Informal Arguments for Volume and Area Formulas

HSG.GMD.A.1 | Geometric Measurement and Dimension

In this lesson:

• Why circumference and area formulas work
• How to justify volume formulas using reasoning — not memorization

Learning Objectives

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

1. Give an informal argument for why
2. Give an informal argument for why
3. Justify for a cylinder
4. Justify for a pyramid
5. Justify for a cone
6. Explain the difference between informal and formal reasoning

Quick Review: Formulas You Already Know

You've used these formulas since middle school:

• Circumference:
• Circle area:
• Cylinder volume:

But why do these formulas work? Today we find out.

Informal Arguments: Making Formulas Plausible

An informal argument uses:

• Visual reasoning and diagrams
• Physical models and hands-on experiments
• Approximation and limiting processes

Goal: convince you a formula makes sense, not prove it rigorously

Formal Proofs: Certainty from First Principles

A formal proof requires:

• Starting from axioms (accepted definitions and postulates)
• Applying logical deduction at every step
• Meeting strict mathematical standards

This standard focuses on understanding, not formal proof-writing.

Informal vs. Formal: What's the Difference?

Quick Check

Which type of reasoning is this?

"Inscribe a polygon in a circle. As the number of sides grows, the perimeter gets closer to the circumference."

Is this an informal argument or a formal proof?

Think before advancing...

What Is π, Really?

• This ratio is the same for every circle — regardless of size
• π ≈ 3.14159... is an approximation of this exact ratio
• Since :

Inscribed Square (n = 4)

: perimeter , less than

More Sides: Hexagon and Octagon (n = 6, 8)

: perimeter ; : perimeter — both closer to

Even More Sides: 12-gon (n = 12)

: perimeter — barely distinguishable from

In the Limit: Polygon Perimeter → 2πr

• As grows, the polygon fits the circle more and more precisely
• π is the exact ratio that emerges from this geometry — not arbitrary
• More sides = closer fit; infinitely many sides = exact circumference

Quick Check: Polygon Perimeters

A circle has radius 1, so .

As the number of sides increases, does the inscribed polygon's perimeter:

• Approach 6.28 from below?
• Stay constant?
• Eventually exceed 6.28?

Think before advancing...

Area Dissection: Cut into Wedges

Imagine cutting a circle into 16 equal wedges (like pizza slices):

• Rearrange them alternating: one pointing up, the next pointing down
• The result approximates a parallelogram or rectangle
• More cuts → the shape fits a rectangle more and more precisely

This is the key idea behind the area formula .

Rearranged Wedges Form a Near-Rectangle

Width ≈ (half the circumference); height ≈ (the radius)

Area = Width × Height = πr²

• Width of the near-rectangle ≈ (half the circumference)
• Height of the near-rectangle = (the radius)
• More wedges → better fit → exact area =
• Note: uses once (linear); uses (quadratic) — different measurements

Another Approach: Concentric Rings

Imagine the circle as many thin rings stacked from center to edge:

• Ring at radius has circumference and thickness
• Area of each ring ≈ (circumference × thickness)
• Sum all rings from 0 to : total area =

Quick Check: The Wedge Rectangle

When circle wedges are rearranged into a near-rectangle:

• What does the width of the rectangle represent?
• What does the height of the rectangle represent?

These two dimensions multiply to give the area formula — think it through.

From 2D Circles to 3D Cylinders

We've established the circle formulas:

• Circumference:
• Area:

Next: Extend to three dimensions — volume of a cylinder

Cylinder: A Shape with Constant Cross-Sections

Every horizontal slice of a cylinder is a circle of radius :

• Area of each circular slice =
• All slices are identical — the cross-section never changes with height
• Compare to a rectangular prism: every slice is the same rectangle

Building a Cylinder from Stacked Disks

• Each disk has area
• Stack units of disks to fill the cylinder

Deriving the Cylinder Volume Formula

• Base area: circular cross-section =
• Height: stack the base from 0 to
• Pattern: Volume = (base area) × height — same as for prisms

Cylinder and Prism: The Same Pattern

All follow: V = (base area) × height

Quick Check: Cylinder Volume

A cylinder has radius units and height units.

What is the volume? Leave your answer in terms of .

Try the calculation before advancing...

Answer: V = 45π ≈ 141.4 units³

• Substitute: ,
• Square the radius first:
• Multiply:

From Cylinders to Pyramids and Cones

Cylinders have constant cross-sections — every slice is the same circle.

Pyramids and cones taper — their cross-sections shrink to a point at the apex.

Question: How does this tapering change the volume formula?

Pyramid: Cross-Sections That Shrink

A pyramid's cross-sections decrease from base to apex:

• At the base: full area
• Halfway up: area
• At the apex: area

The tapering makes the volume less than the matching prism.

Same Base and Height — Different Volumes

A pyramid with base and height next to a prism with the same and

Physical Demonstration: 3 Pyramids Fill 1 Prism

• Fill a pyramid with rice, pour into the matching prism
• Repeat three times — the prism is exactly full
• The 1/3 factor reflects tapering geometry — not a guess

Quick Check: The 1/3 Factor

A pyramid has the same base and height as a prism.
The pyramid's volume is exactly 1/3 of the prism's volume.

Why 1/3? What feature of the pyramid causes this factor?

Think about what changes from base to apex...

Cavalieri's Principle: Equal Cross-Sections, Equal Volume

If two solids have equal cross-sectional area at every height, they have equal volume.

Using this principle:

• A tilted pyramid and an upright pyramid have equal cross-sections at every height
• Therefore they have equal volume — regardless of the tilt
• Allows us to compare pyramids to simpler shapes

Cube Dissection into 6 Congruent Pyramids

• A cube of side has volume
• It splits into 6 identical pyramids, each with volume
• Each pyramid: base , height — so ✓

Cube Dissection Confirms V = (1/3)Bh

• Each pyramid: base , height , volume
• Six such pyramids fill the cube exactly
• General formula is confirmed for this case ✓

Cone: A Pyramid with a Circular Base

• Cross-sections of a cone taper from circle of area to zero
• The same reasoning applies: tapering shape → factor of 1/3
• Replace pyramid's base area with cone's circular base area

Water Demo: 3 Cones Fill 1 Cylinder

Three cones (same base and height as the cylinder) fill it exactly:

• Cone: , → Volume
• Cylinder: , → Volume
• Three cones: = cylinder ✓

Your Turn: Pyramid and Cone Volumes

Calculate the volume of each shape. Leave answers in terms of if needed.

1. Pyramid: base area units², height units
2. Cone: radius units, height units

Pause and try both before advancing

Key Takeaways

✓ Informal arguments explain why — formal proofs establish certainty
✓ from polygon limit; from wedge dissection
✓ Pyramid/cone = × prism/cylinder

π = C/d exactly — 3.14 is only an approximation
(linear in ) ≠ (quadratic) — different measurements
Cylinder: , not
The factor reflects tapering — not arbitrary

What Comes Next

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

• Why is for a sphere?
• Comparing sphere cross-sections to a cylinder minus a double cone

HSG.GMD.A.3 — Apply volume formulas to solve problems:

• Real-world applications of cylinder, pyramid, and cone volumes