Area and Composite Figures
A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.
Area measures the space inside a figure. Composite figures can be split into rectangles, triangles, parallelograms, or trapezoids, then the areas can be added or subtracted. 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
- Find polygon area
- Break apart composite figures
- Use units carefully
Worked example
Problem. A rectangle is 12 by 8. A 3 by 4 rectangle is removed. What area remains?
- Large rectangle area: 12 x 8 = 96.
- Removed area: 3 x 4 = 12.
- Subtract: 96 - 12 = 84.
- Connect the result back to Area and Composite Figures so the geometric relationship is explicit.
Answer: 84
Practice problems
1. Find the area of a rectangle with length 11 and width 6.
Show solution
- Warm-up: First identify exactly what the question is asking: Find the area of a rectangle with length 11 and width 6.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Area = length x width.
- Check the result by substituting or estimating: the response should match 66 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 66
2. Find the area of a triangle with base 14 and height 5.
Show solution
- Warm-up: First identify exactly what the question is asking: Find the area of a triangle with base 14 and height 5.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Area = 1/2 x base x height.
- Check the result by substituting or estimating: the response should match 35 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 35
3. Find the area of a parallelogram with base 9 and height 7.
Show solution
- Core Practice: First identify exactly what the question is asking: Find the area of a parallelogram with base 9 and height 7.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Area = base x height.
- Check the result by substituting or estimating: the response should match 63 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 63
4. Find the area of a trapezoid with bases 8 and 14 and height 5.
Show solution
- Core Practice: First identify exactly what the question is asking: Find the area of a trapezoid with bases 8 and 14 and height 5.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Area = 1/2(b1 + b2)h.
- 1/2(22)(5) = 55.
- Check the result by substituting or estimating: the response should match 55 and make sense in the original problem.
Answer: 55
5. A 15 by 10 rectangle has a 5 by 4 corner removed. What area remains?
Show solution
- Challenge: First identify exactly what the question is asking: A 15 by 10 rectangle has a 5 by 4 corner removed. What area remains?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- 150 - 20 = 130.
- Check the result by substituting or estimating: the response should match 130 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 130
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