Problem list（20 problems）
- Area of the circle? π is simply "how many radius-squares fit in a circle"
- Area of the sector? "Arc × radius ÷ 2" works without knowing the central angle
- Perimeter of the sector? Forgetting to add the two radii is by far the most common mistake in perimeter problems
- How long is the whole spiral? Multiply by π once, at the very end
- Area of the red part? "Take the whole, subtract what you don't want
- Area of the red part? With scattered sectors, add up the central angles first
- What is the total area of the three sectors? When the parts cannot be found one by one, look only at the total
- The circle's area is what percentage of the square's? When a question asks for a percentage, try writing the side as a letter
- How much wider is the big circle than the two small ones? Double the length and the area quadruples; triple it and it goes up ninefold
- Area of the overlap? "Add them up, then subtract the excess
- Total area of the two crescents? Found by Hippocrates 2,200 years ago
- Area of the red part? This shape is called the "shoemaker's knife"
- If the rope is moved 1 m outwards, how much longer must it be? The famous result that the answer does not depend on the original radius
- How far does the centre travel? However many sides the polygon has, the corner arcs always make one full circle
- How many turns does the small circle make? A rolling circle's turn count comes from the path of its centre, not the length it touched
- How long is the rope? Three logs or four, the curved part of a rope going once round is always one full circle
- What is the perimeter of the hexagon? 3
- What is the angle at the centre of the sector? The whole of a cone's net rests on one fact: arc = base circumference
- What is the total surface area? A curved surface flattens out if you cut it open: a cylinder's side becomes a rectangle, a cone's becomes a sector
- How many degrees is angle x? When a triangle appears inside a circle, hunt for "all radii are equal" first
Other ideas
- Shearing — moving a shape without changing its area
- Ratios — answers without fixing a single number
- Add, then subtract — an overlap is a counting problem
- Chasing angles — there are only four tools
- Decide how to count, then count
- Flatten the solid before you think about it
- Write the same thing two different ways
- Fold it, turn it, copy it across
- Back to why the formula works