Scope

The GRE often tests on the knowledge of the geometries of 3-D objects such cylinders, cones, cubes & spheres. The purpose of this document is to summarize some of the important ideas and formulae and act as a useful cheat sheet for such questions

Cube



A cube is the 3-D generalisation of a square, and is characterized by the length of the side, $a$. Important results include :

• Volume = $a^3$
• Surface Area = $6a^2$
• Diagnol Length = $\sqrt{3}a$

Cuboid



A cube is the 3-D generalisation of a rectangle, and is characterized by the length of its sides, $a,b,c$. Important results include :

• Volume = $abc$
• Surface Area = $2(ab+bc+ca)$
• Diagnol Length = $\sqrt{a^2+b^2+c^2}$

Cylinder



A cylinder is a 3-D object formed by rotating a rectangular sheet along one of its sides. It is characterized by the radius of the base, $r$, and the height, $h$. Important results include :

• Volume = $\pi r^2 h$
• Outer surface area w/o bases = $2 \pi r h$
• Outer surface area including bases = $2 \pi r (r+h)$

Cone



A cone is a 3-D object obtained by rotating a right angled triangle around one of its sides. It is charcterized by the radius of its base, $r$, and the height, $h$. The hypotenuse of the triangle formed by the height and the radius (running along the diagnol side of the cone), is known as it lateral height, $l=\sqrt{r^2+h^2}$. Important results include :

• Volume = $\frac{1}{3} \pi r^2 h$
• Outer surface area w/o base = $\pi r l =\pi r \sqrt{r^2+h^2}$
• Outer surface area including base = $\pi r (r+l)=\pi r (r+\sqrt{r^2+h^2})$

Sphere



A sphere is a 3-D generalisation of a circle. It is characterised by its radius, $r$. Important results include :

• Volume = $\frac{4}{3} \pi r^3$
• Surface Area= $4 \pi r^2$

A hemisphere is a sphere cut in half and is also characterised by its radius $r$. Important results include :

• Volume = $\frac{2}{3} \pi r^3$
• Surface Area w/o base = $2 \pi r^2$
• Surface Area with base = $3 \pi r^2$

Some simple configurations

These may appear in various forms on the GMAT, and are good practice to derive on one's own :

• Sphere inscribed in cube of side $a$ : Radius of sphere is $\frac{a}{2}$
• Cube inscribed in sphere of radius $r$ : Side of cube is $\frac{2r}{\sqrt{3}}$
• Cylinder inscribed in cube of side $a$ : Radius of cylinder is $\frac{a}{2}$; Height $a$
• Cone inscribed in cube of side $a$ : Radius of cone is $\frac{a}{2}$; Height $a$
• Cylinder of radius $r$ in sphere of radius $R$ ($R>r$) : Height of cylinder is $2\sqrt{R^2-r^2}$