Find the area of a triangle with base 13 and height 4.
Unit 10 · Lesson 06 · Area, Measurement & Geometric Probability
Geometric Probability by Length and Area
Use length or area ratios under a uniform-selection assumption.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to use length or area ratios under a uniform-selection assumption.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Decompose figures, manage units and precision, and use area ratios to model probability and optimize design constraints.
A=\frac12 bhA=\frac12 aPA=\sqrt{s(s-a)(s-b)(s-c)}P(E)=\frac{\text{favorable measure}}{\text{total measure}}
Worked reasoning
Process: Decompose or select a suitable formula, keep exact values until the final step, attach square units, and assess whether rounding is reasonable.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: Area uses square units and scales by the square of a linear scale factor; it does not scale linearly.
Khaemenes Academy · Open-Age Geometry
Geometric Probability by Length and Area
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
A regular polygon has apothem 5 and perimeter 48. Find its area.
Find the area of a triangle with side lengths 5, 5, and 6.
A figure is dilated by scale factor 3. By what factor does its area change?
A target occupies 18 cm² inside a 60 cm² board. Find the geometric probability of a uniform hit.
Why should exact π or radical form usually be kept until the final step?