The prism, the pyramid and the round solids

A prism is a solid whose two bases are congruent polygons in parallel planes, and whose lateral faces are parallelograms.

• A lateral edge joins two matching vertices of the bases – $\displaystyle AA'$, $\displaystyle BB'$, $\displaystyle CC'$; all the lateral edges are parallel and equal.
• The height $\displaystyle h$ is the distance between the planes of the bases.
• In a right prism the lateral edges are perpendicular to the bases: the lateral faces are rectangles, and the height equals the lateral edge.
• A box is a right prism whose base is a rectangle, and a cube is a box whose edges are all equal.

The surface area is the lateral area (the lateral faces) plus the areas of the two bases.

A pyramid is a solid whose base is a polygon, and whose lateral faces are triangles meeting at one point – the apex $\displaystyle S$.

• The height is the perpendicular $\displaystyle SO$ from the apex to the base’s plane.
• In a right pyramid the base is a regular polygon, and the foot of the height $\displaystyle O$ is the centre of the base. Then all the lateral edges are equal, and the lateral faces are congruent isosceles triangles.
• The height of a lateral face, $\displaystyle SM$, goes down to the midpoint of a side of the base.

• A right cylinder: two round bases of radius $\displaystyle R$, height $\displaystyle h$. Its volume is the prism’s, $\displaystyle {V}={\pi R^2h}$; its lateral area is $\displaystyle 2\pi Rh$.
• A right cone: a round base, and the apex above its centre. The slant height $\displaystyle \ell$ joins the apex to the rim of the base: $\displaystyle {\ell^2}={R^2+h^2}$. Its volume is the pyramid’s, $\displaystyle {V}={\frac{\pi R^2h}{3}}$.
• A sphere of radius $\displaystyle R$: all its points are at most $\displaystyle R$ from the centre.

In a right pyramid $\displaystyle SABCD$ on a square base: the base’s side is 6, and the height $\displaystyle {SO}={4}$.
Find the surface area and the volume.

The solution below first finds the face’s height $\displaystyle SM$, from it the faces’ area, and then the volume.

• $\displaystyle {OM}={3}$, and in the triangle $\displaystyle SOM$, right-angled at $\displaystyle O$: $\displaystyle {SM}={\sqrt{4^2+3^2}}={5}$.
• The area of a face: $\displaystyle {\frac{6\cdot5}{2}}={15}$; the lateral area: $\displaystyle {4\cdot15}={60}$.
• The surface area: $\displaystyle {60+6^2}={96}$.
• The volume: $\displaystyle {V}={\frac{36\cdot4}{3}}={48}$.

The base’s radius of a right cone is 3, and its height is 4.
Find the lateral area and the volume.

• The slant height: $\displaystyle {\ell}={\sqrt{3^2+4^2}}={5}$.
• The lateral area: $\displaystyle {M}={\pi\cdot3\cdot5}={15\pi}$.
• The volume: $\displaystyle {V}={\frac{\pi\cdot3^2\cdot4}{3}}={12\pi}$.

The bases of a right prism are equilateral triangles of side 4, and its height is 10.
Find the prism’s volume, exactly. (The area of an equilateral triangle of side $\displaystyle a$ is $\displaystyle \frac{\sqrt{3}}{4}a^2$.)

In a right pyramid on a square base: the base’s side is 8, and the lateral edge is 9.

The slant height of a right cone is 10, and it makes an angle of $\displaystyle 60^{\circ}$ with the base’s plane.
Round to two decimal places.

The volume of a sphere is $\displaystyle 36\pi$.
Find the radius $\displaystyle R$.