What is a circle inside a square called?

Each of the four congruent shapes is called a minor segment of the circle. A segment is demarcated by a chord of a circle and the arc subtended by the chord. The chord divides the circle into the minor (smaller) and major (larger) segments, except when the chord is a diameter, in which case it divides the circle into two equal semicircles. So the semicircle is the special case of a segment.

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In this case, each segment is subtended by a central right angle, and its area is given by $A = \frac 12 r^2(\theta - \sin\theta) =\frac 12 r^2(\frac{\pi}{2} - 1)$.

(note that the angle measure is in radians).

The perimeter of each segment is $r(\frac{\pi}{2}+ \sqrt 2) $ (the former is the arc length term, the latter is the side of the inscribed square by the Pythagorean theorem).

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In Euclidean geometry, squaring the circle was a long-standing mathematical puzzle that was proved impossible in the 19th century. The term also has been used as a symbol in alchemy, particularly in the 17th century, and it has a metaphorical meaning: attempting anything that seems impossible.

According to mathematicians, "squaring the circle" means to construct for a given circle a square with the same area as the circle. The trick is to do so using only a compass and a straightedge. The devil is in the details:

First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side [square root of] A clearly has the same area. Secondly, we are not saying that [it] is impossible, since it is possible, but not under the restriction of using only a straightedge and compass.

A symbol of a circle within a square within a triangle within a larger circle began to be used in the 17th century to represent alchemy and the philosopher's stone, which is the ultimate goal of alchemy. The philosopher's stone, which was sought for centuries, was an imaginary substance that alchemists believed would change any base metal into silver or gold.

There are illustrations that include a squaring the circle design, such as one in Michael Maier’s book "Atalanta Fugiens," first published in 1617. Here a man is using a compass to draw a circle around a circle within a square within a triangle. Within the smaller circle are a man and a woman, the two halves of our nature that are supposedly brought together through alchemy.

Philosophically and spiritually, to square the circle means to see equally in four directions—up, down, in, and out—and to be whole, complete, and free.

Circles often represent the spiritual because they are infinite—they have no end. The square is often a symbol of the material because of the number of physical things that come in fours, such as four seasons, four directions, and the four physical elements—earth, air, fire, and water, according to ancient Greek philosopher Empedocles—not to mention its solid appearance.

The union of man and woman in alchemy is a merging of spiritual and physical natures. The triangle is then a symbol of the resulting union of body, mind, and soul.

In the 17th century, squaring the circle had not yet been proved impossible. However, it was a puzzle no one had been known to solve. Alchemy was viewed very similarly: It was something few if any had ever fully completed. The study of alchemy was as much about the journey as the goal, as no one might ever actually forge a philosopher’s stone.

The fact that no one was ever able to square the circle explains its use as a metaphor, meaning to attempt to complete a seemingly impossible task, such as finding world peace. It is different from the metaphor of attempting to fit a square peg into a round hole, which implies two things are inherently incompatible.

We've seen that when a square is inscribed in a circle, we can express all the properties of either the square or circle (area, perimeter, circumference, radius, side length) if we know just the length of the radius or the length of the square's side.

Now we'll see that the same is true when the circle is inscribed in the square.

Problem 1

A circle with radius ‘r’ is inscribed in a square. Find formulas for the square’s side length, diagonal length, perimeter and area, in terms of r.

What is a circle inside a square called?

Strategy

In solving the similar problem of a square is inscribed in a circle, the key insight was that the diagonal of the square is the diameter of the circle. 

Here, a similar key insight is that the circle's radius is equal to half the square's side length.

Since the circle is inscribed in the square, the square's side is tangent to the circle. By definition, the radius is perpendicular to the tangent line at the point of tangency.

The radius forms a 90° angle with one side of the square. Thus, it is parallel to the adjacent side of the square. This is true for the other radii, as well. That means that the quadrilateral formed by two radii and two adjacent sides of the square is also a square.

What is a circle inside a square called?

In this new small square, the side is equal to the radius. We can do this for each of the other 4 sides of the larger, outer square. In each one of them, the radii will form smaller squares. In the smaller squares, the sides are all equal to the radius.

The sides of two such adjacent small squares form the side of the larger outer square. Since each one of the smaller square's sides is equal to r, the side of the outer square is 2r.

Formal proof:

What is a circle inside a square called?

(1) OA ⊥ AB //Given, AB is tangent to circle O, a tangent line is perpendicular to the radius(2) m∠OAB=90° //Definition of perpendicular line(3) m∠ABC=90° //All interior angles of a square are right angles(4) OA||CB //Converse Consecutive Interior Angles Theorem(5) OC ⊥ CB //Given, CB is tangent to circle O, a tangent line is perpendicular to the radius(6) m∠OCB=90° //Definition of perpendicular line(7) m∠ABC=90° //All interior angles of a square are right angles

(8) OC||AB //Converse Consecutive Interior Angles Theorem

(9) OC=OA //All radii of a circle are equal(10) OCBA is a parallelogram //(4), (8), definition of a parallelogram

(11) BA=OC=r //opposite sides of a parallelogram are equal

And similarly,

(12) OEDA is a parallelogram (13) AD=OE=r //opposite sides of a parallelogram are equal

(14) |BD|=|BA|+|AD| = r + r = 2r

Ok, now that we have done that, the rest is very easy. The square's area is side length squared, so it is 4r2. The perimeter is 4 times the side length, so it is 8r. And the length of the diagonal can be calculated using the Pythagorean theorem, and it is 2r√2.

Now let’s do the converse, finding the inscribed circle’s properties from the length of the side of the square.

Problem 2

A circle is inscribed in a square, with a side measuring 'a'. Find formulas for the circle's radius, diameter, circumference and area , in terms of 'a'.

What is a circle inside a square called?

As we've shown above, the circle's radius is equal to the half the length of the square's side, so r=a/2. The diameter is twice the radius, so d=a. The circumference is d ·π, so C=πa. Finally, the area is π·r2, so A= π·a2/4.

These type of inscribed shape problems often have a component of finding the area between the shapes, which is irregular, so let's explore an example.

Problem 3

A circle is inscribed in a square, with a side measuring 10 units. Find the area of the shaded region:

What is a circle inside a square called?

Strategy

The key to solving these type of problems is to find the areas of the regular shapes. Then, express the shaded area as the difference between them. In this case, we can easily find the area of the circle and the square.

The difference between them is the 4 corner shapes formed by two edges of the square and an arc which is a quarter of the circle. From symmetry, all these shapes have equal area. So the shaded region is half the difference between the areas of the circle and the square.

Solution

Asquare=a·a=10·10=100
Acircle= π·a2/4=π·102/4=25π
Ashaded=(Asquare-Acircle)/2=(100-25π)/2=12.5(4-π)

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]