# Advanced Volume on the ISEE

#### Middle and Upper Level

**ISEE Middle-Level Volume Question:**

The surface area of a cube is 150 centimeters. What is its volume?

A) 5 cm^{3}

B) 25 cm^{3}

C) 100 cm^{3}

D) 125 cm^{3}

**The solution:**

Step 1: Work backwards from the surface area to determine the length of each side of your cube. Since the cube has 6 faces, we can divide the surface area by 6 to get the area of each face. Find its square root to get the side length:

150 ÷ 6 = 25; √25 = 5

**Helpful tip:** Since you don’t get a calculator, the ISEE usually provides numbers that are easy to work with, such as perfect squares. If you find yourself doing a complicated calculation, double check your work before continuing.

Step 2: Find the volume by taking the side length, 5, to the third power. The answer is D.

5 × 5 × 5 = 125

**Helpful tip:** Don’t stop before you’re finished! The test is tricky and often offers partial solutions, like A and B, as answer choices.

**ISEE Upper-Level Volume Question:**

The height of the cylinder shown is 5 times its diameter. The formula used to find the volume of a cylinder is V = πr^{2}h or V = r^{2}hπ where r is the radius of the cylinder and h is the height of the cylinder. If the diameter of the cylinder is 4 in., what is its volume, in inches^{3}?

A) 80π

B) 100π

C) 160π

D) 320π

**Helpful tip:** If you see an unfamiliar figure on the test and you don’t know the formula, don’t panic! The ISEE often gives the formula for a less common figure within the problem. That said, it’s important to memorize common formulas, such as the volume of cubes and rectangular prisms.

Step 1: Use the information in the problem to figure out each variable in the formula:

r = half the diameter: 4 ÷ 2 = 2 in.

h = 5 times the diameter: 4 × 5 = 20 in.

**Helpful tip:** Don’t mix up diameter and radius! The test often gives you one of these measures when you need the other one to solve the problem.

Step 2: Plug each variable into the equation and solve: V = r^{2}hπ = 2^{2} x 20 x π = 80π. The correct answer is A.