Demonstration 1 of 4
The ratio that never changes
Why does circumference divided by diameter give the same number for a small circle and a large one?
Every circle is drawn the same size here because only the ratio matters. A larger diameter brings a proportionally larger circumference, so C / d stays π. A chord is shorter than the diameter, so the same circumference divided by it is larger.
\[π = C/d\]
\[C = πd\]
C is the circumference, the length once around the boundary. d is the diameter, a segment through the center. π is their exact ratio; 3.14 is only a rounded display of it.
Predict first. Switch from the diameter to a chord that misses the center. Will C divided by that segment be bigger than π or smaller?
Choose an example
Constructed example: the chapter's ideal circles of diameter 2 and 4 and the 1.2 m tabletop.
Calculated values
- Diameter d
- 1.2 m
- Circumference C (exact)
- 1.2π m
- Circumference C (nearest hundredth)
- 3.77 m
- Segment used
- diameter
- C / d
- 1.2π m / 1.2 m = π (no unit)
C / d = 1.2π / 1.2 = π, about 3.14159. The circle has diameter 1.2, so C = π x 1.2 = 1.2π, shown as 3.77. Pencil check: 3.14159 x 1.2 = 3.77 to the nearest hundredth. Change the size and the ratio stays π.
Use the idea
If you know a round lid's diameter, multiply by π for the length of trim that wraps it once; keep π exact until you choose a display.
Where the conclusion applies
An ideal Euclidean circle. A real lid with an uneven edge only fits the model approximately. The ratio fails if the segment does not pass through the center; the chord shown sits 0.6 radius from the center, so its length is 0.8 of the diameter.
Check your understanding: A circle has diameter 10 units. What is its exact circumference, and what is C / d?
Chapter 19 source: section "Pi is the ratio, not the rounded display". Demonstration C19-D01.
Demonstration 2 of 4
Recover the radius first
Given only the circumference, how do you find the area without slipping a diameter into πr²?
Dividing the circumference by 2π gives the radius. Squaring the radius gives the area. Squaring the diameter, which is twice the radius, multiplies the area by 2 x 2 = 4.
\[2\pi r=14\pi\]
\[\pi(7)^2=49\pi\]
r is the radius, from the center to the edge. 2πr is the circumference. πr² is the area inside, in square centimeters.
Predict first. If the diameter is squared instead of the radius, how many times too large will the area be?
Choose an example
Constructed example: the chapter's worked example with circumference 14π centimeters, plus two other circumferences.
Calculated values
- Circumference
- 14π cm
- Radius from 2πr = C
- 14π / 2π = 7 cm
- Diameter
- 14 cm
- Area π x (length used)²
- π x 7² = 49π cm²
- Correct area
- 49π cm²
2πr = 14π, so r = 14 / 2 = 7 cm. Area = π x 7² = 49π cm². Check: 2π x 7 = 14π.
Use the idea
Before using a formula, write the name of the length you were given beside its number: radius or diameter.
Where the conclusion applies
An ideal circle with an exact circumference written as a multiple of π. The method needs 2π, which is not zero, to divide by.
Check your understanding: A circle has circumference 20π centimeters. What are its radius and its area?
Chapter 19 source: section "Worked example: recover the radius before finding area". Demonstration C19-D02.
Demonstration 3 of 4
Double the radius
When the tabletop's radius doubles, why does the area grow more than the circumference?
Scaling every length by a factor multiplies the circumference once by that factor. The area contains r twice, so it is multiplied by the factor squared.
\[A = πr²\]
r is the radius in meters. Circumference 2πr is a length in meters. Area πr² is in square meters, because the radius is used twice.
Predict first. If the radius triples, by what factor does the area grow?
Choose an example
Constructed example: the chapter's 1.2 m tabletop (radius 0.6 m) and its doubling argument.
Calculated values
- Scale factor on lengths
- 2
- Circumference
- 1.2π m to 2.4π m (x 2)
- Area
- 0.36π m² to 1.44π m² (x 4)
- Area factor
- 2² = 4
New radius 2 x 0.6 = 1.2 m. Circumference 2π x 1.2 = 2.4π m, which is 2 times 1.2π. Area π x 1.2² = 1.44π m², which is 2² = 4 times 0.36π. Lengths take one factor of 2; area takes two.
Use the idea
A quick magnitude check: if a radius doubles and a reported area only doubles, the square was probably dropped.
Where the conclusion applies
Ideal circles scaled in every direction by the same factor. Thickness, material and any noncircular details of a real table are outside the formulas.
Check your understanding: A circle's radius is multiplied by 5. By what factors do its circumference and area change?
Chapter 19 source: section "Doubling the radius changes two quantities differently". Demonstration C19-D03.
Demonstration 4 of 4
Measure an angle by its arc
How long is the arc reached by a central angle of 120 degrees?
Convert degrees to a fraction of a turn, then to radians. Multiplying the radius by the radian count gives the arc. A full turn returns the whole circumference.
\[s = rθ\]
\[θ = s/r\]
θ is the central angle in radians. r is the radius. s is the length of the arc the angle reaches. A full turn is 2π radians, which is 360 degrees.
Predict first. At 360 degrees on a radius 3 circle, what should s equal?
Choose an example
Constructed example: the chapter's 120 degree conversion and the full turn check, on circles of declared radius.
Calculated values
- Fraction of a turn
- 120/360 = 1/3
- θ in radians
- 1/3 x 2π = 2π/3
- Arc length s = rθ
- 3 x 2π/3 = 2π ≈ 6.28
- Full circumference
- 2π x 3 = 6π
120/360 = 1/3 of a turn, so θ = 1/3 x 2π = 2π/3 radians. Then s = rθ = 3 x 2π/3 = 2π. Check by fraction of the circumference: 1/3 x 6π = 2π.
Use the idea
For any arc, say its fraction of a full turn first; the arc is that fraction of the circumference.
Where the conclusion applies
The angle's vertex is at the center of an ideal circle. s = rθ needs θ in radians: putting 120 in for θ without converting would treat degrees as radians.
Check your understanding: On a circle of radius 6, how long is the arc for a central angle of 60 degrees?
Chapter 19 source: section "An angle can be measured by the arc it reaches". Demonstration C19-D04.