Measurement and Circles Checkpoint
A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint mixes measurement skills: polygon area, composite area, surface area, volume, circle circumference and area, and fractional circle sectors. 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 area and volume
- Use circle formulas
- Work with arcs and sectors
Worked example
Problem. A 90-degree sector has circle area 80. Find the sector area.
- 90/360 = 1/4.
- The sector is one fourth of the circle.
- 80/4 = 20.
- Connect the result back to Measurement and Circles Checkpoint so the geometric relationship is explicit.
Answer: 20
Practice problems
1. Find the area of a triangle with base 20 and height 7.
Show solution
- Area: First identify exactly what the question is asking: Find the area of a triangle with base 20 and height 7.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- 1/2 x 20 x 7 = 70.
- Check the result by substituting or estimating: the response should match 70 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 70
2. Find the area of a trapezoid with bases 6 and 14 and height 8.
Show solution
- Area: First identify exactly what the question is asking: Find the area of a trapezoid with bases 6 and 14 and height 8.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- 1/2(20)(8) = 80.
- Check the result by substituting or estimating: the response should match 80 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 80
3. A 14 by 9 rectangle has a 3 by 6 section removed. What area remains?
Show solution
- Composite Area: First identify exactly what the question is asking: A 14 by 9 rectangle has a 3 by 6 section removed. What area remains?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- 126 - 18 = 108.
- Check the result by substituting or estimating: the response should match 108 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 108
4. Find the volume of a rectangular prism that is 9 by 4 by 5.
Show solution
- Volume: First identify exactly what the question is asking: Find the volume of a rectangular prism that is 9 by 4 by 5.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- 9 x 4 x 5 = 180.
- Check the result by substituting or estimating: the response should match 180 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 180
5. A cube has side length 7. What is its surface area?
Show solution
- Surface Area: First identify exactly what the question is asking: A cube has side length 7. What is its surface area?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Each face is 49.
- 6 x 49 = 294.
- Check the result by substituting or estimating: the response should match 294 and make sense in the original problem.
Answer: 294
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