A rectangle's area and perimeter, and a box's volume.
Area is length times width; perimeter is twice their sum: S = a × b and P = 2(a + b). For a room of 4 × 3 metres the area is 12 m² and the perimeter 14 m. You need the area for flooring and the perimeter for skirting, and confusing them is expensive.
The diagonal comes from Pythagoras: d = √(a² + b²). For a 4 × 3 metre rectangle it is √(16 + 9) = 5 metres. This is the check builders use to confirm a frame or a foundation is truly rectangular — if both diagonals match, the corners are square.
No, and the gap can be wide. Rectangles of 1 × 9 and 5 × 5 both have a perimeter of 20, but areas of 9 and 25. For a fixed perimeter the area is greatest when the shape is a square, and the more elongated it gets the smaller the area. The same principle is why round shapes are the most economical.
Area units scale by the square, not linearly. One square metre holds 10,000 square centimetres, not 100, because a metre is 100 centimetres and that gets squared. Likewise a hectare is 10,000 m². This is the most common error when converting areas to order materials.