Pi, written as π, is one of the most important constants in geometry because it links every circle's circumference to its diameter. No matter how large or small a circle is, the ratio C/d is always the same number, about 3.14159. This makes π essential for measuring circular objects, from wheels and pipes to planets and waves.
A circle diagram with radius, diameter, and circumference shows why the formula C = πd is so powerful.
Key Facts
- π = C/d, where C is circumference and d is diameter.
- C = πd and C = 2πr, where r is radius.
- d = 2r, so the diameter is twice the radius.
- A = πr^2 gives the area inside a circle.
- π is irrational, so its decimal form never ends and never repeats.
- Common approximations include π ≈ 3.14 and π ≈ 22/7.
Vocabulary
- Pi
- Pi is the constant ratio of a circle's circumference to its diameter, written as π.
- Circumference
- Circumference is the distance around the outside edge of a circle.
- Diameter
- Diameter is a line segment that passes through the center of a circle and connects two points on the circle.
- Radius
- Radius is a line segment from the center of a circle to any point on the circle.
- Irrational Number
- An irrational number cannot be written exactly as a ratio of two integers and has a nonrepeating, nonterminating decimal expansion.
Common Mistakes to Avoid
- Using radius instead of diameter in C = πd is wrong because the formula requires the full width across the circle, not the distance from the center to the edge.
- Thinking π equals exactly 3.14 is wrong because 3.14 is only a rounded approximation of an infinite decimal.
- Confusing circumference with area is wrong because circumference measures distance around a circle, while area measures the surface inside it.
- Assuming larger circles have a larger value of π is wrong because π is the same ratio for every circle, regardless of size.
Practice Questions
- 1 A circular table has a diameter of 1.2 m. Use π ≈ 3.14 to find its circumference.
- 2 A wheel has a radius of 35 cm. Use C = 2πr and π ≈ 3.14 to find the distance it travels in one full rotation.
- 3 Explain why measuring many different circular objects should give nearly the same value for C/d, even if the objects have different sizes.