Geometry Drills
One hundred geometry problems of the kind set by Japanese junior-high entrance exams. Getting the answer is not the main point: this site exists to widen the range of ways you can look at a figure.
One problem, one idea. No padding the count by changing the numbers in the same problem. A hundred problems means a hundred different ways of seeing.
Press a hint and one more line is drawn onto the figure. Unlike a wall of prose, you can watch where the construction line goes and why it goes there. Pick up from wherever you got stuck.
Type an answer and it is checked on the spot, along with how many hints you used. Available in Japanese, English and Chinese.
Browse by kind
- Area Area problems rarely stop at plugging into a formula. Cuttin…
- Angles Angle problems look like simple addition and subtraction, ye…
- Length Lengths get interesting when they cannot be measured. Copy t…
- Volume Solids stop being frightening once they are flattened. Slice…
- Counting Chase them with your eyes and you will always miss some. Fin…
- Hard Problems with no obvious first move. Every one of them colla…
- Normal Problems where you know the formula but cannot say why it wo…
Browse by idea
- Shearing — moving a shape without changing its area
- Ratios — answers without fixing a single number
- Add, then subtract — an overlap is a counting problem
- Circles and sectors — multiply by π once, at the very end
- 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
All 100 problems
- Area of the inner triangle?（Hard）
- Area of triangle DOA?（Hard）
- Area of quadrilateral ABCD?（Hard）
- How many turns does the small circle make?（Hard）
- Area of the overlap?（Hard）
- How many cubes have 2 painted faces?（Hard）
- a + b + c = ?（Hard）
- What is the side of the square?（Hard）
- Total area of the two crescents?（Hard）
- Area swept by the square?（Hard）
- Into how many pieces do 4 lines cut the paper?（Hard）
- What area of the table is covered by paper?（Hard）
- How long is MN?（Hard）
- How big is angle x?（Hard）
- How many degrees is angle x?（Hard）
- Area of the red part?（Hard）
- Area of the overlap?（Hard）
- How long is the rope?（Hard）
- Area of the shaded triangle?（Hard）
- How much larger is イ than ア?（Hard）
- How long is the pole's shadow?（Hard）
- How deep was the water?（Hard）
- How far does the centre travel?（Hard）
- In how many ways can the three chosen rods form a triangle?（Hard）
- What is the volume of the bottom part?（Hard）
- What is the area of the folded-over triangle?（Hard）
- What is A＋B＋C＋D＋E?（Hard）
- Volume of the solid made by one full turn?（Hard）
- Area of triangle ADE?（Hard）
- Area of the red triangle?（Hard）
- How many triangles are there in the figure?（Hard）
- How long is the line from B down to AC?（Hard）
- How big is angle BIC?（Hard）
- How many degrees is angle x?（Hard）
- Area of the shaded triangle?（Hard）
- How many squares are there altogether?（Hard）
- Area of the square?（Hard）
- Area of the shaded part?（Hard）
- Area of the lawn?（Hard）
- How deep will the water be?（Hard）
- How far from the left wall did it bounce?（Hard）
- What is the largest area it can enclose?（Hard）
- What is the total surface area?（Hard）
- How many degrees is angle x?（Hard）
- Area of the sector?（Hard）
- Area of trapezoid ABCD?（Hard）
- What is the volume of the solid produced?（Hard）
- How far is O, where the diagonals cross, from AD?（Hard）
- Where does the halving line cross the bottom edge?（Hard）
- Area of the path?（Hard）
- Total area of the blue and red triangles?（Hard）
- If the rope is moved 1 m outwards, how much longer must it be?（Hard）
- Area of the inner quadrilateral?（Hard）
- How many squares does the diagonal pass through?（Hard）
- What is the angle at the centre of the sector?（Hard）
- Area of the inner square?（Hard）
- What is the total area of the three sectors?（Hard）
- What is the perimeter of the hexagon?（Hard）
- Area of the circle?（Normal）
- Area of the overlap?（Normal）
- Volume of the stone?（Normal）
- How long is DC?（Normal）
- How many degrees is angle x?（Normal）
- Volume of the cuboid?（Normal）
- How many matches are needed?（Normal）
- What is the volume of the box?（Normal）
- Area of triangle ABC?（Normal）
- The circle's area is what percentage of the square's?（Normal）
- Volume of the cylinder?（Normal）
- Area of this shape?（Normal）
- Area of triangle ABC?（Normal）
- Area of the red part?（Normal）
- How long is the whole spiral?（Normal）
- Perimeter of this shape?（Normal）
- Area of the rhombus?（Normal）
- What is the total surface area?（Normal）
- How many degrees is angle x?（Normal）
- How many degrees is angle x?（Normal）
- How many degrees is angle x?（Normal）
- How tall is the tree?（Normal）
- What do the three "?" faces add up to?（Normal）
- How much wider is the big circle than the two small ones?（Normal）
- How long is DE?（Normal）
- Perimeter of the sector?（Normal）
- Area of this triangle?（Normal）
- At 3:40, what angle do the hands make?（Normal）
- How many diagonals can be drawn in total?（Normal）
- How many degrees is angle x?（Normal）
- Area of triangle ABD?（Normal）
- How big is angle AED?（Normal）
- How many degrees is the gap angle x?（Normal）
- Area of the trapezoid?（Normal）
- How many lines of symmetry does it have?（Normal）
- Area of triangle APD?（Normal）
- Through what angle must it turn to match itself again?（Normal）
- How many degrees is angle x?（Normal）
- Area of this triangle?（Normal）
- Sum of the interior angles of a hexagon?（Normal）
- Volume of the cone?（Normal）
- Area of the red part?（Normal）