ShapesProperties
- All four sides are equal.
- Every angle is 90°.
- The two diagonals are equal, cut each other in half and cross at right angles.
- It has four lines of symmetry.
- A square is a special rectangle and also a special rhombus.
Why the formulas work
Lay unit squares inside a square of side 5 cm. You get 5 rows of 5, so 25 squares fit: the area is side times side. The perimeter is the walk around the edge, four equal sides. The diagonal splits the square into two right-angled triangles, and Pythagoras gives its length: d² = a² + a², so d is a times the square root of 2, about 1.414 times the side.
Worked example
Take a square with side 12 cm.
- Area. Area = a2. Here: 12 × 12 = 144 cm².
- Perimeter. Perimeter = 4a. Here: 4 × 12 = 48 cm.
- Diagonal. Diagonal = a√2. Here: 12 × √2 = 16.97 cm.
Finding a missing measurement
From the area, the side is its square root: an area of 144 cm² means a side of 12 cm. From the perimeter, divide by 4: a perimeter of 60 cm means a side of 15 cm, and so an area of 225 cm². From the diagonal, divide by √2.
Mistakes to avoid
- Doubling the side does not double the area. It makes it four times as large.
- Area is in square units (cm²) and perimeter in plain units (cm). Mixing them is the most common slip.
- To find the side from the area, take the square root. To find it from the perimeter, divide by 4.
Where you see it
Floor tiles, chessboard squares, carrom boards, a sheet of origami paper and many window panes are squares.
Try these
Work each one out, then tap to check.
A square has side 7.5 cm. Find its area, perimeter and diagonal.
Area = 56.25 cm²; Perimeter = 30 cm; Diagonal = 10.61 cm.
A square has side 25 cm. Find its area, perimeter and diagonal.
Area = 625 cm²; Perimeter = 100 cm; Diagonal = 35.36 cm.