K Khaemenes GeometryOpen-Age · Printable Visual Lesson
Unit 10 Course Home Student Lesson

Unit 10 · Lesson 03 · Area, Measurement & Geometric Probability

Heron’s Formula and Coordinate Area · Teacher Key

Find triangle area from three side lengths and compare with coordinate or base-height methods.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intention

Use accurate diagrams, definitions, measurements, and reasoning to find triangle area from three side lengths and compare with coordinate or base-height methods.

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

Area geometry diagram A clear labeled diagram supporting area reasoning. base bheight h

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.

  1. Read the figure and rewrite all given information with correct notation.
  2. Identify the exact relationship that connects the known and unknown quantities.
  3. Substitute, construct, or prove one justified step at a time.
  4. 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

Heron’s Formula and Coordinate Area

Find, construct, prove, or explain each result. Show reasoning and label units.

U10 · L03
Name
Date
1

Find the area of a triangle with base 15 and height 12.

Area geometry diagram A clear labeled diagram supporting area reasoning. base bheight h
Answer: 90
Reasoning: A=1/2·15·12.
2

A regular polygon has apothem 6 and perimeter 36. Find its area.

Area geometry diagram A clear labeled diagram supporting area reasoning. base bheight h
Answer: 108
Reasoning: A=1/2·aP.
3

Find the area of a triangle with side lengths 5, 5, and 6.

Area geometry diagram A clear labeled diagram supporting area reasoning. base bheight h
Answer: 12
Reasoning: Heron's formula with s=8 gives √(8·3·3·2)=12.
4

A figure is dilated by scale factor 3. By what factor does its area change?

Area geometry diagram A clear labeled diagram supporting area reasoning. base bheight h
Answer: 9
Reasoning: Area scales by k².
5

A target occupies 18 cm² inside a 60 cm² board. Find the geometric probability of a uniform hit.

Area geometry diagram A clear labeled diagram supporting area reasoning. base bheight h
Answer: 3/10 or 0.3
Reasoning: Probability is favorable area divided by total area.
6

Why should exact π or radical form usually be kept until the final step?

Area geometry diagram A clear labeled diagram supporting area reasoning. base bheight h
Answer: It reduces accumulated rounding error
Reasoning: Round once after the full calculation.