Problem list
- Area of the circle? π is simply "how many radius-squares fit in a circle"
- Area of the overlap? "Sum of the sheets − what you see = the overlap
- Volume of the stone? However complicated the shape, sinking it in water measures its volume — the very thing Archimedes noticed in his bath
- How long is DC? "Same height, so area ratio = base ratio" works in both directions — lengths to areas, and areas back to lengths
- How many degrees is angle x? "Exterior angle = sum of the two non-adjacent interior angles
- Volume of the cuboid? "Base area × height" works for any prism — square, triangular, or circular
- How many matches are needed? "The first one is special; after that the growth is steady
- What is the volume of the box? In cut-and-fold problems the loss happens at both ends
- Area of triangle ABC? "See 30°, halve the slanted side
- 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
- Volume of the cylinder? Cylinders and prisms alike are "base area × height"
- Area of this shape? Most area problems have two or more routes
- Area of triangle ABC? Either diagonal halves it — parallelogram, rectangle, rhombus or square, it makes no difference
- Area of the red part? "Take the whole, subtract what you don't want
- How long is the whole spiral? Multiply by π once, at the very end
- Perimeter of this shape? However many steps there are, the answer is the same
- Area of the rhombus? Any quadrilateral whose diagonals cross at right angles (rhombus, square, kite) obeys this same formula
- What is the total surface area? For surface area, do not count six separate faces — see three pairs of identical faces
- How many degrees is angle x? When a triangle appears inside a circle, hunt for "all radii are equal" first
- How many degrees is angle x? Any zigzag between parallel lines yields to one move: draw a parallel line through the bend
- How many degrees is angle x? The angles of a triangle add to 180° because cutting them off and lining them up makes a straight line
- How tall is the tree? Getting a height you cannot measure from a shadow you can
- What do the three "?" faces add up to? Hunting for which faces are opposite means fresh work for every net
- 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
- How long is DE? The trick with ratios is to convert to "the whole of AB", not DB
- Perimeter of the sector? Forgetting to add the two radii is by far the most common mistake in perimeter problems
- Area of this triangle? For any slanted shape: box it in, then subtract the right triangles around it
- At 3:40, what angle do the hands make? Six degrees a minute for the minute hand, half a degree for the hour hand
- How many diagonals can be drawn in total? Count from one corner, multiply by the number of corners, then halve
- How many degrees is angle x? "Vertically opposite angles are equal" is not a rule to memorise — you can rebuild it from "a straight line is 180°"
- Area of triangle ABD? "Shared apex, bases on one line" means the area ratio equals the base ratio
- How big is angle AED? Whenever a square meets an equilateral triangle, start from the 30° left over between 90° and 60°
- How many degrees is the gap angle x? Equilateral triangles, squares and regular hexagons tile the plane; regular pentagons cannot — 108° does not divide 360° exactly
- Area of the trapezoid? The "÷2" in the formula is there because flipping a second copy of the trapezoid makes a parallelogram
- How many lines of symmetry does it have? Many people count only the corner-to-corner lines and stop at half the answer
- Area of triangle APD? Wherever P sits on BC, the answer is the same
- Through what angle must it turn to match itself again? The turn that brings a shape back onto itself is "a full turn ÷ the number of corners"
- How many degrees is angle x? "Equal sides mean equal angles
- Area of this triangle? "Sliding a vertex along a parallel line does not change the area" — this is shearing
- Sum of the interior angles of a hexagon? The formula is 180 × (corners − 2)
- Volume of the cone? Any pointed solid is one third of the prism with the same base and height — pyramids and cones alike
- Area of the red part? With scattered sectors, add up the central angles first
Area Problems Angle Problems Length Problems Volume Problems Counting Problems Hard Problems