# Do Pi and Geometric Shapes Coexist?

• MHB
• highmath
In summary, $\pi$ is a number that is seen in the domain of angles in geometry, but not in the domain of coordinates. It shows up when an angle or turn is involved. It is present in the area of an ellipse, and in the circumference of a curve like a parabola or hyperbola.
highmath
(1) Is there a pi in ellipse entity?Why not or yes?
(2) Is there a pi in polygons entities (e.g square)? not or yes?
(3) If there is pi in some geometries and other not - What is the reason to that?
(4) How cloud I know that are no hidden formula of pi in a square figure that the expression in formula, his value is: 0 in addition and 1 in multiplicatoin and etc

Last edited:
I really don't understand what you mean by this. "$\pi$" is a number. What do you mean by a number "in" in a geometric figure? If you mean formulas for circumference, area, etc. Then it is true that the area of an ellipse, with semi-axes of lengths a and b is $\pi ab$, very similar to the formula for the area of a circle. On the other hand, while the circumference of a circle is simply $2\pi r$, there is no simple formula for the area of an ellipse.

highmath said:
(1) Is there a pi in ellipse entity?Why not or yes?
(2) Is there a pi in polygons entities (e.g square)? not or yes?
(3) If there is pi in some geometries and other not - What is the reason to that?
(4) How cloud I know that are no hidden formula of pi in a square figure that the expression in formula, his value is: 0 in addition and 1 in multiplicatoin and etc
The Greeks, at least, defined $$\displaystyle \pi$$ as the circumference of a circle divided by its diameter. So $$\displaystyle \pi$$ does come up in a lot of geometric figures. But if you are trying to find some deeper reasoning to it you aren't going to find it. It's just a handy definition.

-Dan

highmath said:
(1) Is there a pi in ellipse entity?Why not or yes?
(2) Is there a pi in polygons entities (e.g square)? not or yes?
(3) If there is pi in some geometries and other not - What is the reason to that?
(4) How cloud I know that are no hidden formula of pi in a square figure that the expression in formula, his value is: 0 in addition and 1 in multiplicatoin and etc

We can roughly divide geometry in angles and coordinates.
We see $\pi$ in the domain of the angles and not in the domain of the coordinates.
In a simple right angle on the unit circle we have the simplest possible coordinates with $0$ and $1$.
Its angle is $\smash{\frac \pi 2}$.
Its hypotenuse is $\sqrt 2$, which is an algebraic number and not transcendental like $\pi$.
So $\pi$ comes along whenever an angle or turn is involved.
To find the area of a circle, we integrate over the angle so that we see $\pi$ back in the result.

An ellipse is a circle scaled in one direction by a factor, which scales its area by the same factor.
So yes, there is $\pi$ in the area of an ellipse.
The circumference of an ellipse is so complicated that an elliptic integral function was invented to describe it.
Still, there is probably a $\pi$ hidden in there somewhere.

The coordinates of a polygon and its sides are typically algebraic numbers.
Its angles contain $\pi$, but $\pi$ only shows up if we map the polygon into the domain of a circle, since that is how angles have been defined. If we would identify angles by their coordinate ratios (slopes) there would be no $\pi$.

Through projective geometry a circle is transformed, or rather is equivalent to an ellipse, a parabola, and a hyperbola. These are the so called conic sections.
We already saw $\pi$ in the area of an ellipse.
A parabola and a hyperbola don't have an area though as they are unbounded. If we bound a parabola with a line at an algebraic coordinate, we don't see $\pi$, but we see an algebraic number. A hyperbola bounded by a line gives an area that contains $e$ instead of $\pi$.
The arc length of a parabola also contains $e$, while the arc length of a hyperbola is too complicated to tell.

We see $\pi$ a lot in physics formulas as well, and usually as $2\pi$ or $4\pi^2$. What these formulas have in common is that they deal with a full turn. As we know, the arc length of a full turn is $2\pi r$. All those physics formulas would be simplified if we used $\tau$ for a full turn instead of $2\pi$.

## 1. What is Pi and how is it related to geometric shapes?

Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14. Pi is closely related to geometric shapes because it is used to calculate the measurements of circles, which are a type of geometric shape.

## 2. How is Pi used to find the area and circumference of a circle?

To find the area of a circle, you can use the formula A = πr², where r is the radius of the circle. To find the circumference, you can use the formula C = 2πr. These formulas use Pi to calculate the measurements of a circle, making it an essential component in geometry.

## 3. Can Pi be used to calculate the measurements of other geometric shapes?

Yes, Pi can be used to calculate the measurements of other geometric shapes, such as cylinders, cones, and spheres. These shapes have a circular base or cross-section, making Pi a necessary constant in their calculations.

## 4. Is there a limit to the number of decimal places in Pi?

As of now, Pi has been calculated to over 31 trillion decimal places. While there is no known limit to the number of decimal places in Pi, it is impossible to accurately measure it to an infinite number of digits due to the limitations of computing power.

## 5. How is Pi related to the concept of infinity?

Pi is often associated with the concept of infinity because it is an irrational number, meaning it has an infinite number of decimal places that do not repeat in a pattern. This makes it a never-ending number, similar to the concept of infinity.

Replies
1
Views
2K
Replies
6
Views
3K
Replies
2
Views
1K
Replies
29
Views
2K
Replies
1
Views
1K
Replies
6
Views
2K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
3
Views
3K
Replies
2
Views
2K