Why measuring is a little off

The idea

Geometry studies ideal lines, which have no thickness, and finds their relationships by reasoning rather than measurement; actual measurements may differ slightly because of measuring errors and the thickness of drawn lines.

Rani gets 181°

Rani draws a straight line and another line across it. She measures the linear pair with a protractor: 118° and 63°. Together: 118° + 63° = 181°.

We reasoned that a linear pair adds up to exactly 180°. So why did Rani get 181°?

• The straight angle is really 181° on her page
• The lines she drew are thick and the readings are a little off
• Her protractor must be broken

Zoom in on a thick line

A drawn line is never infinitely thin. Zoom in on the protractor scale to see what that does to a reading.

Ideal lines and drawn lines

On a thick line the pointer still touches at 119°, 120° and 121°, so the reading could be any of them. The linear partner ∠b (60°) reads 59°, 60° or 61°. So a measured linear pair can add up to anything from 178° to 182°.

Geometry studies ideal lines, which have no thickness, and finds how they are related by reasoning. Measurements on drawn lines can differ a little, because of measuring errors and the thickness of the lines.

• Ideal line | Drawn line
• Has no thickness | Has some thickness
• Exactly 180° in a linear pair | Measures close to 180°
• Angles found by reasoning | Angles found by measuring

Notes

Geometry studies ideal lines, which have no thickness, and finds how they are related by reasoning. Measurements on drawn lines can differ a little, because of measuring errors and the thickness of the lines.

Check yourself

Meena draws two crossing lines and measures the vertically opposite angles ∠b and ∠d. She gets 62° and 59°. She says: “So ∠b = ∠d is false.” Is she right?

In an ideal crossing ∠a = 120°. A protractor shows its linear partner ∠b as 61°. By reasoning, what is ∠b exactly?

Answer: 60 °

∠a + ∠b = 180°, so ∠b = 180° − 120° = 60° exactly. The 61° on the protractor is about 1° off.

Which describes an ideal line?

Measured angles are only close to what reasoning predicts. Why is geometry still used in building, art and science?

• Yes. What the protractor shows is more reliable than reasoning.. The reasoning for ∠b = ∠d uses no measurement at all. A gap of 3° is only the roughness of drawing and measuring.
• No. Ideal lines give exactly equal angles. Thick lines and measuring errors explain the gap. — correct. Yes! Reasoning with ideal lines gives exactly ∠b = ∠d. A drawing is thick and a protractor reading is never perfect, so a small gap is expected.
• Yes. A gap of 3° means the two angles are really different.. For ideal lines the angles are exactly equal. The 3° gap comes from the drawing and the protractor.
• It is only an idea: it has no thickness at all. — correct. Yes! An ideal line has no thickness, so it crosses a protractor scale at exactly one mark.
• It is as thin as the sharpest pencil can draw it.. Even a very sharp pencil leaves a line with some thickness. An ideal line has none.
• It is a line drawn neatly with a ruler.. A ruler makes a line straight, but the drawn line still has some thickness.
• Measuring is exact, so reasoning is not needed.. No measurement is exact. The zoomed protractor showed that.
• Drawn lines are ideal, so they always agree with reasoning.. Drawn lines have thickness, so measurements and reasoning can differ a little.
• Real measurements come very close to what reasoning predicts. — correct. Yes! The measurements land very close to the predictions, so geometry is used in physics, art, engineering and architecture.