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Advanced Composite Measurement

A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.

Composite measurement problems ask you to build a figure from simpler parts or remove a missing part. Keep each shape's formula separate before combining results. In Measurement, Circles, and 3D Solids, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

What you'll learn

Why it matters: Architects and fabricators combine and subtract shapes when estimating material for cutouts, arches, and custom parts.

Worked example

Problem. A 10 by 8 rectangle has a semicircle with diameter 8 attached to one side. Using pi = 3.14, find the total area.

  1. Rectangle area is 10 x 8 = 80.
  2. Semicircle radius is 4, so half the circle area is 1/2 x 3.14 x 16 = 25.12.
  3. Total area is 80 + 25.12 = 105.12.
  4. Connect the result back to Advanced Composite Measurement so the geometric relationship is explicit.

Answer: 105.12

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. A 12 by 9 rectangle has a 4 by 3 rectangle removed. What area remains?

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  1. 108 - 12 = 96.

Answer: 96

2. A shape combines rectangles of areas 24 and 35. What is the total area?

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  1. Add the areas.

Answer: 59

3. A 6 by 8 rectangle has a triangle with base 6 and height 4 attached. What is the total area?

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  1. Rectangle area is 48.
  2. Triangle area is 12.

Answer: 60

4. A prism has two rectangular-prism parts with volumes 72 and 45. What is the total volume?

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  1. Add the volumes.

Answer: 117

5. A 14 by 10 rectangle has a semicircle of radius 5 removed. Using pi = 3.14, what area remains?

Show solution
  1. Rectangle area is 140.
  2. Semicircle area is 39.25.
  3. 140 - 39.25 = 100.75.

Answer: 100.75

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