Using Volume Formulas to Solve Problems

HSG.GMD.A.3

In this lesson:

• Apply the right formula to any 3D shape
• Solve problems with composite solids and unit conversions
• Find missing dimensions when volume is known

Learning Objectives

By the end of this lesson, you should be able to:

1. Select and apply the correct volume formula for any real-world problem
2. Solve composite solid problems by decomposing shapes and adding or subtracting volumes
3. Perform unit conversions within volume calculations
4. Set up and solve equations where volume is given and a dimension is unknown
5. Interpret volume calculations in real-world contexts

From Why to How

You already know why volume formulas work — today we use them.

• Before: "Give an informal argument for why "
• Now: "A conical funnel holds 1.57 liters — find its dimensions"

The same formulas. A new kind of thinking: which one? what values? what do the units mean?

Four Volume Formulas — Reference

These formulas come from HSG.GMD.A.1 — the reasoning is behind you; the application is ahead.

Three-Step Problem-Solving Strategy

Every volume problem follows the same three steps:

1. Identify the shape — what 3D shape does the object resemble?
2. Extract dimensions — what are , , or ? Is the measurement a radius or diameter?
3. Compute and interpret — calculate, attach correct units, explain what the answer means

This strategy applies to every problem in this lesson — and to every exam problem you'll see.

Warm-Up: Cylindrical Soup Can

Problem: A cylindrical soup can has radius 4 cm and height 11 cm. What is its volume?

Step 1: Cylinder →

Step 2: cm, cm

Step 3:

Interpretation: the can holds about 553 mL of soup — is that reasonable for a soup can?

Quick Check: Shape Identification

A hardware store sells traffic cones for road construction.

• The cone is 75 cm tall with a base diameter of 35 cm.
• Which formula applies? Name it before computing anything.
• What's your first step after choosing the formula?

Identify both before advancing.

Cylinders and Cones: Real-World Applications

Real cylinders: water pipes, tanks, cans, silos, columns

Real cones: funnels, conical piles (sand, grain), traffic cones, ice cream cones

The formula is a tool — the skill is recognizing when to use it.

Problem 1: Cylindrical Water Pipe

Problem: A water pipe has interior diameter 15 cm and length 200 m. How many liters of water does it hold?

Step 1: Cylinder →

Step 2: Diameter = 15 cm → cm m; m

Step 3:

Unit Conversions: The Cubic Trap

• 1 m = 100 cm (linear)
• 1 m³ = 100 × 100 × 100 cm³ = 1,000,000 cm³ (cubic)
• 1 m³ = 1,000 L; 1 L = 1,000 cm³; 1 ft³ ≈ 7.48 gallons

Problem 2: Conical Funnel

Problem: A conical funnel has top diameter 20 cm and depth 15 cm. How much liquid can it hold?

Step 1: Cone →

Step 2: Diameter = 20 cm → cm; cm

Step 3:

Problem 3: Traffic Cone Volume

Problem: A traffic cone is 75 cm tall with base diameter 35 cm. How much plastic is needed for a solid cone?

Step 2: cm; cm

About 24 liters of plastic — reasonable for a heavy traffic cone.

Quick Check: Swimming Pool Setup

A cylindrical above-ground pool has:

• Diameter = 18 feet, Depth = 4 feet

Before computing: What is the radius? Write the formula with values substituted.

Set it up — don't compute yet. Advance for the solution.

Problem 4 Solution: Cylindrical Pool

Setup: ft, ft

Is this reasonable? A standard backyard pool holds 5,000–15,000 gallons — ✓

Common Errors: Cylinders and Cones

• Diameter vs. radius: When a problem gives diameter, always write as the first line
• Cubic unit conversion: — the linear factor gets cubed
• Missing units: A volume in cm³ is not the same as a volume in m³ or liters — label every answer

These errors don't vanish without practice — look for them in every problem you solve.

