Using Area, Volume, and Circumference to Problem Solve
Using Area, Volume, and Circumference to Problem Solve
Objectives
In this unit, students will use radius, diameter, and perimeter to solve for area and volume. They will solve problems using attributes of triangles, quadrilaterals, and circles. They will find, confirm, and use relationships involving perimeter, circumference, area, and volume to solve real-world problems. Students will:
- discover the relationship between the diameter and circumference of a circle.
- discover the relationship between the radius and area of a circle.
- calculate the area and circumference of a circle.
- calculate the volume of right prisms and cylinders.
- apply circumference, area, and volume calculations to solve real-world problems.
Essential Questions
- How can patterns be used to describe relationships in mathematical situations?
- How can recognizing repetition or regularity assist in solving problems more efficiently?
- How are spatial relationships, including shape and dimension, used to draw, construct, model, and represent real situations or solve problems?
- How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving?
- How can geometric properties and theorems be used to describe, model, and analyze situations?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- Enter any value for radius to have circumference and area of a circle instantly calculated
http://www.calculatorsoup.com/calculators/geometry-plane/circle.php
- Mini-lesson on using the area formula to find the area of a circle without a calculator
(5 minutes)
https://www.youtube.com/watch?v=OBnKYoOpdsM&NR=1
- A song about parts of a circle and the formulas for area and circumference; use as introduction, review, or memory tool (1:35 minutes)
https://www.youtube.com/watch?v=lWDha0wqbcI&feature=related
- Area of a Circle
http://www.mathgoodies.com/lessons/vol2/circle_area.html
- Area, Circumference, Radius, and Diameter Calculator
http://www.mathsisfun.com/geometry/circle-area.html
- Trapezoids as Composite Figures
http://www.mathsteacher.com.au/year8/ch12_area/09_comp/fig.htm
- Breaking Down Composite Figures
http://www.icoachmath.com/math_dictionary/Composite_Figures.html
- Composite Solids
http://www.ck12.org/concept/Composite-Solids/#all
Formative Assessment
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View
Multiple-Choice Items:
1. The diameter of a circle is 16 inches. What is the radius of the circle?
A
2 inches
B
4 inches
C
8 inches
D
32 inches
2. A circle has a radius of 6 centimeters. What is the circumference of the circle?
A
3π centimeters
B
6π centimeters
C
12π centimeters
D
36π centimeters
3. A circle has a circumference of 100π inches. What is the diameter of the circle?
A
10 inches
B
25 inches
C
50 inches
D
100 inches
4. The radius of a circle is 3 inches. What is the area of the circle?
A
3π square inches
B
6π square inches
C
9π square inches
D
12π square inches
5. What is the area of the circle shown below?
A
576π square inches
B
144π square inches
C
24π square inches
D
12π square inches
6. A circle has an area of 64π square inches. What is the diameter of the circle?
A
8 inches
B
16 inches
C
32 inches
D
64 inches
7. What is the area of the figure shown below?
A
60 square inches
B
160 square inches
C
190 square inches
D
220 square inches
8. What is the volume of the solid shown below?
A
42 cubic inches
B
768 cubic inches
C
928 cubic inches
D
23,040 cubic inches
9. What is the volume of the shaded part of the figure shown below?
A
288 cubic inches
B
2,592 cubic inches
C
9,120 cubic inches
D
12,000 cubic inches
Multiple-Choice Answer Key:
1. C
2. C
3. D
4. C
5. B
6. B
7. C
8. B
9. B
Short-Answer Items:
10. A circle has a circumference of 25π units. Explain how the radius and diameter can be calculated from this circumference.
11. A circle has a diameter of 14 units. Explain the steps needed to determine the area of the circle. Find the area of the circle.
12. Explain two different ways to find the volume of the solid shown below. For each way, clearly indicate the volumes of the parts needed and show the volume of the solid.
Short-Answer Key and Scoring Rubrics:
10. A circle has a circumference of 25π units. Explain how the radius and diameter can be calculated from this circumference.
The diameter can be found by dividing 25π by π. The radius can be found by dividing the diameter by 2 (or by dividing 25π by 2π).
Points
Description
2
The student clearly explains how to find both the radius and diameter of the circle.
1
The student clearly explains how to find either the radius or diameter of the circle (but not both).
0
The student does not clearly explain how to find the radius nor the diameter of the circle.
11. A circle has a diameter of 14 units. Explain the steps needed to determine the area of the circle. Find the area of the circle.
First, divide the diameter by 2 to find the radius, which is 7. Then, square the radius, which gives 49. Finally, multiply that value by π, so the area is 49π square units.
Points
Description
2
The student clearly explains how to find the area and correctly finds the area.
1
The student either clearly explains how to find the area or correctly finds the area but not both.
0
The student does not clearly explain how to find the area nor does the student correctly find the area.
12. Explain two different ways to find the volume of the solid shown below. For each way, clearly indicate the volumes of the parts needed and show the volume of the solid.
There are three possible methods (of which the student must provide two):
Points
Description
2
The student provides 2 different correct solution methods.
1
The student provides 1 correct solution method.
0
The student does not provide a correct solution method.
Performance Assessment:
The student should construct/draw a model of a castle wall in three dimensions that consists of at least 4 rectangular prisms. The dimensions of the wall should be labeled. Students are encouraged to be creative (i.e., putting in doors, etc.), but the volume of the entire figure should be able to be computed.
Also, each castle should have a flag in the shape of a rectangle, square, or triangle (with the base and height given so the area can be computed). Each flag should have a composite figure design on it to represent the “kingdom.”
Students should present their drawings to one another in a small or large group. They can either explain how to find the volume and area of the parts of their drawing or it can be “assigned” to other group members. (In either case, the creator of the drawing should have already calculated the volume and areas.)
Performance Assessment Scoring Rubric:
Points
Description
4
- The student creates a castle wall with at least 4 rectangular prisms.
- The student creates a flag with a geometric design.
- The student correctly finds the volume of his/her castle wall.
- The student correctly finds the area of the geometric design of his/her flag.
- The student demonstrates an advanced understanding of finding areas and volumes of composite figures.
3
- The student completes 3 of the 4 tasks shown above.
- The student demonstrates some understanding of finding areas and volumes of composite figures.
2
- The student completes 2 of the 4 tasks shown above.
- The student demonstrates a limited understanding of finding areas and volumes of composite figures.
1
- The student completes 1 of the 4 tasks shown above.
- The student demonstrates a very limited understanding of finding areas and volumes of composite figures.
0
- The student does not complete any of the 4 tasks shown above.
- The student demonstrates no understanding of finding areas and volumes of composite figures.
Final 05/24/2013
