Sphere

A sphere is a solid formed by rotating a semicircle 360 degrees around its diameter. It’s a perfectly symmetrical geometric shape.

The complete rotation of the semicircle creates the spherical surface.

Point C is the center of the sphere and is equidistant from every point on the sphere’s surface.

The radius of the semicircle becomes the radius of the sphere, which is the distance from the center to any point on the surface.

Similarly, you can think of a sphere as a solid of revolution formed by rotating a circle 180 degrees. The result is the same.

A sphere can also be defined as the set of all points in space (a geometric locus) that are at a distance less than or equal to a given point called the center (C).

A spherical surface, by contrast, is defined as the set of all points in space that lie at a distance from the center equal to the radius of the sphere.

The study of spheres dates back to ancient times. Greek mathematicians like Plato and Archimedes were among the first to explore their properties. Notably, Archimedes discovered key formulas for calculating the volume and surface area of spheres. Spheres have also held significant roles in philosophy and cosmology. To Greek philosophers, the sphere symbolized perfection, harmony, and the universe. This symbolism is also found in other cultures, where the sphere is often associated with concepts of completeness and infinity.

Formulas

Here are the key formulas for calculating a sphere's properties:

• Surface Area
  The surface area of a sphere is given by the formula $$ 4πr^2 $$ where r is the radius of the sphere.

  Note. The surface area of a sphere is four times the area of its largest cross-sectional circle. $$ 4 \cdot \pi r^2 $$
  
  It can also be shown that the surface area of a sphere is equal to the lateral surface area $ A_L $ of the cylinder that just contains it. $$ A_L = C \cdot h = 2 \pi r \cdot 2r = 4 \pi r^2 $$
  

• Volume
  The volume of a sphere is proportional to the cube of its radius, given by the formula: $$ \frac{4}{3}πr^3$$

Spherical Caps

A spherical cap is the portion of a sphere’s surface bounded by a secant plane that cuts through the sphere.

When a plane (π) intersects a sphere, it divides its surface into two distinct regions. Each of these regions is called a cap or spherical cap.

Each cap is defined by three key elements:

• The base: the circular cross-section created by the intersection of the plane and the sphere. This circle is centered at point C.
• The vertex (V): the point on the sphere that lies farthest from the plane, aligned with the center of the base along the sphere’s diameter.
• The height (h): the perpendicular distance from the vertex (V) to the center (C) of the base.

The diameter VV′ that passes through the base’s center (C) also serves as the axis of symmetry for the cap. This vertical axis bisects the cap into two mirror-image halves.

Special Cases

If the plane (π) passes through the center (O) of the sphere, the resulting caps are congruent hemispheres.

In this particular case, each cap comprises exactly half the surface area of the sphere, rather than just a curved segment.

If the plane (π) is tangent to the sphere, the cap degenerates into a single point V on the sphere’s surface.

Note. A spherical cap preserves many of the sphere’s geometric properties - for instance, its curvature and symmetry - while being defined by a distinct base and vertex.

Surface Area of a Spherical Cap

The surface area of a spherical cap can be calculated using the formula

$$ A= 2 \pi R h $$

where $ R $ is the radius of the sphere and $ h $ is the height of the cap.

Example

Consider a sphere with a radius of \( R = 5\,cm \). If we slice the sphere exactly in half, the resulting cap has a height of $ h = R = 5\,cm $.

Applying the formula, we find the surface area of the cap:

\[ A = 2 \pi R h = 2 \pi \cdot 5 \cdot 5 = 50 \pi \approx 157.08\,cm^2 \]

This is precisely half the total surface area of the sphere.

