Learning Math: Measurement
Circles and Pi (π) Homework
Session 7, Homework
A circle is inscribed in a square. What percentage of the area of the square is inside the circle?
Take It Further
Imagine that a giant hula hoop is fitted snugly around the Earth’s equator. The diameter of the hula hoop is 12,800 km. Next, imagine that the hula hoop is cut and its circumference is increased by 10 m. The hula hoop is adjusted around the equator so that every part of the hula hoop lies the same distance above the surface of the Earth. Would you be able to crawl under it? Walk under it standing upright? Drive a moving truck under it? Determine the new diameter of the hoop, and find out the distance between the Earth and the hula hoop.
A new car boasts a turning radius of 15 ft. This means that it can make a complete circle with a radius of 15 ft. and return to its original spot. The radius is measured from the center of the circle to the outside wheel. If the two front tires are 4.5 ft. apart, how much further do the outside tires have to travel than the inside tires to complete the circle?
Take It Further
Your dog is chained to a corner of the toolshed in your backyard. The chain measures 10 ft. in length. The toolshed is rectangular, with dimensions 6 ft. by 12 ft. Draw the picture showing the area the dog can reach while attached to the chain. Compute this area.
Draw a diagram of the shed and the possible areas that the dog could reach on its chain. Then divide the space into different sections and calculate the area of each section.
An annulus is the region bounded by two concentric circles.
- If the radius of the small circle is 10 cm and the radius of the large circle is 20 cm, what is the area of the annulus?
- A dartboard has four annular rings surrounding a bull’s-eye. The circles have radii 10, 20, 30, 40, and 50 cm. How do the areas of the annular rings compare? Suppose a dart is equally likely to hit any point on the board. Is the dart more likely to hit in the outermost ring or inside the region containing the bull’s-eye and the two innermost rings? Explain.
Express the areas in terms of π and then compare them.
An oval track is made by erecting semicircles on each end of a 50 m-by-100 m rectangle. What is the length of the track? What is the area of the region enclosed by the track?
Zebrowski, Ernest (1999). A History of the Circle (pp. 48-49). Piscataway, N.J.: Rutgers University Press.
Reproduced with permission from the publisher. � 1999 by Rutgers University Press. All rights reserved.
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A History of the Circle
Area of circle: AC = πr2
Area of square: AS = (2r)2 = 4r2
AC/AS = πr2/4r2 = π/4
The fractional part of the area is π/4. This is equal to 0.785, and therefore, expressed as a percentage, is approximately 78.5%.
Since the diameter of the hula hoop is 12,800 km, the circumference is approximately 40,212.386 km. If we cut the hula hoop and add 10 m (0.01 km), the circumference is now 40,212.396 km. The new diameter of the hula hoop is found by dividing 40,212.396 by π; it is 12,800.003 km. The difference between the two diameters is 0.003 km, or 3 m. Dividing this difference in half (since d = 2r) results in a 1.5-meter height change between the Earth and the hula hoop. You could easily crawl under it or walk under it in a crouched position, but you could not drive a truck under it!
The interesting fact about this problem is that the distance added to the diameter (and radius) is independent of the original diameter and circumference:
C + addition to C = π(d + addition to diameter) [new]
C = π • d [original]
addition to C = π • (addition to diameter)
(addition to C)/π = addition to diameter
Since the radius of the circle formed by the outside tires is 15 ft., the radius formed by the inside tires is (15 – 4.5) = 10.5 ft. The circumference of the two circles can be calculated and subtracted:
15 • 2 • π – 10.5 • 2 • π = 4.5 • 2 • π,
which is approximately 28.3 ft. Note that the radius of the circle formed by the outside tires was not important to the final result (which only used the 4.5-foot difference). This means that the calculation is valid for any car whose wheels are 4.5 ft. apart.
The area is a three-quarters circle with radius 10 ft., plus a quarter-circle with radius 4 ft. (The dog can reach this area by stretching along the six-foot wall and then pointing into the exposed area.) The total area is
3/4 • π • (10)2 + 1/4 • π • (42) = 75 • π + 4 • π = 79 • π,
or approximately 248 ft2.
- The area of the annulus is the difference between the circles’ areas. For these circles, the area is 300π cm2, or approximately 943 cm2.
- If the smallest circle has a radius of 10 cm, then the area of that bull’s-eye circle is 100π cm2. The area of the first annular ring is the 400π – 100π, or 300π cm2. Since the second interior circle has a radius of 20 cm, we can find the area of it and then subtract the area of the bull’s-eye. Using this line of reasoning, the area of the second annular ring is 900π – 400π, or 500π cm2; the area of the third annular ring is 1,600π – 900π, or 700π cm2; and the area of the fourth (outer) annular ring is 2,500π – 1,600π, or 900π cm2. The probability of a dart thrown at random hitting the outermost ring or a dart hitting the bull’s-eye and the two innermost rings is exactly the same; both regions have an area of 900π cm2.
The length and area can be more easily calculated by isolating the circular sections, which then form a complete circle of radius 25 m.
Length: 2 • π • 25 + 2 • 100, or approximately 357 m
Area: π • 252 + 100 • 50, or approximately 6,963 m2
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.