how many right angles are there in a grid of 10 x 10 squares?

Question

Apparently, there are more than 400 so the teacher must be asking for all additional squares / rectangles (eg. 2 x 2, 3 x 3, 2 x 1 etc) within the 10 x 10 grid to be included.

Answer 1

You can only count the same right angle once, but each square has 4 right angles. So there are 400 individual right angles. However, if you count the external angles of the whole grid as right angles (reflex angles) there would be 404 right angles. Any squares or rectangles formed by sets of squares can only use the already identified right angles.

Take the top left corner right angle. It can be expressed in different ways if all the vertices of the squares are labelled. So labels along the top edge of the grid could be labelled A₁, A₂, A₃, ..., A₁₁. The labels on the next row would be B₁, B₂, B₃, ..., B₁₁. The bottom edge would be labelled K₁, K₂, K₃, ..., K₁₁. The top left right angle could be referred to as ∠B₁A₁A_2, ∠B₁A₁A₁₁, ∠C₁A₁A₃, ∠K₁A₁A₁₁, etc., but it's still the same angle and can't therefore be counted twice despite its many references. Angle A₁ can have 100 such references depending on which other vertex labels are used to frame it. Angle A₂ can have 90 vertex references; B1 can have 81 references, and so on. However, it still implies only one individual angle. Maybe the tutor was asking for the total number of right angle references rather than the number of right angles.

The answer is different if you were required to find the number of rectangles or squares in the grid.