# Chapter 2 notebook: Geometry: Estimate Shape Before Measuring It

**Goal:** Use scaling and feasibility checks before committing to a geometric calculation.

**Start with:** Rules 2.1.1, 2.1.3, 2.2.1. Use a calculator or the optional Python lab. Programming is optional for the workbook; the Jupyter version requires a Python 3 kernel. All lab numbers are constructed practice inputs, not observed data.

**Source:** [Finished Chapter 2](../skills/math-thumb-geometry/references/chapter.md) from *Mathematical Rules of Thumb*. Numbered rules, classifications, and the decision path come from the book. The lab, exercises, answer key, and worksheet prompts are companion additions.

## 1. Frame your decision

Write a question in which an answer would change something you do. Gather: Lengths with units; evidence of similarity; vertex order or perpendicular height; permitted error.

- My question and intended decision:
- Known inputs and units:
- Required accuracy or threshold:
- What I expect before calculating:
- What I still need to find out:

Use the lab as a worked starting point if you do not yet have your own problem. For notation or prerequisites, ask the chapter skill to explain only the concept blocking the next step.

## 2. Choose a route

- **Are two figures scaled copies?** Find the linear factor first. Raise it to the second power for area and the third for volume; subtract powers when forming ratios such as perimeter over area.
- **Is the change small?** Express it as a dimensionless fraction. For circle area, double the fractional radius change and compare its square with the required accuracy.
- **Could the data be impossible?** Test the triangle inequality before any three-side formula and check units before interpreting coordinates or lengths.
- **Do you need only a plausibility range?** Enclose the shape in a box or bracket a Euclidean diagonal before performing the exact computation.
- **Is the boundary circular or polygonal?** Use chord-radius geometry for shallow sagitta, a full turn for regular-polygon angles, and the $1/n^2$ law to predict the effect of refinement.
- **Is area encoded in coordinates?** Keep vertices in cyclic order and use shoelace. If a perpendicular height is already known, $bh/2$ is usually the shorter route.
- **Is the result a weighted location?** With nonnegative weights, require the centroid to remain inside the convex hull.

**Chapter-specific stop check:** Do not apply square-cube laws to shape changes that are not similar, or use a necessary geometric bound as a sufficient construction test.

## 3. Work the lab

A circular opening's radius increases by 3%. Estimate the area increase without calculating either area. The estimate is 2(0.03)=0.06, or 6%; the exact relative increase is 1.03^2-1=0.0609, or 6.09%. The omitted term is 0.03^2=0.0009, equal to 0.09 percentage points. This distinction between fractions and percentage points matters when specifying a tolerance.

Predict the sign and scale before running the code. Then change one input and explain why the result moves. The code checks the constructed example; it does not prove the rule for every possible input. Code assertions may describe the example's chosen regime, so inspect them before changing that regime.

```python
fractional_radius_change = 0.03
assert fractional_radius_change > -1
estimate = 2*fractional_radius_change
exact = (1+fractional_radius_change)**2-1
correction = fractional_radius_change**2
print(f"Area increase: estimate {100*estimate:.3f}%; exact {100*exact:.3f}%")
print(f"Difference: {100*correction:.3f} percentage points")
assert abs(exact-estimate-correction) < 1e-12
```

**My prediction, observed result, and explanation:**

_Record your work here._

## 4. Practise without the answers

### Exercise 1

Scale a similar solid by 1.5 in every length. Find its area and volume factors.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 2

Can lengths 2, 3, and 6 metres form a triangle? What changes at 2, 3, and 5?

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 3

A designer doubles a cylinder's radius while preserving its height and claims eight times the volume using similarity. Repair the argument.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

**Coaching prompt:** “Use the Chapter 2 skill to help me with Exercise 2. Ask for my attempt, give one useful hint if I need it, and help me check the assumptions before showing the answer.”

## 5. Answer key and reasoning

Read this after attempting the exercises, or use it immediately if you prefer a complete walkthrough. An answer is complete only when its assumptions and stopping point are clear.

### Answer 1

Area factor 1.5^2=2.25; volume factor 1.5^3=3.375. Similarity must hold in every dimension.