Proof. The total surface area of a sphere is given by: \[ A_{\text{total}} = 4 \pi R^2 \] This formula represents the entire surface area of a sphere of radius \( R \). If we cut the sphere through its center, we obtain a hemisphere, whose surface area is exactly half that of the full sphere: \[ A_{\text{hemisphere}} = 2 \pi R^2 \] Now, let's examine a special case: a spherical cap whose height \( h \) is exactly equal to the radius \( R \) of the sphere. In other words, the cap covers an entire hemisphere. Substituting \( h = R \) into the general formula for the surface area of a cap or zone, we get: \[ S = 2 \pi R h \] which, with \( h = R \), becomes: \[ S = 2 \pi R \cdot R = 2 \pi R^2 \] matching exactly the surface area of the hemisphere as derived earlier. This confirms that the formula \( S = 2 \pi R h \) remains valid even in limiting cases, such as a hemispherical cap.

Thus, the surface area of a spherical zone is equivalent to the lateral surface area of a cylinder with the same radius as the sphere and a height equal to that of the cap. The lateral surface area of a cylinder is likewise given by $ S_L = 2 \pi R h $

Spherical Zone

A spherical zone is the portion of a sphere's surface bounded between two parallel planes cutting through the sphere.

These planes intersect the sphere in two circles, known as the bases of the zone, dividing the sphere’s surface into three distinct regions.

The region lying between the two planes is called a spherical zone. It forms a curved, belt-like strip between two circular cuts across the sphere.

Note. To visualize it, imagine the sphere as a globe: drawing two different lines of latitude creates a ring-shaped area between them - that’s the spherical zone.

The diameter of the sphere passing through the line segment $ PQ $, which connects the centers $ P $ and $ Q $ of the two bases, is known as the axis of symmetry of the zone. The distance between the two centers is called the height of the zone and is denoted by \( h \).

Surface Area of a Spherical Zone

The surface area \( S \) of a spherical zone is found using the same formula as for a spherical cap:

\[ S = 2 \pi R h \]

where \( R \) is the radius of the sphere and \( h \) is the distance between the two planes.

Note. A spherical cap is simply a special case of a spherical zone, where one of the bounding planes is tangent to the sphere’s surface. Therefore, the same formula applies: in both cases, the surface area depends solely on the radius of the sphere and the height of the curved section.

Example

Suppose we have a sphere with a radius of \( R = 10\,cm \) and two parallel planes cutting through it, creating a spherical zone with a height of \( h = 4\,cm \).

The surface area of the spherical zone is approximately \( 251.33\,cm^2 \)

$$ S = 2 \pi R h = 2 \pi \cdot 10 \cdot 4 = 80 \pi \approx 251.33\,cm^2 $$

Note. The surface area of the spherical zone can also be compared to the lateral surface area of a cylinder with the same radius as the sphere and a height equal to that of the zone. The lateral surface area of a cylinder is likewise given by $ S_L = 2 \pi R h $

Spherical Wedge

A spherical wedge is a section of a sphere’s surface bounded by two half-planes passing through the sphere’s center.

The key components of a spherical wedge are:

• Dihedral angle: the solid angle (α) formed by the two half-planes.
• Equatorial arc: the arc of a great circle (analogous to the Earth's equator) contained within the wedge.
• Wedge sides: the semicircular arcs where the two half-planes intersect the sphere’s surface.

In simpler terms, a spherical wedge looks much like an orange slice, cut along its natural segments.

Surface Area of a Spherical Wedge

The surface area of a spherical wedge depends on the measure of its dihedral angle. It can be calculated in two ways:

• If the dihedral angle is measured in radians: \[ S_f = 2 \alpha_{\text{rad}} R^2 \]
• If the dihedral angle is measured in degrees: \[ S_f = \frac{\alpha^\circ}{90^\circ} \pi R^2 \]

Here, \( S_f \) represents the area of the wedge, \( R \) is the radius of the sphere, \( \alpha_{\text{rad}} \) is the dihedral angle in radians, and \( \alpha^\circ \) is the angle in degrees.

It’s important to note that the wedge’s area is directly proportional to the size of the dihedral angle (α) relative to the sphere’s entire surface area.

