Spheres and Spherical Geometry — Worked Examples

A spherical ball has radius $0.150\,\text{m}$. Determine its volume.

A sphere has total surface area $100\pi\,\text{m}^2$. Determine its radius.

A hollow spherical vessel has outer radius $0.500\,\text{m}$ and inner radius $0.450\,\text{m}$. Determine the geometric volume of shell material.

A spherical cap is cut from a sphere of radius $R=5.00\,\text{m}$. The cap height is $h=2.00\,\text{m}$. Determine the cap volume, curved spherical area, base-disk area, and total closed-cap area.

A cap is cut from a sphere of radius $R=8.00\,\text{m}$ and has height $h=3.00\,\text{m}$. Determine the radius $r$ of the circular cutting section.

A spherical segment of height $h=2.00\,\text{m}$ has circular base radii $a=4.00\,\text{m}$ and $b=3.00\,\text{m}$. Determine its volume.

A sphere has radius $R=10.0\,\text{m}$. Two parallel planes bound a spherical zone of height $h=3.00\,\text{m}$. Determine the zone area.

A spherical sector is associated with a zone of height $1.50\,\text{m}$ on a sphere of radius $4.00\,\text{m}$. Determine the sector volume.

Two great-circle planes on a sphere of radius $6.00\,\text{m}$ meet at angle $30.0^\circ$. Determine the area of the spherical lune and the volume of the corresponding wedge.

A spherical triangle lies on a sphere of radius $10.0\,\text{m}$ and has interior angles $100^\circ$, $110^\circ$, and $120^\circ$. Determine its area.

For a spherical Earth approximation with radius $R=6371\,\text{km}$, two points subtend a central angle of $12.0^\circ$. Estimate their great-circle arc distance.

A solid metal cylinder of radius $6.00\,\text{cm}$ and height $8.00\,\text{cm}$ is melted and recast into one sphere with no material loss. Determine the sphere radius.