mathematics

Perfect rectangle

Perfect rectangle is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Perfect rectangle rather than just read about it. In short: A perfect rectangle is a rectangle that can be divided into squares of different sizes. If a perfect rectangle is specifically a square, it is analogously called a perfect square.

Key takeaways

Perfect rectangle belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Perfect rectangle to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Perfect rectangle from memory before moving on to harder problems.

Reference excerpt

A perfect rectangle is a rectangle that can be divided into squares of different sizes. If a perfect rectangle is specifically a square, it is analogously called a perfect square. A rectangle that is not perfect is also called an imperfect rectangle.

Discoverers of Perfect Rectangles (Selection) Many mathematicians have been involved in the discovery of perfect rectangles and perfect squares. Below is a selection of important discoveries in this field.

1925: Zbigniew Moroń decomposed a perfect smallest possible 33x32 rectangle into nine squares. 1939: The German mathematician Roland Sprague published a large perfect square with 55 squares. 1978: A. J. W. Duijvestijn dissected a perfect square into 21 squares with a total side length of 112, where 21 is the lowest possible number of subsquares of perfect squares.

Perfect Rectangles with Special Properties Among the numerous perfect rectangles and squares, the following selected examples are intended to highlight some special features. (The numbers in the squares indicate their respective side lengths.)

References

External links Perfect Rectangle Maths2Mind Perfect Rectangle Michael Holzapfel's Homepage "Did you know...?" (Perfect Rectangle) Math projects from the University of Giessen for elementary school students Perfect Rectangles Extensive collection of perfect rectangles on iread.it

Illustrations

Worked examples

Example 1 — a first encounter with Perfect rectangle

Start with the simplest possible case. Write down what Perfect rectangle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Perfect rectangle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Perfect rectangle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Perfect rectangle

In research: Perfect rectangle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Perfect rectangle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Perfect rectangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Perfect rectangle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Perfect rectangle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Perfect rectangle means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Perfect rectangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Perfect rectangle in simple terms?

A perfect rectangle is a rectangle that can be divided into squares of different sizes. If a perfect rectangle is specifically a square, it is analogously called a perfect square.

Why does Perfect rectangle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Perfect rectangle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Perfect rectangle.