Thirty-Six

Unit 6 · GeometryG4 · Area & Volume · Aerodynamics & Navigation

Volume, Surface Area & Scale

Solve prisms, cylinders, cones, spheres, displacement, and scale-factor problems.

  • 30 min
  • Unlocks: Payload Bay
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

volume
the three-dimensional space inside a solid

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Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

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The big idea

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The decision that matters

Match the formula to the solid and cube a length scale factor for volume. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Match the formula to the solid and cube a length scale factor for volume. Keep exact values through the setup; round only when the stem asks.
Solid formulas and scale factors
Vprism=BhVcylinder=πr2hVcone=13πr2hVsphere=43πr3SArectangular prism=2(lw+lh+wh)SAsphere=4πr2\begin{aligned} V_{prism}&=Bh & V_{cylinder}&=\pi r^2h\\ V_{cone}&=\frac13\pi r^2h & V_{sphere}&=\frac43\pi r^3\\ SA_{rectangular\ prism}&=2(lw+lh+wh) & SA_{sphere}&=4\pi r^2 \end{aligned}

If every length is multiplied by kk, surface area is multiplied by k2k^2 and volume by k3k^3. Identify whether the question asks for covering (square units) or capacity (cubic units) before choosing a formula.

Water displacement is another volume measurement. If a solid is submerged in a tank with constant base area BB, then Vsolid=B(hafter−hbefore)V_{solid}=B(h_{after}-h_{before}). For example, a tank with base area 30 square inches that rises from 8 inches to 10 inches displaces 30(10−8)=6030(10-8)=60 cubic inches.

For a sphere of radius 3, the exact volume is 43π(3)3=36π\frac43\pi(3)^3=36\pi cubic units and the surface area is 4π(3)2=36π4\pi(3)^2=36\pi square units. The numerical coefficients happen to match here, so the units and the requested measurement still matter.

Algebra or Desmos?

Keep π\pi symbolic when the choices are exact. Desmos is useful when the question asks for a decimal volume or compares several formulas, but selecting the solid and the correct squared or cubed dimension comes first.

Explore in Desmos

Compare a cylinder and cone with the same base

A cylinder and cone each have radius 4 and height 11. Evaluate both expressions, record their decimal volumes, and verify that the cone's result is exactly one third of the cylinder's.

The calculator is live: change anything and see what happens.

Compare a cylinder and cone with the same base

Degree mode

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Trap

The main trap

Doubling every length multiplies surface area by 4 and volume by 8.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1

Problem

A cylinder has radius 4 and height 11. What is its exact volume?
Square the radius: r2=16r^2=16.

Before multiplying, which expression correctly substitutes the radius and height?

V=π(16)(11)=176πV=\pi(16)(11)=176\pi cubic units.

Why are the units cubic?

Worked example

Example 2: Surface area of a prism

Problem

A closed rectangular shipping box is 5 feet long, 4 feet wide, and 3 feet high. How many square feet of cardboard cover its six faces?
The three different face areas are lw=20lw=20, lh=15lh=15, and wh=12wh=12.
Each face has an opposite congruent face.

Which expression counts all six faces?

The surface area is 2(47)=942(47)=94 square feet.

Why is 5(4)(3)=605(4)(3)=60 not the requested value?

Faded example · you finish it

Example 3: You finish it

Problem

A spherical display has surface area 54 square inches. A similar display has every length tripled. Find the larger display’s surface area.
Surface area uses the scale factor k2=32=9k^2=3^2=9.

Multiply the original surface area by the correct scale factor.

Desmos way

Decimal volume for the radius-4 solids

Enter pi*4^2*11 for the cylinder and (1/3)*pi*4^2*11 for the matching cone.
The readouts are about 552.92552.92 and 184.31184.31 cubic units. Divide the first by the second to confirm the factor of 3; use 176π176\pi when the choices request an exact cylinder volume.

Decimal volume for the radius-4 solids

Degree mode

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Spot the mistake

Where did it go wrong?

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.