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Pi

Pi  π

The ratio of the circumference of a circle to its diameter. Pi is written π and is a transcendental number.

π ≈ 3.14159 26535 89793...

 

See also

Greek alphabet

Key Formula

π=Cd\pi = \frac{C}{d}
Where:
  • π\pi = Pi, approximately 3.14159
  • CC = Circumference of the circle
  • dd = Diameter of the circle

Worked Example

Problem: A circular pond has a diameter of 10 meters. Find the circumference and area of the pond using π ≈ 3.14159.
Step 1: Find the circumference using C = πd.
C=π×10=31.4159 mC = \pi \times 10 = 31.4159 \text{ m}
Step 2: Find the radius. The radius is half the diameter.
r=d2=102=5 mr = \frac{d}{2} = \frac{10}{2} = 5 \text{ m}
Step 3: Find the area using A = πr².
A=π×52=π×25=78.5398 m2A = \pi \times 5^2 = \pi \times 25 = 78.5398 \text{ m}^2
Answer: The pond has a circumference of approximately 31.42 meters and an area of approximately 78.54 square meters.

Another Example

Problem: A wheel has a circumference of 62.83 cm. What is its diameter?
Step 1: Start from the definition of π and solve for the diameter.
π=Cd⇒d=Cπ\pi = \frac{C}{d} \quad \Rightarrow \quad d = \frac{C}{\pi}
Step 2: Substitute the known circumference.
d=62.833.14159≈20 cmd = \frac{62.83}{3.14159} \approx 20 \text{ cm}
Answer: The wheel has a diameter of approximately 20 cm.

Frequently Asked Questions

Why does π never end?
Pi is an irrational number, meaning it cannot be written as a fraction of two integers. Because of this, its decimal expansion goes on forever without settling into a repeating pattern. This was proven rigorously by Johann Lambert in 1761. Even stronger, π is transcendental — it is not a solution to any polynomial equation with integer coefficients.
Is π exactly equal to 22/7?
No. The fraction 22/7 is a useful approximation (it gives 3.142857…), but it is not equal to π. The true value of π is 3.14159265…, so 22/7 is slightly too large. Another common approximation, 355/113, is accurate to six decimal places but is still not exact.

π (pi) vs. τ (tau)

Some mathematicians advocate for tau (τ), defined as τ = 2π ≈ 6.28318. While π relates a circle's circumference to its diameter, τ relates the circumference to the radius: C = τr. Tau simplifies certain formulas (for example, a full rotation is τ radians instead of 2π), but π remains the standard constant used in textbooks and exams.

Why It Matters

Pi is essential in every formula involving circles, spheres, and cylinders — from calculating the area of a pizza to designing satellite orbits. It also appears in areas far beyond geometry, including trigonometry, probability, and physics. Engineers, scientists, and programmers use π constantly, making it one of the most important constants in all of mathematics.

Common Mistakes

Mistake: Confusing the diameter with the radius when using π in formulas.
Correction: Circumference = πd uses the diameter (the full width), while area = πr² uses the radius (half the diameter). Mixing these up doubles or halves your answer.
Mistake: Treating π as exactly 3.14 and rounding too early in a calculation.
Correction: The value 3.14 is a rough approximation. Use your calculator's π button or at least 3.14159 to avoid significant rounding errors, especially when π is multiplied by large numbers or squared.

Related Terms