Thursday
Thursday: Wedges Make a Rectangle
Rocket traces the window lid onto paper and cuts the circle out. "The window needs a clear plastic sheet. How much sheet?"
"That is area, not circumference," Raven says. "Space inside, not distance around."
She folds the paper circle in half, then in half again, then once more. She cuts along the folds. "Eight wedges."
Rocket lays them points up, points down, points up. "What do you notice?" Raven asks.
"A bumpy rectangle," he says. "As tall as a radius. As long as half the circle's edge."
Nova hovers over the wedges, her light flattening the bumps. "Half the circumference times the radius," she says. "Write it with π."
Rocket writes. "Half of 2πr is πr. Times r. πr²." Nova hums. "Now measure the lids."
From wedges to a formula
Cut a circle into many thin wedges and lay them tip to tip. They make a shape close to a rectangle.
Its height is the radius, r. Its length is half the circumference, half of 2 × π × r, which is π × r.
Area of a rectangle is length × height: π × r × r. We write it r², so A = πr².
Worked example: the plastic lid has radius 5 cm. A = 3.14 × 5 × 5 = 78.5 square centimeters.
Raven's lid table for the window
The crew measured the radius of four lids at the Blueprint Table. These are their own made-up measurements, in centimeters.
| Lid | Radius (cm) | Diameter (cm) | Circumference (cm) | Area (sq cm) |
|---|---|---|---|---|
| small lid | 3 | 6 | 18.84 | 28.26 |
| plastic lid | 5 | 10 | 31.4 | 78.5 |
| can lid | 6 | 12 | 37.68 | 113.04 |
| window lid | 10 | 20 | 62.8 | 314 |
Read a row: the window lid has radius 10 cm. Its circumference is 2 × 3.14 × 10 = 62.8 cm. Its area is 3.14 × 10 × 10 = 314 square centimeters.
Notice the units. Circumference is a length, in centimeters. Area covers space, in square centimeters.
- Read the question.
- Tap your answer.
A circle has radius 3 cm. What is its area in square centimeters? (Use 3.14 for π.)
A circle has radius 5 cm. What is its area in square centimeters? (Use 3.14 for π.)
A circle has radius 6 cm. What is its area in square centimeters? (Use 3.14 for π.)
The window lid has radius 10 cm. How much clear sheet covers it, in square centimeters? (Use 3.14 for π.)- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| The wedges of a circle lay out as a shape close to a rectangle. | ? |
| The near-rectangle is as long as the whole circumference. | ? |
| 3.14 × 5 × 5 = 78.5 | ? |
| Area is measured in square centimeters, circumference in centimeters. | ? |
Lay the eight wedges point up, point down. Measure the height and the length of your bumpy rectangle.
Pick up every paper scrap when you finish so no one slips.
Half the circumference times the radius. Tomorrow you find circles everywhere and review the week.