Circle Sector and Segment

Circle Sector and Segment

Slices There are two main "slices" of a circle: ·         The "pizza" slice is called a Sector. ·         And the slice made by a chord is called a Segment.

SECTOR OF A CIRCLE

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

Half a circle is 
a Semicircle.

Quarter of a circle is
a Quadrant.

Area of a Sector

A circle has an angle of 2π and an Area of:		πr2
A Sector with an angle of θ (instead of 2π) has an Area of:		(θ/2π) × πr2
Which can be simplified to:		(θ/2) × r2

Area of Sector = ½ × θ × r²   (when θ is in radians)

Area of Sector = ½ × (θ × π/180) × r²   (when θ is in degrees)

Arc Length By the same reasoning, the arc length (of a Sector or Segment) is: L = θ × r   (when θ is in radians) L = (θ × π/180) × r   (when θ is in degrees)

Area of Segment The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here). There is a lengthy reason, but the result is a slight modification of the Sector formula:
Area of Segment = ½ × (θ - sin θ) × r2   (when θ is in radians) Area of Segment = ½ × ( (θ × π/180) - sin θ) × r2   (when θ is in degrees)

ARC LENGTH, PARAMETRIC CURVES

ARC LENGTH, PARAMETRIC CURVES

Arc Length, Parametric Curves

Parametric Curves. A parametric curve can be thought
of as the trajectory of a point that moves trough the plane with coordinates
(x, y) = (f(t), g(t)), where f(t) and g(t) are functions of the
parameter t. For each value of t we get a point of the curve. Example:
A parametric equation for a circle of radius 1 and center (0, 0) is:
x = cos t, y = sin t .
The equations x = f(t), y = g(t) are called parametric equations.

Given a parametric curve, sometimes we can eliminate t and obtain
an equivalent non-parametric equation for the same curve. For instance
t can be eliminated from x = cos t, y = sin t by using the trigonometric
relation cos2 t + sin2 t = 1, which yields the (non-parametric) equation
for a circle of radius 1 and center (0, 0):
x2 + y2 = 1 .

Example: Find a non-parametric equation for the following parametric
curve:
x = t2 − 2t, y = t + 1 .
Answer: We eliminate t by isolating it from the second equation:
t = (y − 1) ,
and plugging it in the first equation:
x = (y − 1)2 − 2(y − 1) .
i.e.:
x = y2 − 4y + 3 ,
which is a parabola with horizontal axis.

Transformative Technology in Education

HOW TO WRITE AN ESSAY OUTLINE?

Writing an essay can feel like driving a car in the dark; if you can’t see what’s ahead of you, then how do you know where to steer? Fortunately, creating an outline can make the essay writing process easier, no matter what your assigned word count is. Creating an outline will give you a clear plan from which to work and will also result in a logical essay with a strong foundation.

Begin at the End or Start in the Middle?

The experience of reading an essay should be smooth and logical, but the process of writing one is not always as orderly! When creating an outline, you may wish to try writing out your thoughts in the order that you plan to write them, but be advised that this method does not work for all writers. You may find that listing out your main ideas first helps you get a sense of what the bulk of your essay will look like. Or you may wish to think about how to construct your introduction and conclusion first and then fill in the middle with the path from A to B. Either way, your outline should eventually list out the basic structure of your essay with a short description of each paragraph to give yourself a clear plan.

Order, Details and Transitions

When constructing an essay outline, consider the order in which you’d like to present your paragraphs. In a narrative essay, you will typically go in chronological order, but a descriptive essay may jump around a bit more in time. Your outline is the place to figure out what you’d like to present first, next and last, so allow yourself enough time to move the pieces around until it feels right.

Once you have the order set, take a look at the main ideas of each paragraph and add at least one detail that you plan to include. For an expository or persuasive essay, this can be a fact or a piece of evidence that will further your argument. For a narrative essay, this detail might be a particular moment you want to focus in on during the paragraph. Finally, for a descriptive essay, this can be a sensory detail that you want to make sure really stands out to your reader.

Once you have your order and details listed out, you can consider including your ideas for how to transition from one paragraph to the next. Adding transitional phrases or sentences can be done in the last stage of writing your essay, after you have a first draft complete, but it can be helpful to jot down your transition thoughts as early as possible, so you don’t get stuck when you’re ready to transition from one paragraph to the next.

