord length (c = 12) cm. Convert (θ) to radians: (θ = \frac{120}{180} \pi = \frac{2}{3} \pi).

Step 2:

Calculate the area of the sector using (A_{\text{sector}} = \frac{1}{2} r^2 θ): [A_{\text{sector}} = \frac{1}{2} \times 10^2 \times \frac{2}{3} \pi = \frac{100}{3} \pi \approx 104.72 \, \text{cm}^2].

Step 3:

Calculate the area of the triangle using its base and height: [A_{\text{triangle}} = \frac{1}{2} \times c \times r = \frac{1}{2} \times 12 \times 10 = 60 \, \text{cm}^2].

Step 4:

Calculate the area of the segment using (A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}): [A_{\text{segment}} = \frac{100}{3} \pi - 60 = \frac{100}{3} \pi - \frac{180}{3} = \frac{100 - 180}{3} \pi = -\frac{80}{3} \pi \approx -83.77 \, \text{cm}^2].

Real-World Applications

Areas related to circles have numerous applications in various fields. Some examples include:

Construction:

Architects and engineers use circle-related calculations to design circular structures like domes, arches, and circular bridges.

Manufacturing:

Circular patterns and components are common in manufacturing industries. Calculations related to areas of circular sections are crucial for designing parts such as gears and bearings.

Landscaping:

Circular lawns, fountains, and other landscaping features often require calculations of circular areas.

Physics:

Circular motion calculations are crucial in physics, where understanding areas of circular sections helps analyze rotational motion, angular velocity, and more.

Art and Design:

Artists and designers often use circular patterns and shapes in their creations. Calculating areas related to these circles contributes to achieving desired aesthetics.

Areas related to circles hold a significant place in mathematics due to their practical applications and intricate geometric properties. In this article, we explored two methods for calculating areas of sectors and segments, providing step-by-step examples for each. These calculations find relevance in various real-world scenarios, underscoring the importance of mastering these concepts. From construction to physics, circular calculations are essential tools that enable us to understand and shape the world around us. So whether you're a student aiming to ace your maths exams or a professional looking to apply mathematical principles in your field, maths.ai is a valuable asset that opens doors to a myriad of possibilities.

#mathsai #math struggles #ai #mathsguide #mathematics

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sanguinarysanguinity · 5 years

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Navigatio Britannica: Planar Trigonometry

Chapter 4: Of Trigonometry, Sections I - V

The first half of this chapter is Planar Trigonometry, which I learned in high school and have used on-and-off ever since; the second half of this chapter is Spherical Trigonometry, which I know nothing about. Consequently, I’m dividing this chapter into two parts -- before we let John Barrow attempt to teach me spherical trig (wish me luck!), I want to do a fast recap of what he has to say about planar trig...

Section I: Definitions

Everything is defined geometrically, on the unit circle, via a diagram that I have yet to find in this scan. (Also, even if I do find the page, I don’t have much hope that it will have been scanned correctly, since Google’s scanning machine can turn pages but not unfold them.) Happily, Barrow is pretty good about describing his figures in enough detail that I can reconstruct them as I go, which is the only reason I was able to understand anything in this chapter.

Not many surprises here, although I did learn that cosine, cotangent, and cosecant are the sine, tangent, and secant of the complementary angle, and likewise that the tangent of an angle is called the tangent because its physical instantiation lives on a line tangent to the circle. Also, Barrow defines the Verse-Sine, which was new to me: geometrically, it is the part of the radius that isn’t the cosine. (Algebraically, it is 1 - cosine.) Wikipedia says the versine was important to navigation, so I assume it will come up later.

Section II: Geometrical Constructions of the Tables of Sines, Tangents, Secants, &c.

In which we are instructed to build ourselves a unit circle, mark it off in 1-degree intervals, and construct ourselves a... well, it’s gonna look a bit like a ruler, but it’s going to measure 1 to 90 degrees, on several parallel scales: chords, sines, versed-sines, tangents, etc. To make this thing, you use your compass and measure the length of a chord for a 10-degree angle, then mark it on your chords-scale, and label it “10 deg.” Repeat for the other 89 degrees, and ta-dah, you have a chords-scale! Then do it again for sines, versed-sines, tangents, and so on. When we get to actually solving trig problems, how to use this scale is one of the three standard methods that Barrow is going to teach us.

Section III: Arithmetical Constructions of the Tables of Sines, &c.

First off, Barrow reassures us this is going to be easy-peasy, no need to panic -- which is our first cue that panicking will be required before we’re done.

But true to his word, Barrow starts out easy, using similar triangles to prove all the basic trigonometric identities: tan = sin/cos, sec = 1/cos, etc. All well and good, except it’s all done in proportions and nothing is called out by name, only by referring to various line-segments in his nowhere-to-be-found unit circle diagram. All of which makes it difficult to absorb at a glance, but once you finish decoding everything this is basically just SOHCAHTOA.

