--- name: math-thumb-geometry description: "Apply Chapter 2 (Geometry: Estimate Shape Before Measuring It) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to use scaling and feasibility checks before committing to a geometric calculation." --- # Geometry: Estimate Shape Before Measuring It Use this chapter to help the reader make a checked mathematical decision. All 12 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Lengths with units; evidence of similarity; vertex order or perpendicular height; permitted error. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **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. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Do not apply square-cube laws to shape changes that are not similar, or use a necessary geometric bound as a sufficient construction test. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 2.1.1, 2.1.3, 2.2.1 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C02-D01: Separate length, area, and volume scaling**: rule 2.1.1. - **C02-D02: Check whether an arc is shallow enough**: rule 2.3.1. - **C02-D03: Refine a polygon toward a circle**: rule 2.3.3. - **C02-D04: Check the triangle inequality before triangle algebra**: rule 2.2.1. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.