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 \(2\sqrt{R^2-r^2}\)