Then he proves two variants on the standard trigonometric sum/difference formulas (although he expresses them as proportions and via verbal descriptions, talking about the means of equi-different angles and the differences between them):

cos x = (1/2) (sin y + sin (y + 2x)) / sin (y + x)

sin x = (1/2) (sin (y + 2x) - sin y) / cos (y + x)

You can verify those via the standard trigonometric sum/difference identities if you want. (I did.) But they’re also pretty straightforward geometrically, if you take the time to very carefully reconstruct what his diagram must have been: in the end, it’s all just similar triangles. He then proves several corollaries -- which in hindsight are simple enough (just straightforward algebraic manipulations, multiplying everything by two, or both sides by the denominator), but sadly, I lost MANY HOURS to a rash of typos in them.

Then.

Oh, then.

All hell breaks loose as he endeavours to prove that a semi-circle has an arc-length of pi. I admit to not following this bit: I haven’t seen Newton’s notation for calculus since I was seventeen, when that one weirdo physics professor used it in lectures, and I didn’t really feel like re-teaching it to myself for this. Nor did I really want to get into re-teaching myself binomial expansions. Also, the type-face on all the fractions in the expansions was super-squinchy to read, and you know what, fuck it, I think it’s well-established that a semi-circle has an arc-length of pi, let’s move on.

The point of establishing that a semi-circle has an arc-length of pi is so that we can calculate the arc-length of one minute (simply divide pi by 10,800 minutes, easy-peasy), which we will then use as an approximation of the sine of one minute. ... Which, okay, I suppose if your angle is small enough and your applications are practical enough you can get away with that? But it makes the mathematician in me cry, I’m just saying. (Even as I admit that you really can get away with it for most purposes: according to my handy-dandy TI-84 Plus, pi/10800 differs from sin(1′) in the ninth significant digit. But Dr. Roberts and Dr. Chrestenson would never have let me get away with that shit, never mind that I also have an engineering degree and thus should be okay with this kind of ruthless practicality. In my soul there is a mathematician and an engineer battling to the death over questions like these, you simply don’t know how much shit like this wounds me.)

Anyway, once I finally got over my fit of vapours...

Now that we have an approximation for the sine of one minute, we can calculate the cosine of one minute via the pythagorean trigonometric identity, and then...

And now I want to cry again, because now we get to build our table of sines (and along with it, our table of cosines), minute by freaking goddamn minute, by using the above equations like so:

2 cos (1′) sin (1′) - sin (0′) = sin (2′)

2 cos (2′) sin (2′) - sin (1′) = sin (3′)

2 cos (3′) sin (3′) - sin (2′) = sin (4′)

...

Continue until I cry blood and the seas boil dry.

(At one point Barrow admits that it’s possible to build this table in 5-minute increments and interpolate the intervening minutes when you need them. While this reduces the task to 1/5th of the original, I still want to hug and rock myself and cry.)

Happily, I don’t need to cry, because Barrow includes these tables in the book? But someone cried blood to make those tables, and John Barrow wants us all to know it.

Section IV: Actual Trigonometry Problems, At Long Last!

A ton of sample problems, all worked three ways:

Geometrically: Basically, use a compass and straight edge and your scale-thingie of chords/sines/secants that you made earlier, and draw a triangle of the correct proportions. Then just read/measure your answer right off the actual triangle in question, ta-dah! No abstract math required, just pretty pictures!

Arithmetically: What you learned in high-school, using the tables that someone cried blood over but without calculators (although you can use logs if you want to skip ahead to chapter five for them!) God, it looks miserable and grindingly awful, and I admit I don’t have the strength of character to follow any of these calculations through to the end.

Magic, I mean, Gunter’s Scale: The instructions here are amazingly low-key -- use your chart-dividers (what tumblr calls a pointy-leg-man what it likes to make walk on tippy-toes across charts) to measure off an interval on one scale, and then drop that same interval across the appropriate second scale, and voila! You have found your answer!

Of course I wanted to know what this magical tool is!

Apparently it was a slide rule without the slidey parts -- you used your dividers to accomplish what the slidey bit does on a slide rule -- but with some extra scales especially chosen for the convenience of navigators.

Apparently these things were so common among navigators that they were simply called Gunters, and I WANT ONE SO BAD. Here’s a nifty article about them, complete with pictures, and did I say? I WANT ONE SO BAD. I collect old-school mathematical tools and I WANT ONE SO BAD.

Ahem.

Anyway, Section V is more trig homework, except now we’re no longer dealing with right triangles. I admit it, I skimmed this like fuck.

And ta-dah! That’s Planar Trigonomometry, according to John Barrow in Navigatio Britannica, or, A Complete Guide to Navigation, pub. 1750!

Up next: Section VI - Spherical Trigonometry, what makes William Bush cry. Do I have the fortitude to teach myself spherical trig? PLACE YOUR BETS NOW.

