โญ Circles: Circumference
Measuring around a circle
๐ฏ Introduction
Circles are everywhere - wheels, pizzas, clocks, coins! Let's learn how to measure around a circle. This might surprise you, but it's actually a proportional relationship!
The circumference (distance around a circle) is always ฯ times the diameter. This makes circumference and diameter proportional!
๐ Visual Guide
๐ก Key Concepts
Parts of a Circle
A circle has special parts with special names: the center (middle point), radius (center to edge), diameter (edge to edge through center), and circumference (distance around).
๐ Real Example:
On a bicycle wheel: the axle is at the center, a spoke is like a radius, and the tire is the circumference.
๐ญ Radius is like a spoke going from the center to the edge. Diameter is like a spoke that goes ALL the way across, passing through the center.
Radius and Diameter Relationship
The diameter is TWICE the radius. Or: the radius is HALF the diameter. d = 2r, or r = d/2
๐ Real Example:
If a pizza has a radius of 6 inches (center to crust), its diameter is 12 inches (crust to crust through center).
๐ญ Think of diameter as two radii (radiuses) placed end to end.
Pi (ฯ) - The Magic Number
Pi is a special number (approximately 3.14) that shows up whenever you work with circles. The circumference divided by the diameter ALWAYS equals pi!
๐ Real Example:
Take ANY circle - a quarter, a frisbee, Earth! - and divide its circumference by its diameter. You'll always get about 3.14!
๐ญ ฯ โ 3.14159... It goes on forever, but we usually just use 3.14 or the ฯ button on a calculator.
Circumference Formula
C = ฯd (Circumference = pi times diameter) or C = 2ฯr (Circumference = 2 times pi times radius). Both work!
๐ Real Example:
A bike wheel with diameter 26 inches: C = ฯ ร 26 โ 3.14 ร 26 โ 81.64 inches around.
๐ญ The circumference is a little more than 3 times the diameter. If you rolled a circle on a ruler, it would travel about 3.14 diameters.
๐ Step-by-Step Examples
Example 1: Finding circumference from diameter
๐ Problem: A cereal bowl has a diameter of 16 cm. What is its circumference?
Write the formula
C = ฯd
Substitute the diameter
C = ฯ ร 16
Calculate (use 3.14 for ฯ)
C = 3.14 ร 16 = 50.24 cm
Or leave in terms of ฯ
C = 16ฯ cm (exact answer)
โ Answer: About 50.24 cm (or exactly 16ฯ cm)
Example 2: Finding circumference from radius
๐ Problem: A circular garden has a radius of 7 feet. What is the circumference?
Write the formula using radius
C = 2ฯr
Substitute the radius
C = 2 ร ฯ ร 7
Simplify
C = 14ฯ
Calculate using 3.14
C = 14 ร 3.14 = 43.96 feet
โ Answer: About 43.96 feet (or exactly 14ฯ feet)
Example 3: Finding diameter from circumference
๐ Problem: A circular track has a circumference of 400 meters. What is its diameter?
Write the formula
C = ฯd
Solve for d by dividing both sides by ฯ
d = C รท ฯ
Substitute and calculate
d = 400 รท 3.14 โ 127.39 meters
โ Answer: About 127.39 meters
โ ๏ธ Common Mistakes to Avoid
โ Confusing radius and diameter
Why it's wrong: Using radius in the diameter formula (or vice versa) gives an answer that's off by a factor of 2.
โ How to avoid: Always check: diameter goes ALL the way across through the center. Radius is only HALF of that.
โ Forgetting to multiply by 2 when using radius
Why it's wrong: C = ฯr is WRONG! You need C = 2ฯr when using radius.
โ How to avoid: Remember: C = ฯd OR C = 2ฯr. If you use r, don't forget the 2!
โ Using the wrong value for ฯ
Why it's wrong: ฯ is not exactly 3! Using 3 instead of 3.14 will make your answers too small.
โ How to avoid: Use 3.14 (or your calculator's ฯ button) unless told otherwise.
You've unlocked the secrets of circles! ๐ต Pi is one of the coolest numbers in all of math. It shows up everywhere in nature and science!