### Answer 2

No: 2+3<6. At 2+3=5 the figure is degenerate, so neither set forms a nondegenerate triangle.

### Answer 3

This is not a similar enlargement: only radius changes. V=pi × r^2 × h increases by 2^2=4. An eightfold change requires all three linear dimensions to double.

## 6. Build the complete chapter toolkit

The new lab samples the chapter; the following checklist covers all 12 rules. Study one thematic group at a time. A large group can take several sessions.

- **Predict How Shape Responds to Scale:** work with rules 2.1.1, 2.1.2, 2.1.3.
- **Bound First, Compute Second:** work with rules 2.2.1, 2.2.2, 2.2.3.
- **Extract Information from Curves and Coordinates:** work with rules 2.3.1, 2.3.2, 2.3.3, 2.3.4, 2.3.5, 2.3.6.

For each selected rule, read its equation, explanation, and worked use in the source. Reproduce that example; change one input; then change one assumption so the rule is no longer justified. Record the result and what check catches the failure. Historical examples remain labeled and qualified as in the source.

Read each complete numbered profile in the [chapter reference](../skills/math-thumb-geometry/references/chapter.md) before applying it. The cues below abbreviate the graph metadata; they are not complete conditions. Change the status only after doing the practice described below.

| Rule | Book role | First assumptions to inspect | Practice status |
|---|---|---|---|
| 2.1.1: Remember the square-cube laws for similar shapes | Independent | geometric similarity;  uniform linear scale factor | new |
| 2.1.2: Expect boundary-to-area ratio to shrink like inverse size | Independent | geometrically similar shapes;  uniform length scaling | new |
| 2.1.3: Double the fractional radius change to estimate circle-area change | Independent | small relative radius change;  positive reference radius | new |
| 2.2.1: Check the triangle inequality before doing triangle algebra | Independent | metric or normed distance;  nonnegative lengths | new |
| 2.2.2: Use a bounding box for a fast area or volume ceiling | Independent | object contained in box;  box dimensions known | new |
| 2.2.3: Bracket a right-triangle diagonal before taking a square root | Independent | nonnegative component lengths;  euclidean diagonal | new |
| 2.3.1: Estimate shallow-arc sagitta by chord squared over radius | Independent | sagitta small relative to radius;  circular arc | new |
| 2.3.2: Estimate regular-polygon angles from the exterior turn | Independent | simple planar polygon;  interior angles counted consistently | new |
| 2.3.3: Expect regular-polygon circle error to fall quadratically | Independent | regular inscribed or circumscribed polygon;  large side count | new |
| 2.3.4: Use the shoelace formula for polygon area from coordinates | Independent | simple planar polygon;  vertices in cyclic order | new |
| 2.3.5: Prefer base times perpendicular height when height is available | Workflow | triangle or decomposable shape;  known perpendicular height | new |
| 2.3.6: Use the convex hull as a centroid sanity check | Workflow | nonnegative weights;  weights sum to one | new |

For a completed row, record: **rule number / my new input / mathematical claim type / verified assumptions / calculation / check / valid use / rejected use / next step**. “Practised” means you worked an example. “Demonstrated” means you can explain a valid use, transfer it, and reject a misuse without the answer key.

## 7. Apply it to your own problem

Return to your opening question. Choose the smallest rule set that can settle it. Use the book's independent/workflow/specialized classification separately from the claim type (exact, approximate, bound, diagnostic, or heuristic).

- Selected rule number(s) and reason:
- Assumptions that hold, fail, or remain uncertain:
- Substitution with units:
- Result and error, uncertainty, or bound:
- Independent check or limiting case:
- Decision this supports:
- Stop here, do a named next calculation, or gather missing information:

## 8. Transfer and continue

Useful nearby chapters: Chapter 1: algebra; Chapter 4: calculus; Chapter 16: measurement. Bring the question, units, assumptions, result type, and uncertainty to the next chapter. Choose a bridge only when it supplies an operation you actually need.

**Completion check:** Explain this chapter's lab in your own words; solve one changed-input exercise; reject one invalid use; and produce a decision record for your own problem. If one check fails, revisit that part of the chapter rather than marking every rule complete.