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Geometry 9-12 / Week 11 / Monday
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Week 11 · Volume: Where the Formulas Come From

Monday

Wedges and stacked discs
// Paint, foam and a jug of water
⏱ about 20 min

Monday: Wedges and Stacked Discs

Comet rolls the bolt of backdrop fabric to the end of the bench to clear a space. "Column next. It needs filling."

Wren stands the cardboard column on the plywood row and reads the tape. "Radius 20, height 90. How much foam fits inside?"

"Base area times height," Comet says. "What can we make of the base? It is a circle, so πr²."

"But why is a circle πr²?" Wren asks. "What do you notice if you cut one into wedges?"

Nova projects a circle slicing itself into sixteen thin wedges, then lays them tip to tail. "Would you like a hint?"

"It is almost a parallelogram," Comet says. "Half the circumference along the bottom, the radius for the height."

"πr times r," Wren says. "Then stack discs to the top. Our designer, that is the whole formula."

Comet unrolls the orange fabric bolt on the bench while Wren checks the plywood row under the honeycomb panel, Nova between them.

Where πr² comes from

Cut a circle into many thin wedges and lay them in a row, points up, points down. The row is almost a parallelogram.

Its base is half the circumference, πr. Its height is the radius, r. So the area is πr × r = πr².

With more and thinner wedges the bumpy edge flattens out. That is an informal limit argument: the fit gets as close as you like.

The column prop: a cylinder with radius 20 cm and height 90 cm.

Where πr²h comes from

A cylinder is a stack of thin discs, each the same circle. Every disc adds the same area, so the volume is base area times height.

Pushing the stack sideways into a leaning tower does not change the volume. Same discs, same heights, same volume.

A solved problem to study

Comet works out the column. Base area: 3.14 × 20 × 20 = 1,256 square centimeters. Use 3.14 for π.

Volume: 1,256 × 90 = 113,040 cubic centimeters of foam.

Why does this work? Each 1 centimeter slice of the column is a disc with that base area. Ninety slices stack to the top.

THE COLUMN PROP
  • Read the question.
  • Tap your answer.
A circle with radius 20 cmThe column's base has radius 20 cm. What is its circumference in centimeters? Use 3.14 for π.
A circle with radius 20 cmThe column's base has radius 20 cm. What is its area in square centimeters? Use 3.14 for π.
A cylinder with radius 20 cm and height 90 cmThe column is a cylinder with radius 20 cm and height 90 cm. What is its volume in cubic centimeters? Use 3.14 for π.
WHY THE FORMULAS WORK
  • Read the question.
  • Tap your answer.
A circle is cut into many thin wedges and laid in a row, tips up and down. What shape is the row close to?
In the wedge parallelogram, what is the length of the base?
A cylinder is thought of as a stack of thin discs. What is each disc's area?
StatementTrue or false?
Thinner wedges make the row closer to a true parallelogram.?
The height of the wedge parallelogram is the diameter.?
The volume of a cylinder is base area times height.?
Leaning a stack of discs sideways changes its volume.?
The column holds more than 100000 cubic centimeters of foam.?
WHY THIS EXERCISEKnowing why a formula works lets you rebuild it when you forget it, and trust it when you use it.
The column's base area is 1,256 square centimeters and its height is 90 cm. What is its volume? Type the number.
WHY THIS EXERCISEBase area times height is the whole cylinder formula, once you know the base is πr².
Try it
On paper, draw a circle and divide it into eight wedges. Cut the wedges out with scissors and lay them tip to tail.
Try sixteen wedges. Which row looks more like a parallelogram?
Draw the column as a stack of thin discs. Label the radius of one disc and the height of the stack.

Strong start, designer. Tomorrow a cube splits into three pyramids and the cone learns its one third.