Pi


Figure ABCD is a square. Points A, B, C, and D are all located on the circumference of the circle. Figure ADE is an isosceles right triangle. One side measures 1 meter.

What is the area of the square? What is the area of the circle?  (Use 3.14 for ∏.)

What prior knowledge/skills do students need to bring to this problem?

Building on the last two problems, what do students need to notice in the diagram, what shapes, properties, and relationships do they need to consider to solve this problem? Then solve it.

The area of rectangle ABCD is 588 cm². The left side is 21 cm. What is the circumference of the circle? Use Π = 22/7.

1. Look at Pi Chart #1. What is the measurement of the radius? Is each radius the same length? What shape is created when the radius is squared? What is the area of that shape in square centimeters?

2. Look at Pi Chart #2. What fraction of the circle is shaded? If you multiply 35 cm x 35 cm, will that tell you the number of square centimeters in the shaded part of the circle? Why is it so hard to count how many square centimeters are in the shaded part of the circle? (Shapes?) According to our count, there are approximately 962.5 square centimeters in the shaded part of the circle?

3. Look at Pi Chart #3. If you know how many square centimeters are in the shaded part of the circle, how can you figure out how many square centimeters are inside the entire circle? What is the approximate area of the entire circle?  What is the area of the radius squared?

A ratio is a comparison of two amounts, which can be written in the form of a fraction. What is the ratio of the area of the entire circle (write it on top) to the area of the radius squared (write it on the bottom)? Can this ratio be reduced? A ratio can look like an improper fraction. An improper fraction can look like a division problem (top number divided by the bottom number). Divide the area of the circle by the area of the radius squared and see what you get.

The last two problems of the week established that π is less than 4 and greater than 3, and less than 4 and greater than 2 by looking at the ratios of the perimeter/width of a square and the circumference/diameter of a circle, and the area and radius of a circle. We invite you now to use these three figures to reason why pi must be approximately 3.14.

Imagine that you had never been told that π ≈ 3.14. Use Pi Charts 1, 2, & 3 to prove that the value of π ≈ 3.14. Each little box represents 1 square centimeter. The solution will be posted next week.

1. What is the area of the shaded part of Figure 1? What is the area of the entire figure? What is the ratio of the whole area to the shaded area?

2. How many squares are there in Figure 2? What is the area of Figure 2 (the whole thing)? What is the area of one of the shaded triangles? What is the area of the shaded square (the square-within-a-square)? What is the ratio of the whole area to the shaded area?

3. In Figure 3, what is the area of the square-within-a-square (see Figure 2)? What is the area of the small shaded square in the corner? What is the ratio of the area of the square-within-a-square to the area of the small shaded square in the corner?

4. What fraction of the circle in Figure 4 is shaded? What is the ratio of the area of the entire circle to the area of the shaded part of the circle?

5. In Figure 5, what is the area of the shaded triangle? What is the area of the 10 cm x 10 cm square in the lower right corner? Is the area of the shaded part of the circle more or less than the area of the shaded triangle? Is the area of the shaded part of the circle more or less than the area of the square in the lower right corner?

6. In Figure 6, which is greater: the area of the 20 cm x 20 cm square, or the area of the circle that is drawn (inscribed) within the square? Which is greater: the area of the circle, or the area of the square-within-a square (see the shaded part of Figure 2)—which is inscribed within the circle? What is the ratio of the area of the 20 cm x 20 cm square to the area of the 10 cm x 10 cm square in the corner? What is the ratio of the area of the square-within-a square to the area of the 10 cm x 10 cm square in the corner? Do we know the exact ratio of the area of the circle to the area of the 10 cm x 10 cm square in the corner? Is it less than 4 to 1 (the ratio of the area of the 20 cm x 20 cm square to the area of the 10 cm x 10 cm square in the corner)? Is it greater than 2 to 1 (the ratio of the area of the square-within-a square to the area of the 10 cm x 10 cm square in the corner? What is the radius of the circle? What is the area of the radius squared? How does the area of the radius squared compare to the shaded area in Figure 1? Estimate: what is the ratio of the area of the circle to the area of the radius squared?

