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.
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.
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.
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.
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.
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.
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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