Pyramids, Spheres, and Composite Solids

Pyramids: architecture, packaging, monuments — key: compute the base area first

Spheres: balls, tanks, bubbles, planets — key: volume grows as , so doubling radius multiplies volume by 8

Composite solids: real objects are combinations — decompose, compute each part, add or subtract

Problem 1: Square Pyramid

Problem: A square pyramid has base side length 8 cm and height 12 cm. What is its volume?

Step 1: Pyramid →

Step 2: ; cm

Step 3:

cm² — area, not the side length 8

Problem 2: Spherical Balloon

Problem: A spherical balloon is inflated to a diameter of 30 cm. What volume of air does it contain?

Step 2: Diameter = 30 cm → cm

Doubling the radius to 30 cm would give the volume — not

Quick Check: Basketball Volume

A standard basketball has diameter 24 cm.

1. What is the radius?
2. Write the formula with the radius substituted.
3. Compute the volume — leave in terms of , then approximate.

Try all three steps before advancing.

Decomposing Composite Solids

Real objects are combinations — identify each component, then add or subtract.

Problem 3: Capsule-Shaped Container

Problem: Cylinder (, ) + hemisphere caps on each end (). Find total volume.

Two hemispheres = one full sphere →

Problem 4: Cone Removed from Cylinder

Problem: A cone (, ) is scooped out from a cylinder (, ). What volume of material remains?

Cone removed → subtract, not add

Quick Check: Grain Silo

Problem: Grain silo = cylinder ( m, m) + cone roof ( m, m).

1. Decision: add or subtract? Why?
2. Compute the total volume. Leave in terms of .

Try both parts before advancing.

Grain Silo Solution

Decision: add — cone sits on top of cylinder, separate regions

A typical grain silo holds 500–2000 m³ — this is right in range ✓

Add vs. Subtract — The Decision Rule

Use this before computing any composite solid:

• "Do these shapes fill separate, non-overlapping regions?" → Add their volumes
• "Is one shape removed or carved out of the other?" → Subtract the inner from the outer

Before computing, say out loud: "I am adding/subtracting because ___."

Don't default to addition — look at the geometry first.

Working Backward: Finding Dimensions

The reverse problem: volume is given; find a missing dimension.

Strategy:

1. Write the volume formula
2. Substitute all known values
3. Isolate the unknown (algebra)
4. Verify by substituting back

Design problems always start from a target volume and work backward to dimensions.

Problem 1: Optimal Cylinder Design

Problem: A cylindrical can must hold exactly 1,000 cm³. The manufacturer requires height = diameter. Find the radius.

Since :

Problem 2: Spherical Storage Tank

Problem: A spherical tank must hold exactly 10 m³ of liquid. What interior radius is required?

Verify: ✓

Quick Check: Conical Gravel Pile

Problem: A gravel pile has height = half its base radius. A delivery drops 50 m³ of gravel. What is the radius of the resulting pile?

Hint: if , substitute into to write volume in terms of alone.

Write the simplified formula, then solve for .

Try before advancing.

Problem 3 Solution: Conical Gravel Pile

Substituting into the cone formula:

Verify: ✓

Always Verify Your Answer

Two verification habits:

1. Substitute back: plug your answer into the original formula and check you recover the target volume
2. Reasonableness check: does the magnitude make sense in context?

Key Takeaways

✓ Three-step strategy: identify shape → extract dimensions → compute and interpret
✓ Composite solids: decompose, compute each part, add if separate, subtract if carved out
✓ Unit conversions: cube the linear factor ()
✓ Working backward: write formula → substitute → isolate → verify

Always write before substituting — diameter ≠ radius
in pyramid formula is area (cm²), not side length (cm)
Composite: don't default to addition — look at the geometry first
Cubic unit trap:
Always verify: substitute back and check reasonableness

What Comes Next

HSG.GMD.A.2 — Cavalieri's Principle for Spheres:

• Why is ? Cross-section argument using Cavalieri's principle
• The same reasoning that justified the cone formula, extended to 3D

HSG.MG.A.2 — Density and Modeling:

• Density = mass ÷ volume — volume is in the denominator
• Today's volume calculations become the engine for density problems

The formulas you applied today reappear in every modeling context.