3D shapes

The three-dimensional (“3D” or “3-d”) shapes you need to know for the GRE are cubes, rectangular solids, and right circular cylinders. Every 3D shape you see on the GRE will be one of these shapes or a combination of them. Even if a shape looks unfamiliar at first, you can usually break it into these basic building blocks.

It’s worth memorizing the formulas so you can apply them quickly. It’s also useful to understand where they come from, because composite shapes often require you to combine or adapt these ideas.

Cube

Definitions

Cube formulas

Volume: $s^3$

Surface area: $6s^2$

A cube is a 3D shape made entirely of squares. The base is a square, and the height is the same as the side length of that square. Because every face is a square, all edges have the same length $s$.

The volume of a cube is $s^3$. Here’s why: the area of the square base is $s\cdot s=s^2$, and the cube extends upward a height of $s$. Volume is base area times height, so $s^2\cdot s=s^3$.

The surface area of a cube is $6s^2$. A cube has $6$ faces, and each face is a square with area $s^2$. So the total surface area is $6\cdot s^2=6s^2$.

Let’s try a simple cube question:

What is the surface area of a cube that has a volume of 125?

Try it using the formulas above, then check your work.

We’re given the volume, so start by solving for the side length $s$.

$$\begin{array}{rl}V& =s^3\\ 125& =s^3\\ 5& =s\end{array}$$

Now plug $s=5$ into the surface area formula.

$$\begin{array}{rl}\text{SA}& =6s^2\\ & =6(5^2)\\ & =6(25)\\ & =150\end{array}$$

This is the same general process you’ll use whenever you’re given one measurement (like volume) and need to find another (like surface area).

Rectangular solid

Definitions

Rectangular solid formulas

Volume: $lwh$

Surface area: $2lw+2lh+2wh$

A rectangular solid has a rectangular base that extends upward to a height. A cube is a special case of a rectangular solid where all dimensions are equal: $l=w=h$. In general, though, a rectangular solid can have different values for length, width, and height.

The volume of a rectangular solid is $l\cdot w\cdot h$ (often written as $lwh$). The base area is $l\cdot w$, and multiplying by the height $h$ gives the volume.

A rectangular solid has six rectangular faces, in three matching pairs:

• The top and the bottom ($2\ast lw$)
• The front and the back ($2\ast lh$)
• The close side and the further side ($2\ast wh$)

Adding these areas gives the total surface area: $2lw+2lh+2wh$. The order of the terms doesn’t matter, as long as you include all three pairs.

Now try a fundamental rectangular solid question:

What is the surface area of a rectangular solid that has a length of 10, a width of 12, and a volume of 720?

Solve it, then check your work.

We’re given $V$, $l$, and $w$, so first solve for the missing dimension $h$.

$$\begin{array}{rl}V& =lwh\\ 720& =10\ast 12\ast h\\ 720& =120h\\ 6& =h\end{array}$$

Now plug $l=10$, $w=12$, and $h=6$ into the surface area formula.

$$\begin{array}{rl}\text{SA}& =2lw+2wh+2lh\\ & =2\ast 10\ast 12+2\ast 12\ast 6+2\ast 10\ast 6\\ & =240+144+120\\ & =504\end{array}$$

Right circular cylinder

Definitions

Right circular cylinder formulas

Volume: $\pi r^{2}h$

Surface area: $2\pi r^2+2\pi rh$

A right circular cylinder has a circular base that extends straight upward to a height $h$.

To find the volume, start with the area of the circular base. A circle with radius $r$ has area $\pi r^2$. Extending that base upward a height $h$ gives volume $\pi r^2\cdot h=\pi r^{2}h$.

For surface area, break the cylinder into its surfaces:

• A top circle (area $\pi r^2$)
• A bottom circle (area $\pi r^2$)
• A “wrapped” side surface

The side surface becomes a rectangle if you unwrap it. One side of that rectangle is the height $h$. The other side is the circumference (perimeter) of the base circle, which is $2\pi r$. So the side area is $(2\pi r)\cdot h=2\pi rh$.

Add the two circles and the side:

$$2\pi r^2+2\pi rh.$$

Now try a cylinder question:

Cylinders A and B have the same volume. Cylinder A has a height of 3 and Cylinder B has a radius of 3, and the radius of Cylinder A is equivalent to the height of Cylinder B. What is the surface area of Cylinder B?

Work through it before checking the answer and explanation.

The key idea is that the volumes are equal, so set the volume expressions equal to each other. Using subscripts helps keep the dimensions straight.

$$\begin{array}{rl}{V}_A& =V_B\\ \pi (r_A^2)(h_A)& =\pi (r_B^2)(h_B)\\ (r_A^2)(h_A)& =(r_B^2)(h_B)\\ (r_A^2)(3)& =(3^2)(h_B)\\ (h_B{)}^2(3)& =(3^2)(h_B)\\ (h_B)(3)& =3^2\\ h_B& =3\end{array}$$

Now you have Cylinder B’s dimensions: $r_B=3$ and $h_B=3$. Plug them into the surface area formula.

$$\begin{array}{rl}{\text{SA}}_B& =2\pi (r_B^2)+2\pi (r_B)(h_B)\\ & =2\pi (3^2)+2\pi (3)(3)\\ & =2\pi (9)+2\pi (9)\\ & =18\pi +18\pi \\ & =36\pi \end{array}$$

When you work with multiple similar variables, go step by step and keep the subscripts consistent so you don’t mix up which radius or height belongs to which cylinder.