← Back to course
Geometry 9-12 / Week 11 / Thursday
4/6
Week 11 · Volume: Where the Formulas Come From

Thursday

The prop table
// Paint, foam and a jug of water
⏱ about 20 min

Thursday: The Prop Table

Wren pins a page to the corkboard: five props, five shapes, five volumes. "Everything we carved this month."

"The column is the big one," Comet says, scanning it. "What do you notice about the ball and the pyramid?"

"Close," Wren says. "The ball is a bit bigger, and it is only 15 across each way from the center."

"Spheres are sneaky," Comet says. "The r is cubed. What can we make of the scrap from the ball?"

Nova projects the foam block the ball was carved from, a box 40 by 30 by 30. "Would you like a hint? Box minus ball."

"And the model props are half size," Wren says. "Every length doubles to full size. Our designer, what happens to the volume?"

"Not double," Comet says. "Work it out and see."

The crew's prop volumes

Every number here is the crew's own Scene Shop measurement in centimeters. Volumes use 3.14 for π.

PropShapeMeasurements (cm)Volume (cubic cm)
columncylinderr = 20, h = 90113,040
foam pyramidsquare pyramidbase 30 by 30, h = 4513,500
foam coneconer = 15, h = 409,420
foam ballspherer = 1514,130
stair unitprismB = 1200, h = 4048,000

The column has the largest volume, 113,040 cubic centimeters.

The foam ball was carved from a box 40 by 30 by 30. Box 36,000 minus ball 14,130 leaves 21,870 cubic centimeters of scrap.

The foam block the ball was carved from: 40 by 30 by 30 cm.

Scaling a prop

The crew's model props are half size. Doubling every length doubles the radius, doubles the height, and doubles the depth.

Three lengths multiply in a volume, so the volume is multiplied by 2 × 2 × 2 = 8. A model cone of 1,177.5 cubic centimeters becomes 9,420 at full size.

READ THE PROP TABLE
  • Read the question.
  • Tap your answer.
A cylinder with radius 20 cm and height 90 cmThe column: radius 20 cm, height 90 cm. What is its volume in cubic centimeters? Use 3.14 for π.
A sphere with radius 15 cmThe foam ball: radius 15 cm. What is its volume in cubic centimeters? Use 3.14 for π.
The foam block is 36,000 cubic centimeters and the ball carved from it is 14,130. How much scrap is left?
HALF SIZE TO FULL SIZE
  • Read the question.
  • Tap your answer.
Every length of a model prop is doubled to full size. What is the volume multiplied by?
A second theater builds the props at three times every length. What is the volume multiplied by?
The full-size cone is 9,420 cubic centimeters. What is the half-size model's volume?
Which prop has the largest volume? Type its name.
WHY THIS EXERCISEA table of volumes tells the crew which prop needs the most foam before any carving starts.
The stair unit prism has base area 1200 and height 40. What is its volume? Type the number.
The foam pyramid is 13,500 cubic centimeters. What is the volume of the prism with the same base and height? Type the number.
StatementTrue or false?
The foam ball has a larger volume than the foam pyramid.?
Doubling every length doubles the volume.?
Doubling every length multiplies the volume by 8.?
The cone and the pyramid together have more volume than the stair unit.?
WHY THIS EXERCISEReading and comparing volumes is how the crew decides what to carve from which block.
Draw the foam block with the ball inside it. Mark the radius of the ball and the three edges of the block.

Sharp reading, designer. Tomorrow you slice solids, spin flat shapes and review the week.

← Wednesday