Introduction and Conclusions

When writing an essay outline, you may decide to hold off on thinking about your introduction and conclusion until you actually begin writing. It’s important not to forget about these important “bookends” altogether, though, so you may wish to think about how you’re going to introduce and close your essay when you plan your outline. You can introduce your essay in a number of different ways, from including an anecdote to asking a question or offering a vivid description. To close your essay, you’ll want to sum up your main points or ideas and avoid bringing in any new information.

Essay Outline Example

If you get stuck while creating your outline, try using the following template.

1. Introduction
   1. I will introduce my essay by …
2. Body Paragraph 1
   1. My main idea in this paragraph is …
   2. One important detail I want to include is …
   3. I will transition to the next paragraph by …
3. Body Paragraph 2
   1. My main idea in this paragraph is …
   2. One important detail I want to include is …
   3. I will transition to the next paragraph by …
4. Body Paragraph 3
   1. My main idea in this paragraph is …
   2. One important detail I want to include is …
   3. I will transition to the next paragraph by …
5. Conclusion
   1. I will sum up my essay by …

TYPES OF ESSAYS

TYPES OF ESSAYS

Putting your ideas into writing can be challenging, but the structure of an essay can help you organize and express your thoughts more clearly. An essay can vary in length, but it is typically a nonfiction piece of writing organized into paragraphs that includes a clear introduction, a body, and a conclusion. Essays are useful for communicating information, persuading an audience, sharing a personal story, or analyzing the various aspects of a piece of literature or art or a moment in history or politics. The ability to write a clear, structured essay is a skill students typically begin to build during their grade school and high school years. College courses will continue to build this skill so that students can write longer and more complicated essays and be better prepared to transfer these skills into professional settings or the pursuit of graduate degrees.

When writing an essay, it’s important to understand which type of essay format will work best for your topic. Read on for a brief description of some of the most common types of essays.

Expository Essays

One of the most common types of essay, the expository essay is used when the writer wants to clearly communicate information. An expository essay can be strictly informative, such as a research report, but expository essays can also include examples and analysis to support a larger point. When writing an expository essay, you can tailor your essay in several different ways. Two of the most common approaches are the “compare and contrast” essay and the “cause and effect” essay. The compare and contrast essay is structured to allow the writer to present information and examples and then point out the similarities and differences. This is a good way to provide information without inserting any opinion or argument. The cause and effect essay also relies on the presentation of information, but the information is typically structured to explain the root cause of a later effect. This type of essay is useful when providing background information on a historical topic and can even be appropriate for constructing lab reports in some science courses.

Descriptive Essays

While less common in high school and college, descriptive essays are frequently used as a way to begin thinking creatively in elementary and middle school classrooms. A descriptive essay uses creative language to bring the subject of the essay to life. These essays can focus on a person, a place, an animal, a moment in time — there are nearly limitless possibilities for a descriptive essay. Descriptive essays typically rely on figurative language to capture the reader’s interest and can be a great opportunity for students to apply some of the more interesting tools in their writing tool kits, including simile, metaphor and personification.

Narrative Essays

A narrative essay can focus on the writer’s personal experience or take a close look at someone else’s experience. These essays typically sound similar to a fictional story in that they have a clear beginning, middle and end and usually include some sort of main event that prompts new understanding or a change in the subject of the essay. A narrative style offers more room to bring in emotion and unique descriptive details than an expository essay, but it usually includes more of a “story” than a descriptive essay. Prompts for narrative essays often ask students to consider an important moment in their lives and then create an essay that walks the reader through that moment.

Argumentative (Persuasive) Essays

Argumentative essays are all about presenting an opinion to readers and backing it up with evidence to persuade them to agree with your position. Argumentative essays, also known as persuasive essays, can be used to discuss an opinion on a work of literature, art or film or to discuss a current event or moment in history. The main feature of this type of essay is a clearly articulated argument in the first paragraph of the essay. This is often called the “thesis statement” of the essay. The goal of an argumentative essay is to include enough evidence and explanation to prove that the argument in the opening paragraph is sound. Writers do this by providing evidence, explaining the source and credibility of the evidence, and even explaining why the counter-argument is not believable. These essays should have a clear conclusion that summarizes the argument and makes a final appeal to the reader to consider and accept the writer’s opinion.

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