Proportional Reasoning and the Meaning of Pi Answer Key

Worksheet 2 of 5 in the Grade 7 circles series. Discover that circumference divided by diameter is always the same number. Next: Circumference.

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Lesson Notes

Take any round object. Measure the distance around it (the circumference, $C$) and the distance across it through the center (the diameter, $d$). Now divide: $C \div d$. No matter what size the object is, you get the same answer, a little more than 3. That number is pi, written $\pi$.

$\pi = C \div d$$\pi \approx 3.14$$\pi \approx 22/7$

Why every circle gives the same ratio

All circles are the same shape, just scaled up or down. When you double the diameter, the distance around also doubles, so the ratio stays fixed. That makes $C$ proportional to $d$, and $\pi$ is the constant of proportionality. A table of measurements makes this easy to see:

Object	$d$ (cm)	$C$ (cm)	$C \div d$
Coin	2	6.28	3.14
Mug	8	25.12	3.14
Plate	25	78.5	3.14

About the number itself

$\pi$ is not exactly 3.14. Its decimals go on forever without repeating (3.14159265...). In 7th grade we round to 3.14.
The fraction $22/7$ (about 3.1429) is another handy approximation. Use it when the diameter or radius is a multiple of 7, because the 7s cancel.
Real measurements are never perfect. If you measure a can and get 3.1 or 3.2, that is measuring error, not a different kind of circle.

Worked example

A lid has $C = 21.98$ cm and $d = 7$ cm. Find $C \div d$.

$21.98 \div 7 = 3.14$, so the ratio is $\pi$.

Turn the ratio around. Since $\pi = C \div d$, you also know $C = \pi \times d$. If a problem gives you the ratio and one measurement, multiply or divide to get the other.

25 problems. Work each one on the canvas under it (pen, eraser and clear are in the canvas toolbar; drag the handle below the canvas for more room). Enter your answer in the box (the keypad under it has fractions, exponents, square roots, pi, the degree sign and the symbols < > ≤ ≥ ≠). When a Unit row is shown, tap the unit for your answer. Then press Check. Use 3.14 or 22/7 for $\pi$ when a problem asks you to. The eye at the right of the row shows or hides the worked solution.