If $A_0$ is the total area of the four corner pieces, and $A_1$ is the area of the square, they want $\frac{A_0}{A_1}$: that fraction is by definition the ratio of $A_0$ to $A_1$. Note also that for the square you have $w=h=2r$.
I can suggest some names, which I have learned from others who research these things.
- A 4-simplex, or 4D analogue of a triangle, is called a Pentachoron , describing a regular, 5-sided 4D polytope. Also called a 5-cell. These are the n-simplex.
There are 4 types of ring torus objects in 4D, which can be seen visualized here. The general name of hypertorus works well:
Spheritorus : sphere-bundle over the circle :$S^2$ x $S^1$ $$\left(\sqrt{x^2+y^2} -a\right)^2 +z^2+w^2 = b^2$$
Torisphere : circle-bundle over the sphere : $S^1$ x $S^2$
$$\left(\sqrt{x^2+y^2+z^2} -a\right)^2 +w^2 = b^2$$
3-torus : circle over circle over circle : $T^3$
$$\left(\sqrt{\left(\sqrt{x^2+y^2}-a\right)^2+z^2}-b\right)^2+w^2 = c^2$$
Tiger : circle-bundle over the flat 2-torus (Clifford torus)
$$\left(\sqrt{x^2+y^2} -a\right)^2 +\left(\sqrt{z^2+w^2} -b\right)^2 = c^2$$
As for the bar -> cylinder -> duocylinder, I'm not sure exactly what sequence you are using here. The best fit I can see is describing a specific bisecting rotation around an n-1 plane into n+1 dimensions. In this case, the next 5D shape is called a Cylspherinder , a cartesian product of a $D^2$ and $D^3$ (solid disk times solid sphere). But, you can also make a Spherinder (sphere prism, another type of 4D cylinder) from a rotation of a cylinder into 4D.
Cylinder: $\left|\sqrt{x^2+y^2} -z\right|+\left|\sqrt{x^2+y^2} +z\right| = a$
Duocylinder: $\left|\sqrt{x^2+y^2} -\sqrt{z^2+w^2}\right|+\left|\sqrt{x^2+y^2} +\sqrt{z^2+w^2}\right| = a$
Spherinder : $\left|\sqrt{x^2+y^2+z^2} -w\right|+\left|\sqrt{x^2+y^2+z^2} +w\right| = a$
Cylspherinder : $\left|\sqrt{x^2+y^2+z^2} -\sqrt{w^2+v^2}\right|+\left|\sqrt{x^2+y^2+z^2} +\sqrt{w^2+v^2}\right| = a$
I guess you can call these n-cylinders, but there are even more types of these than just product of n-balls and n-cubes. You can also include the product of n-ball and n-simplex as well. In fact, any shape with both flat and curved cells can fit into this group (product of 2-ball (and higher) with any genus-0 object)
Cyltrianglinder : $\left|\big||x|+2y\big|+|x| -2\sqrt{z^2+w^2}\right|+\left|\big||x|+2y\big|+|x| +2\sqrt{z^2+w^2}\right| = a$
Best Answer
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.
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).