Key takeaway: Pi is the fixed number, about 3.14159, that results from dividing any circle’s rim length by its width, and its decimal digits never end or repeat.
Grade 77.G.B.4Geometry and Measurement
Definition of Pi
Pi, written with the Greek letter π, is the quotient you obtain by dividing the distance around a circle by the distance straight across it. Big circle or small circle, the quotient comes out the same, which is why mathematicians gave it its own name.
That sameness is the surprise. A coin and a merry-go-round are different sizes, yet the rim of each is a little more than 3 diameters long. The extra piece is about 0.14 of a diameter. The digits that follow the decimal point go on forever with no pattern, so we use an approximation in calculations.
Formula and notation
The definition and the two usual approximations are:
\[\pi = \frac{C}{d} \approx 3.14 \approx \frac{22}{7}\]
Here \(C\) is the circumference and \(d\) the diameter. The calculator key gives 3.14159265 and more digits.
Key facts
- The decimal for this constant has no end and no repeating block, so it is irrational.
- Both 3.14 and the fraction 22/7 are close estimates, never exact values.
- Use 22/7 when the radius or diameter is a multiple of 7.
- Leaving the answer in terms of π keeps it exact, as in “36π square feet”.
Pi example
A round fountain basin in a city park has a diameter of 35 feet. Estimate its circumference two ways, with \(\frac{22}{7}\) and with 3.14.
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The distance around is the diameter times the constant: \(C = \pi d\).
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With 22/7: \(C \approx \frac{22}{7} \times 35 = 22 \times 5 = 110\) feet.
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With 3.14: \(C \approx 3.14 \times 35 = 109.9\) feet.
The basin is about 110 feet around, and both estimates agree to within a tenth of a foot.
Common mistake
Watch out
Gabe writes π = 3.14 and says a circle with diameter 100 cm has a circumference of exactly 314 cm.
Why it is wrong
The value 3.14 is only a rounded copy. The true figure is slightly larger, 314.159… cm, so 314 is an approximation. Writing the exact circumference as \(100\pi\) cm avoids the rounding.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Geometry and Measurement