Circles and disks: a constant ratio and measuring a curve

Where we are on the map

Distance from one point now defines an entire curve.

Begin with a familiar image

Any wheel travels one circumference per revolution, and circumference divided by diameter is constant.

Precise meaning

A circle is the boundary; a disk includes its interior. d=2r, C=2πr=πd, and A=πr².

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

center → r or d → boundary or area? → exact π form → approximation and units

Worked example

For r=4 cm, d=8 cm, C=8π≈25.13 cm, and A=16π≈50.27 cm².

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Writing A=2πr is wrong: that measures the boundary; area is πr².

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. A circular flower bed has radius 3 m and a 1 m path around it. Find the path area as a difference of disks.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

Right triangles inside circles lead to sine, cosine, and tangent.

Course contents