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Everyone knows what a square is -- but they''re surprisingly rare in nature. In fact, one of the few examples of a perfect square in nature is the preferred form of sodium chloride crystals (common table salt). A few other minerals also have a cubic crystal form, including gold and pyrite. And although you might know a square when you see one, you might not be aware of some of its most interesting geometrical properties:

Although this might sound counterintuitive, a square is actually a special case of a rectangle. That''s right -- squares fall into this general classification because they have four right angles and pairs of parallel sides.

But as we all know, squares are much more special compared to rectangles because all of their sides are "congruent." In the geometrical world, this is another word for "equal."

Interestingly, squares can also be classified as rhombi. Remember that a rhombus is a parallelogram with four congruent sides. The difference between a regular rhombus and a square is that all the angles of a square are right angles. By comparison, a "normal" rhombus has a mixture of acute and obtuse angles.

We know that a rhombus is really a square if it has a right vertex angle. We know that a rectangle must be a square if it has two adjacent equal sides. A rhombus with all equal angles is automatically a square. A quadrilateral with four equal sides and four right angles must be a square.

We all know what a square looks like -- but let''s take a look at a diagram to help us visualize its attributes:

As we can see, all four of its angles are right. We can also see that all four of its sides are congruent. Therefore, it''s safe to say that we''re looking at a square!

In order to find the perimeter of a square, all we need to do is add up all of its four sides. Alternatively, we can simply multiply a *single* side by four, since we know that all four sides are the same. We can write this formula as:

P = 4s

In this formula, P = perimeter and s = the length of one side of the square.

But what about the area of a square? In order to find the area, we need to multiply its length by its height. But since we know that the length and the height are the same we can simply say that one side

A = s^2

There are many interesting properties of squares:

- All four internal angles must equal 360 degrees: Each angle of a square measures 90 degrees, and their sum is always 360 degrees.
- A square has a larger area than any other quadrilateral with the same perimeter: This property is based on the isoperimetric inequality, which states that among all shapes with the same perimeter, the one with the largest area is the one that forms a circle. In the case of quadrilaterals, a square is the closest to a circle in terms of its shape and thus has the largest area.
- A square can be inscribed (drawn) inside any regular polygon: A square can be placed inside any regular polygon such that all four vertices of the square touch the sides of the polygon.
- Diagonals are congruent and bisect each other: The two diagonals of a square are equal in length, and they intersect each other at their midpoints, dividing each other into two equal parts.
- Diagonals are perpendicular to each other: The diagonals of a square intersect at a 90-degree angle, forming four right angles at their intersection point.

Two Dimensional Views of Three Dimensional Objects

Common Core: High School - Geometry Flashcards

Common Core: High School - Geometry Diagnostic Tests

Basic Geometry Diagnostic Tests

Squares can be deceptively complex, especially when students start to venture into more complex geometrical problems. With help from a tutor, your student can gain a solid foothold on these foundational concepts, allowing them to move forward with greater confidence. Tutors can use examples that match your student''s hobbies and interests to make math more engaging. They can even help advanced students challenge themselves with new concepts. Varsity Tutors will match your student with a qualified tutor, so reach out today to get started.

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