Proof. The total surface area of a sphere is given by: \[ A_{\text{total}} = 4\pi R^2 \] A spherical wedge represents a fraction of this area, proportional to its dihedral angle. When the angle is measured in radians, the proportion is between \( \alpha_{\text{rad}} \) and \( 2\pi \) (the measure of a full circle in radians): \[ A_{\text{wedge}} = 4\pi R^2 \cdot \frac{ \alpha_{\text{rad}} }{2 \pi} =2\alpha_{\text{rad}} R^2 \] When measured in degrees, the proportion is between \( \alpha^\circ \) and \( 360^\circ \): \[ A_{\text{wedge}} = 4\pi R^2 \cdot \frac{ \alpha^\circ }{360^\circ} = \frac{\alpha^\circ}{90^\circ} \pi R^2 \]

Example

Let's find the surface area of a spherical wedge cut from a sphere with a radius of \( R = 5 \, \text{cm} \), and a dihedral angle of \( 60^\circ \). Applying the formula:

\[ S_f = \frac{60^\circ}{90^\circ} \pi R^2 = \frac{2}{3} \pi (5)^2 \]

\[ S_f = \frac{2}{3} \pi \times 25 = \frac{50}{3} \pi \, \text{cm}^2 \]

Thus, the wedge’s surface area is approximately \( 52.36 \, \text{cm}^2 \).

Example 2

Consider a sphere with a radius of \( 10 \, \text{cm} \), and a wedge with a dihedral angle of \( \frac{\pi}{4} \, \text{radians} \). We calculate the surface area using the formula for radians:

\[ S_f = 2 \alpha_{\text{rad}} R^2 = 2 \times \frac{\pi}{4} \times 10^2 \]

\[ S_f = \frac{\pi}{2} \times 100 = 50\pi \, \text{cm}^2 \]

Therefore, the surface area of the wedge is approximately \( 157.08 \, \text{cm}^2 \).

Sphere in Analytic Geometry

In analytic geometry, a sphere is represented by an equation that defines all points in space at a constant distance from a fixed point, the center.

The standard equation of a sphere with its center at (x₀,y₀,z₀) and radius r is:

$$ (x−x_0)^2+(y−y_0)^2+(z−z_0)^2=r^2 $$

Where (x,y,z) are the coordinates of any point in space.

This equation is a three-dimensional extension of the circle equation in the Cartesian plane.

If the sphere is centered at the origin O(0,0,0), the equation simplifies to:

$$ x^2 + y^2 + z^2 = r^2 $$

Here’s an example of a sphere centered at the origin of the Cartesian plane with a unit radius.

The equation of a sphere can also be expressed in polar coordinates:

$$ \begin{cases} x = x_0 + r \sin \theta \cos \theta \\ \\ y = y_0 + r \sin \theta \sin \theta \\ \\ z = z_0 + r \cos \theta \end{cases} $$

Additional Notes

Here are a few more details about spheres:

• Sections
  Any intersection between a plane and a sphere forms a circle. If the plane passes through the center, the circle is at its largest.
  
• Chord
  A chord of a sphere is a line segment connecting any two distinct points on the surface.
• Diameter
  The diameter of a sphere is twice the radius. It’s a chord that passes through the center of the sphere. $$ d = 2r $$
• External, Tangent, and Secant Planes
  A plane α in relation to a sphere Σ with center C and radius r can be:
  • External if the distance between plane α and sphere Σ is greater than the radius r. The plane and the sphere do not intersect.
  • Tangent if the distance between plane α and sphere Σ is equal to the radius r.
  • Secant if the distance between plane α and sphere Σ is less than the radius r.
• Hypersphere (or n-Sphere)
  A generalization of the sphere to higher dimensions. A 4-sphere is the four-dimensional analogue of an ordinary sphere.
• Theorem of Tangent Segments to a Sphere from an External Point
  Tangent segments to a sphere drawn from an external point P are congruent: $ \overline{AP} \cong \overline{BP} \cong \overline{CP} \cong \overline{DP} $. All tangent segments to the sphere from the same point P form a cone with a circular base.
  

And so on.