The last problem of the week established that π is less than 4 and greater than 3 by looking at the ratios of the perimeter and width of a square, and circumference and diameter of a circle. This week we consider the area and radius of a circle.

A comparison of two amounts is called a ratio. The ratio of the area of a circle to its radius squared is called π (pi) and it can be expressed as a quotient  .

Again this explanation can sound like Greek to many students – who also need to memorize and apply, pi ≈ 3.14. Like last week, we offer a problem to guide students (with simple geometry and good questions) to reason for themselves why pi must be less than 4 and in this case more than 2.

Imagine that you had never been told that π ≈ 3.14. Use the squares and circles below to prove that the value of π must be less than 4 and greater than 2. For this exercise, all corners that look square are square. The solution will be posted next week.

If you’re new to our blog, take a look at our earlier post, Why Guided Solutions for the Problems of the Week?. Here’s the solution for last week’s problem. Share other approaches.

1. Look at the square on the upper left. How wide is it across the middle? How wide is it across the top? How wide is it across the bottom? What is the length of the left side? What is the length of the right side? What is the perimeter (the distance around) of the square?

2.  A comparison of two amounts is called a ratio. Tell this ratio: the perimeter of the square compared to its width. Write the ratio so that it looks like a fraction. Now reduce the fraction. What is the reduced ratio?

3. What is the width (diameter) of the circle on the upper right? Is it wider than the square that it is in? Is it wider than the square on the upper left?

4. Trace your finger around the circumference of the circle. Trace your finger around the perimeter of the square that contains the circle. Which distance is longer— the circumference of the circle, or the perimeter of the square? If you are not sure, think about walking around a corner on a sidewalk: if you are in a hurry, would you go straight and then turn right, or would you “cut the corner” by leaving the sidewalk and taking a curving path?

5. Think about the ratio of perimeter of the square to its width: the width times _____ (what number) = the perimeter? Think about the ratio of circumference of the circle to its width: the diameter times _____ (what number) = the circumference? Do we know? If you multiplied the width of the square times 4, would you get its perimeter? If you multiplied the diameter of the circle times 4, would you get its circumference? Or would you have multiply by some number smaller than 4?

6. Look at the hexagon on the middle left. How many equilateral triangles does it contain? What exactly is an equilateral triangle? How many sides does it have? Are all the sides of equal length? The hexagon is 7 cm wide.  What is half of that distance? How wide is the base of the upper left triangle? How wide is the base of the upper right triangle? Is every side of every triangle that same length? What is the perimeter (the distance around) of the hexagon?

7.  A ratio is a comparison of two amounts. Tell this ratio: the perimeter of the hexagon compared to its width. Write the ratio so that it looks like a fraction. Now reduce the fraction. What is the reduced ratio?

8. What is the width (diameter) of the circle on the middle right? Is it wider than the hexagon that is in it? Is it wider than the hexagon on the middle left?

9. Trace your finger around the perimeter of the hexagon contained in the circle. Trace your finger around the circumference of the circle. Which distance is longer—the circumference of the circle, or the perimeter of the hexagon?

10. Think about the ratio of perimeter of the hexagon to its width: the width times _____ (what number) = the perimeter? Think about the ratio of circumference of the circle to its width: the diameter times _____ (what number) = the circumference? Do we know? If you multiplied the width of the hexagon times 3, would you get its perimeter? If you multiplied the diameter of the circle times 3, would you get its circumference? Or would you have multiply by some number larger than 3?

11. Look at the figure on the bottom: a square with a circle inside it, and a hexagon inside the circle.
Which is longer, the perimeter of the square, or the circumference of the circle?
Which is longer, the perimeter of the hexagon, or the circumference of the circle?
What is the ratio of the perimeter of the square, compared to its width?
What is the ratio of the perimeter of the hexagon, compared to its width?
What is the ratio of the circumference of the circle, compared to its diameter?
Is it more than? Is it less than? Is it more than? Is it less than?

Next Page »