Problem list
- Area of the inner triangle? Divide all three sides in the same ratio and the middle triangle is always a fixed fraction of the whole — 21/63 here — regardless of the triangle's shape
- Area of triangle DOA? Find a ratio once, then use it again somewhere else in the same figure
- Area of quadrilateral ABCD? Shearing is the tool for "change the shape, keep the area"
- 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
- Area of the overlap? Spotting "these two match when you rotate" gives you the area without ever knowing the angle
- How many cubes have 2 painted faces? For N×N×N: 3 faces is always 8, 2 faces is 12×(N−2), 1 face is 6×(N−2)², 0 faces is (N−2)³
- a + b + c = ? Count the same area two ways — split into three, and all at once
- What is the side of the square? If the legs are a and b, the inscribed square's side is always a×b÷(a+b)
- Total area of the two crescents? Found by Hippocrates 2,200 years ago
- Area swept by the square? "I cannot find the length, but I can find length × length
- Into how many pieces do 4 lines cut the paper? Do not try to count the regions
- What area of the table is covered by paper? Overlap problems are really about how many times each region got counted
- How long is MN? "Join the midpoints of two sides and you get half the third side
- How big is angle x? Repeat one move and the numbers line up
- How many degrees is angle x? Every folding problem runs on two facts: overlapping angles are equal, and the strip's parallel edges give equal alternate angles
- Area of the red part? This shape is called the "shoemaker's knife"
- Area of the overlap? "Add them up, then subtract the excess
- How long is the rope? Three logs or four, the curved part of a rope going once round is always one full circle
- Area of the shaded triangle? A regular hexagon is "six equilateral triangles"
- How much larger is イ than ア? When a problem asks for a difference, you may add or subtract the same thing from both sides — which lets you delete an awkward shared region without ever computing it
- How long is the pole's shadow? The sun casts parallel rays; a lamp spreads from a point
- How deep was the water? Every tilting problem rests on one fact: the volume does not change
- How far does the centre travel? However many sides the polygon has, the corner arcs always make one full circle
- In how many ways can the three chosen rods form a triangle? Whether three lengths make a triangle rests on one test: the longest must be less than the other two added
- What is the volume of the bottom part? Double the lengths and areas quadruple, volumes go up eightfold
- What is the area of the folded-over triangle? Folding changes none of the area, the lengths or the angles
- What is A＋B＋C＋D＋E? Nowhere does this argument use "because it is regular"
- Volume of the solid made by one full turn? For a solid of revolution, "the farthest point from the axis is the radius"
- Area of triangle ADE? Know the ratios on two sides and the triangle between them is just two multiplications
- Area of the red triangle? It works wherever the point is, and whatever shape the rectangle has
- How many triangles are there in the figure? To count shapes, do not chase them with your eyes — find out what has to be chosen to pin one down
- How long is the line from B down to AC? Write the same area two ways — with two different base-and-height pairs — and set them equal
- How big is angle BIC? You never learn B and C separately — but knowing their sum is enough to know the sum of their halves
- How many degrees is angle x? The dented vertex equals the sum of the other three angles
- Area of the shaded triangle? One median halves it, three medians sixth it — always, whatever the triangle's shape
- How many squares are there altogether? Do not count in the order you spot things — count in the order they are determined
- Area of the square? Chasing the side length is a dead end at elementary level
- Area of the shaded part? Spot a pair of parallel lines and try sliding a vertex
- Area of the lawn? Even if the path zigzags, as long as its width is the same the answer does not change
- How deep will the water be? Levelling problems are always "all the water ÷ total base area"
- How far from the left wall did it bounce? For a bounce, reflect and straighten
- What is the largest area it can enclose? The same perimeter can enclose wildly different areas
- 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? A regular pentagon is built out of just 108° and 36°
- Area of the sector? "Arc × radius ÷ 2" works without knowing the central angle
- Area of trapezoid ABCD? It is no accident that 9 and 25 are square numbers
- What is the volume of the solid produced? A solid of revolution is just cylinders added step by step
- How far is O, where the diagonals cross, from AD? The "hourglass" made by crossing diagonals is always similar, as long as there are parallel sides
- Where does the halving line cross the bottom edge? A rectangle is halved by any line through its centre
- Area of the path? For any path of constant width — circular, square or wiggly — the area is "length of the centre line × width"
- Total area of the blue and red triangles? Put P anywhere inside and the two opposite triangles always sum to half
- 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
- Area of the inner quadrilateral? However lopsided the original quadrilateral, joining the midpoints always gives a parallelogram, and its area is always half
- How many squares does the diagonal pass through? Counting crossings is the quick way
- What is the angle at the centre of the sector? The whole of a cone's net rests on one fact: arc = base circumference
- Area of the inner square? A tilted square is "the boxing square minus the four corner triangles"
- What is the total area of the three sectors? When the parts cannot be found one by one, look only at the total
- What is the perimeter of the hexagon? 3
Area Problems Angle Problems Length Problems Volume Problems Counting Problems Standard Problems