How do you find the surface area of a sphere using spherical coordinates?
On the surface of the sphere, ρ = a, so the coordinates are just the two angles φ and θ. The area element dS is most easily found using the volume element: dV = ρ2 sin φ dρ dφ dθ = dS · dρ = area · thickness so that dividing by the thickness dρ and setting ρ = a, we get (9) dS = a2 sin φ dφ dθ.
How do you find the area of a part of a sphere?
So the area of the section of the sphere really is 2πrh. This is the same as the surface area of a cylinder of the same radius as the sphere and height h. We can check this with our special cases: if h=r, then this formula gives 2πr2, and if h=2r, we get 4πr2.
How do you rotate in spherical coordinates?
To plot a dot from its spherical coordinates (r, θ, φ), where θ is inclination, move r units from the origin in the zenith direction, rotate by θ about the origin towards the azimuth reference direction, and rotate by φ about the zenith in the proper direction.
What is the area of the sphere?
4πr²
Surface Area of Sphere = 4πr², where r is the radius of sphere. A sphere is a solid figure bounded by a curved surface such that every point on the surface is the same distance from the centre.
Is PHI always between 0 and pi?
Note the subtle change: ϕ is from 0 to 2π and θ is from 0 to 1π. If you plug this in to the grapher, you find that what you get resembles a sphere. However, when you integrate p2sin(ϕ) over p from 0 to 1, ϕ from 0 to 2π, and θ from 0 to π, you get 0.
Why is the area of a sphere?
It is also called lateral surface area. Surface Area of Sphere = 4πr², where r is the radius of sphere. A sphere is a solid figure bounded by a curved surface such that every point on the surface is the same distance from the centre.
What is the area and volume of a sphere?
There are four main formulas for a sphere which include sphere diameter formula, sphere surface area, and sphere volume….Formulas of a Sphere.
| Sphere Formulas | |
|---|---|
| Surface Area of a Sphere | A = 4 π r2 |
| Volume of a Sphere | V = (4 ⁄ 3) π r3 |
What is theta and phi?
Phi Angle, Theta Angle The phi angle (φ) is the angle from the positive y-axis to the vector’s orthogonal projection onto the yz plane. The angle is positive toward the positive z-axis. The phi angle is between 0 and 360 degrees. The theta angle (θ) is the angle from the x-axis to the vector itself.