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Geometry 9-12 / Week 06 / Wednesday
3/6
Week 06 · Similar Triangles and Proof

Wednesday

Shop Lab: the afternoon of shadows
// Shadows, side-splitters and the altitude
⏱ about 20 min

Wednesday: Shop Lab, the Afternoon of Shadows

Comet chalks a mark at the tip of the pole's shadow. "Two o'clock. Pole shadow 4.0, stick shadow 1.0."

"Same height as Monday?" Wren asks. "What does the evidence say?"

"One times four over one. Four meters. Same pole." Comet grins.

An hour later the shadows have stretched. "Stick 1.5, pole 6.0," Wren reads. "Four meters again."

Nova projects the three pairs of triangles side by side, each pair a different shape. "Would you like a hint?"

"The shape changes with the hour," Comet says. "But at any one moment the two triangles match each other."

"That is the whole lab," Wren says. "Our designer, pick a sunny hour and measure something tall."

What you need

  • A meter stick or a ruler to stand upright, a tape measure, chalk or a pebble to mark shadow tips.
  • Something tall with a shadow on flat ground: a post, a pole, a basketball hoop or a tall fence.
  • A sunny hour, your Shop Book and a helper to hold the stick straight.
Safety first
Never look at the Sun, even for a moment. Look at the shadows on the ground only.
Never climb the object you are measuring. The shadow method is how you avoid climbing.
Stay off the road or a driveway while measuring. Nothing heavy is lifted, and no ladder is used.

Run the lab

  1. Stand the stick upright and mark its shadow tip. Measure the shadow at once, before it moves.
  2. Mark and measure the tall object's shadow at the same moment, from its base to the tip.
  3. Divide the object's shadow by the stick's shadow. Multiply the stick's height by that factor.
  4. Repeat an hour later, and again an hour after that. Compare the three heights you get.

The crew's shadow table

The meter stick is 1 m tall. These are the crew's own readings, in meters.

WhenStick shadowPole shadowPole ÷ stickPole height
Monday, late afternoon0.83.244
Wednesday, mid afternoon1444
Wednesday, late afternoon1.5644

Three different afternoons and hours, three different triangle shapes, one pole height. The method does not depend on the hour.

It does depend on measuring both shadows at the same moment. The angle at the shadow tips changes as the afternoon goes on.

READ THE SHADOW TABLE
  • Read the question.
  • Tap your answer.
Mid afternoon: stick shadow 1 m, pole shadow 4 m. How tall is the pole, in meters?
Late afternoon the pole's shadow was 6 m and the stick's was 1.5 m. What is the scale factor from stick to pole?
Mid afternoon a paint can rack cast a 2.5 m shadow while the stick's was 1 m. How tall is the rack, in meters?
Late afternoon the stick's shadow was 1.5 m. A stack of flats cast a 3 m shadow. How tall is the stack, in meters? Type the number.
WHY THIS EXERCISEOne stick shadow at the right moment measures everything in the yard.
What the lab showsTrue or false?
The pole height came out the same at every hour.?
The shadow triangles had the same shape at every hour.?
6 ÷ 1.5 = 4 ÷ 1?
Both shadows must be measured at the same moment.?
WHY THIS EXERCISEKnowing which thing must stay fixed is what makes a measurement method trustworthy.
Try it
Measure your own shadow and the stick's shadow at the same moment. Work out your height from them.
Compare with a tape measure. How close did the shadow method come?
Draw the stick and the pole with their shadows at one moment, both triangles labeled with your own numbers.

Fine lab work, designer. Tomorrow is Proof Day: the altitude of a right triangle proves the Pythagorean theorem.

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