Area of the overlap?
A hard-level "Area" problem.Spotting "these two match when you rotate" gives you the area without ever knowing the angle. Rotating pieces onto each other is as powerful a tool as shearing.
GivenA second square has one vertex placed at the centre of the 8 cm square and is turned at an angle
Hints
- The shape of the overlap changes as you turn it — but the area does not. The key is that the small square's corner sits exactly at the centre of the big one.
- Draw a cross through the centre, perpendicular to the sides. The big square splits into four identical squares. Focus on one of them (blue).
- Compare the overlap with that quarter square. There is an orange triangle sticking out beyond it, and a red triangle inside it that the overlap has not reached.
- Compare those two triangles. Turn one 90° about O and the orange piece lands exactly on the red one — because the cross and the square both have 90° corners.
- Since what sticks out equals what is missing, the overlap has exactly the area of one quarter square — whatever the angle. 8 × 8 ÷ 4
Answer 16cm²
More "Area" problems
- Area of the inner triangle?
- Area of triangle DOA?
- Area of quadrilateral ABCD?
- Total area of the two crescents?
- Area swept by the square?
- What area of the table is covered by paper?