ISEE Upper Level

Volume & Surface Area

Calculating volume and surface area of prisms, cylinders, cones, and spheres

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Imagine you have a giant cardboard box. If you want to paint the outside of the box to look like a Minecraft creeper, you need to know the Surface Area. But if you want to fill that same box to the brim with green slime, you need to know the Volume! 📦🟩

Surface Area is the total area of all the outside faces of a 3D shape. It's like wrapping a present with wrapping paper. Volume is how much space is inside the 3D shape. It's like pouring water into a fish tank. On the ISEE, you will see questions about these measurement concepts in both the Mathematics Achievement section (where you just solve the math) and the Quantitative Reasoning section (where you might compare two different shapes).

The best part about Volume is that it usually follows one simple rule for shapes like prisms and cylinders: just find the area of the bottom (the base) and multiply it by how tall it is (the height). Think of it like a stack of pancakes! 🥞 If you know how big one pancake is (the base area), you just multiply by the number of pancakes in the stack (the height) to find out exactly how much breakfast you have. Let's dive into some formulas and crush these ISEE shapes!

Key Formula

Volume of prisms and cylinders:

V = B \cdot h

. For cones and pyramids:

V = \frac{1}{3} \cdot B \cdot h

. Surface Area of a rectangular prism:

S A = 2 (l w + l h + w h)

. Surface Area of a cube:

S A = 6 s^{2}

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

The formula for the surface area of a cylinder is

S A = 2 π r^{2} + 2 π r h

, where

r

is the radius and

h

is the height. A cylinder has a radius of 4 cm and a height of 6 cm. What is the surface area of the cylinder?

48 π

^{2}

64 π

^{2}

80 π

^{2}

96 π

^{2}

Show Solution

Substitute $r = 4$ and $h = 6$ into the formula: $S A = 2 π (4)^{2} + 2 π (4) (6) = 2 π (16) + 2 π (24) = 32 π + 48 π = 80 π$ cm $^{2}$ .

Answer: C

Q2 Hard

The formula for the volume of a sphere is

V = \frac{4}{3} π r^{3}

. A sphere has a volume of

36 π

cubic inches. What is the radius of this sphere?

A 2 in.

B 3 in.

C 6 in.

D 9 in.

Show Solution

Set the volume formula equal to $36 π$ : $\frac{4}{3} π r^{3} = 36 π$ . Divide both sides by $π$ : $\frac{4}{3} r^{3} = 36$ . Multiply both sides by $\frac{3}{4}$ : $r^{3} = 36 \times \frac{3}{4} = 27$ . Take the cube root: $r = 327 = 3$ inches.

Answer: B

Q3 Hard

The formula for the volume of a cone is

V = \frac{1}{3} π r^{2} h

. Cone P has a radius of 3 cm and a height of 8 cm. Cone Q has a radius of 6 cm and the same height as Cone P. What is the ratio of the volume of Cone P to the volume of Cone Q?

\frac{1}{2}

\frac{1}{4}

\frac{1}{8}

\frac{3}{8}

Show Solution

Volume of Cone P: $V_{P} = \frac{1}{3} π (3)^{2} (8) = \frac{1}{3} π (9) (8) = 24 π$ . Volume of Cone Q: $V_{Q} = \frac{1}{3} π (6)^{2} (8) = \frac{1}{3} π (36) (8) = 96 π$ . The ratio is $\frac{V _{P}}{V _{Q}} = \frac{24 π}{96 π} = \frac{1}{4}$ . Note: since the heights are equal, only the radii differ. The ratio of volumes equals the ratio of $r^{2}$ values: $\frac{3 ^{2}}{6 ^{2}} = \frac{9}{36} = \frac{1}{4}$ .

Answer: B

Q4 Hard

A rectangular box (rectangular prism) has a length of 10 cm, a width of 4 cm, and a height of 5 cm. If each dimension is doubled, how many times greater is the volume of the new box than the volume of the original box?

A 2

B 4

C 6

D 8

Show Solution

Original volume: $V = 10 \times 4 \times 5 = 200$ cm $^{3}$ . New volume with doubled dimensions: $V = 20 \times 8 \times 10 = 1, 600$ cm $^{3}$ . Ratio: $\frac{1 , 600}{200} = 8$ . In general, if all three dimensions of a rectangular prism are multiplied by a factor of $k$ , the volume is multiplied by $k^{3}$ . Doubling each dimension gives $2^{3} = 8$ times the original volume.

Answer: D

Tips & Strategies

When comparing shapes with Pi ( $π$ ) on the Quantitative Reasoning section, remember that $π$ is just a number slightly bigger than 3. You can usually just multiply by 3 to estimate!
If you forget a formula, try to draw the shape. If it's a box or a tube, Volume is always the area of the bottom shape multiplied by how tall it is.
Pay attention to 'radius' vs 'diameter'. The ISEE loves to give you the diameter when the formula needs the radius. Always cut the diameter in half to get your radius!

Common Mistakes

Watch out for forgetting the $\frac{1}{3}$ in cone and pyramid volume formulas! Pointy shapes hold exactly one-third the amount of stuff as straight-up-and-down shapes with the same base.
Don't forget that on the ISEE there is NO penalty for guessing! If a volume question looks too scary, pick your favorite letter and move on. Never leave it blank!

Frequently Asked Questions

Do I need to memorize all these formulas for the ISEE?

Yes, mostly! The ISEE doesn't give you a formula sheet. But don't panic! Just remember the pancake rule: Area of the base times the height. That works for cubes, boxes, and cylinders!

What is a Quantitative Comparison question?

It's a special puzzle on the ISEE where you don't just solve one math problem, you solve two! You figure out the number for Column A, the number for Column B, and decide which one is bigger. If they match, pick C. If it's impossible to know, pick D.

How do I remember the difference between Volume and Surface Area?

Volume is what goes INSIDE (like filling a water balloon). Surface Area is the OUTSIDE (like the rubber of the balloon itself). Volume answers are always 'cubic' (like $c m^{3}$ ) and Surface Area answers are always 'square' (like $c m^{2}$ ).

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