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.
What you'll learn
- Combine area and volume strategies
- Find missing dimensions
- Use composite solids
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.
- Rectangle area is 10 x 8 = 80.
- Semicircle radius is 4, so half the circle area is 1/2 x 3.14 x 16 = 25.12.
- Total area is 80 + 25.12 = 105.12.
Answer: 105.12
Practice problems
1. A 12 by 9 rectangle has a 4 by 3 rectangle removed. What area remains?
Show solution
- Warm-up: First identify exactly what the question is asking: A 12 by 9 rectangle has a 4 by 3 rectangle removed. What area remains?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- 108 - 12 = 96.
- Check the result by substituting or estimating: the response should match 96 and make sense in the original problem.
Answer: 96
2. A shape combines rectangles of areas 24 and 35. What is the total area?
Show solution
- Warm-up: First identify exactly what the question is asking: A shape combines rectangles of areas 24 and 35. What is the total area?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Add the areas.
- Check the result by substituting or estimating: the response should match 59 and make sense in the original problem.
Answer: 59
3. A 6 by 8 rectangle has a triangle with base 6 and height 4 attached. What is the total area?
Show solution
- Core Practice: First identify exactly what the question is asking: A 6 by 8 rectangle has a triangle with base 6 and height 4 attached. What is the total area?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Rectangle area is 48.
- Triangle area is 12.
- Check the result by substituting or estimating: the response should match 60 and make sense in the original problem.
Answer: 60
4. A prism has two rectangular-prism parts with volumes 72 and 45. What is the total volume?
Show solution
- Core Practice: First identify exactly what the question is asking: A prism has two rectangular-prism parts with volumes 72 and 45. What is the total volume?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Add the volumes.
- Check the result by substituting or estimating: the response should match 117 and make sense in the original problem.
Answer: 117
5. A 14 by 10 rectangle has a semicircle of radius 5 removed. Using pi = 3.14, what area remains?
Show solution
- Challenge: First identify exactly what the question is asking: A 14 by 10 rectangle has a semicircle of radius 5 removed. Using pi = 3.14, what area remains?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Rectangle area is 140.
- Semicircle area is 39.25.
- 140 - 39.25 = 100.75.
- Check the result by substituting or estimating: the response should match 100.75 and make sense in the original problem.
Answer: 100.75
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