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"
- Area of the overlap? Spotting "these two match when you rotate" gives you the area without ever knowing the angle
- 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
- What area of the table is covered by paper? Overlap problems are really about how many times each region got counted
- Area of the red part? This shape is called the "shoemaker's knife"
- Area of the overlap? "Add them up, then subtract the excess
- 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
- What is the area of the folded-over triangle? Folding changes none of the area, the lengths or the angles
- 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
- Area of the shaded triangle? One median halves it, three medians sixth it — always, whatever the triangle's shape
- 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
- 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
- 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
- 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
- Area of the inner quadrilateral? However lopsided the original quadrilateral, joining the midpoints always gives a parallelogram, and its area is always half
- 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
- 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
- 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
- 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
- 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 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 this triangle? For any slanted shape: box it in, then subtract the right triangles around it
- Area of triangle ABD? "Shared apex, bases on one line" means the area ratio equals the base ratio
- Area of the trapezoid? The "÷2" in the formula is there because flipping a second copy of the trapezoid makes a parallelogram
- Area of triangle APD? Wherever P sits on BC, the answer is the same
- Area of this triangle? "Sliding a vertex along a parallel line does not change the area" — this is shearing
- Area of the red part? With scattered sectors, add up the central angles first
Angle Problems Length Problems Volume Problems Counting Problems Hard Problems Standard Problems