Track quiz

Nine ideas, nine checks. The table comes first, so you have somewhere to look.

Nine ways to measure a curve or a corner you cannot count squares under
Idea	How it goes
The fixed quotient	Way round ÷ way across is one number for every circle. This track uses 22/7.
Not a fraction	A fraction's decimal ends or repeats. π's decimal does neither.
Measured misses	22/7, 3.1416 and 355/113 are too big by about 0.00126, 0.0000073 and 0.00000027.
The arc	An arc is a share of the way round. The share is the angle's share of a turn.
The wedge strip	The strip is half the way round long. It is the radius wide.
The sector	A sector's area is a share of the circle's area. The share is the angle's share of a turn.
Heron	Half the perimeter. Take each side from it. Multiply the four numbers. Then find the number that squares to the product.
Brahmagupta	Half the perimeter. Take each of the four sides from it. Multiply the four brackets. Then find the number that squares to the product.
Rectangle to square	Carry the width past the long side. Swing a semicircle on the whole length. Raise a perpendicular where the long side ends. That length is the square's side.

A bangle's way round divided by its way across gives one number. What does the same division give for a tank?

A bigger number, since the tank is bigger
A smaller number, since the way across grows faster
The same number, 22/7, since both grow together

Why is every fraction or ending decimal only an approximation of π?

Because π is not a fraction. Each ending decimal is one
Because π's decimal has too many digits to store
Because nobody has measured a circle carefully enough yet

22/7, 3.1416 and 355/113 each miss π. Which order is right?

Each is too small, and the misses grow as the digits do
All too big, each closer: 0.00126, then 0.0000073, then 0.00000027
The misses are equal, and all three are famous

A circle of radius 7 has a way round of 44. A 60° angle sits at its centre. How long is its arc?

6 units
11 units, the same as a quarter turn
About 7.33 units, 44 × 60 ÷ 360

Eight wedges of a disc are laid into a strip. What does this honestly show?

A rectangle with slightly more area than the disc, from the bumps
A strip with the disc's area, still short of a rectangle
A rectangle with exactly the disc's area

A sector of the radius-7 circle holds 90°. Which pair of answers is right?

The arc is 11 units and the sector is 38.5 square units
The arc is 11 units and the sector is 5.5 square units
The arc is 38.5 units and the sector is 11 square units

A triangular field has sides 13, 14 and 15. No height is given. What does Heron's rule give?

Nothing, since there is no height to use
42, since that is the perimeter
84, since 21 × 8 × 7 × 6 = 7056 and 84 × 84 = 7056

A cyclic quadrilateral has sides 3, 4, 5 and 6. What happens to the rule when the last side shrinks to 0?

It becomes Heron's rule, since the last bracket equals the half-perimeter
The rule fails, since a side has vanished
The area stays √360, since the sides still add the same

A bed 8 by 2 becomes a square of the same space. What does the drawing give?

A square of side 5, keeping the edge and changing the space
A square of side 4. It does not keep the bed's edge length
A square of side 10, the whole baseline

Chapter 6, Measuring Space: Perimeter and Area, is next. The nine ideas in the table are the chapter's main moves. Read it and see where each one hides.

Next: The Mathematics of Maybe: Introduction to Probability. This track measured space. That one measures chance, as a share.

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