#navigatio britannica #john barrow #age of sail #trigonometry #long post

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entrancei0012 · 3 years

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class 9 maths

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One will be brought to the note of areas in geometrical shapes. The examples of the median can be used in various examples of this in class 9 maths of chapter.

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ss1986us · 5 years

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ncert solutions for class 9 maths

Brace up like scholars

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While going through the maths subject Maths formula can help you a lot so do read and revise entire maths formula from entrancei.

As general pattern, the class 9 maths should be taken very seriously as it helps in building strong concepts for further boards in 10th. The complete categorization of the syllabus of class 9 maths has been done in chapters, which helps students to easily go through it.

Ø Chapter 1: Number systems

Ø Chapter 2: Polynomials

Ø Chapter 3: Coordinate Geometry

This chapters in class 9 maths help the students in understanding the concepts of Cartesian place and coordinates.

Ø Chapter 4: Linear Equations in two Variables

Ø Chapter 5: Introduction to Euclid’s Geometry

Ø Chapter 6: Lines and Angles

The chapter as lines and angles is generally asked as theorems and postulates in class 9 maths examinations. The level of difficulty is very minor compared to other units.

Ø Chapter 7: Triangles

Ø Chapter 8: Quadrilaterals

This chapter of class 9 maths comprises of only one theorem as proof. Questions asked in this chapter are relatively very easy.

Ø Chapter 9: Areas of Parallelograms and Triangles

One will be brought to the note of areas in geometrical shapes. The examples of the median can be used in various examples of this in class 9 maths of chapter.

Ø Chapter 10: Circles

Ø Chapter 11: Constructions

This chapter throws light upon bisecting and creating line segments to angles and lines. The student’s class 9 maths would be learning the creation of triangles with different properties and different base angles.

Ø Chapter 12: Heron’s formula

This chapter of class 9 maths comprises of only three exercises. This chapter gives an extension to the application of heron’s formula in polygons and quadrilaterals and triangles.

Ø Chapter 13: Surfaces Areas and Volumes

Ø Chapter 14: Statistics

Ø Chapter 15: Probability

With only about 1 exercise in class 9 maths, this chapter is very easy. This is based on observation and frequency of day to day examples.

Why choose Entrancei

#https://www.entrancei.com/topics-topics-class-9-mathematics

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udemy-gift-coupon-blog · 5 years

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Learn Trigonometry and It's Application+Hundreds of Examples ##FreeCourse #ApplicationHundreds #Examples #Learn #Trigonometry Learn Trigonometry and It's Application+Hundreds of Examples In a sense, trigonometry sits at the center of high school mathematics. It originates in the study of geometry when we investigate the ratios of sides in similar right triangles, or when we look at the relationship between a chord of a circle and its arc. It leads to a much deeper study of periodic functions, and of the so-called transcendental functions, which cannot be described using finite algebraic processes. It also has many applications to physics, astronomy, and other branches of science. It is a very old subject. Many of the geometric results that we now state in trigonometric terms were given a purely geometric exposition by Euclid. Ptolemy, an early astronomer, began to go beyond Euclid, using th~ geometry of the time to construct what we now call tables of values of trigonometric functions. Trigonometry is an important introduction to calculus, where one studies what mathematicians call analytic properties of functions. One of the goals of this book is to prepare you for a course in calculus by directing your attention away from particular values of a function to a study of the function as an object in itself. This way of thinking is useful not just in calculus, but in many mathematical situations. So trigonometry is a part of pre-calculus, and is related to other pre-calculus topics, such as exponential and logarithmic functions, and complex numbers. The interaction of these topics with trigonometry opens a whole new landscape of mathematical results. But each of these results is also important in its own right, without being "pre-" anything. We have tried to explain the beautiful results of trigonometry as simply and systematically as possible. In many cases we have found that simple problems have connections with profound and advanced ideas. Sometimes we have indicated these connections. In other cases we have left them for you to discover as you learn more about mathematics. Triangle is basically a geometric structure of any existing phenomenon having three sides and three angles. Application of triangle has vast area of interest in everyday life. The description of this course is very practical and emerging. All the concepts are described logically to make the well understanding. The sequence of all videos are categorized in step by step learning. After the basics concept of triangle I have described the complete description of trigonometry. I have tried to put each contents related to trigonometry in this course. I hope that the students will enjoy this course Who this course is for: Students who have problems in understanding of triangle, their types and how to solve triangle? 👉 Activate Udemy Coupon 👈 Free Tutorials Udemy Review Real Discount Udemy Free Courses Udemy Coupon Udemy Francais Coupon Udemy gratuit Coursera and Edx ELearningFree Course Free Online Training Udemy Udemy Free Coupons Udemy Free Discount Coupons Udemy Online Course Udemy Online Training 100% FREE Udemy Discount Coupons https://www.couponudemy.com/blog/learn-trigonometry-and-its-applicationhundreds-of-examples/

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