A square is one of the most important shapes in geometry because it combines the properties of a rectangle and a rhombus. It has four equal sides and four right angles, which makes it highly regular and easy to measure. Squares appear in grids, tiles, coordinate planes, architecture, and many physics diagrams.
Learning the square well builds a strong foundation for area, perimeter, symmetry, and the Pythagorean theorem.
The diagonal of a square connects opposite vertices and splits the square into two congruent right triangles. Because both legs of each triangle have length s, the diagonal length is found using d = s√2. A square also has strong symmetry, including four lines of symmetry and rotational symmetry every 90°.
These properties make squares useful for solving problems involving distance, transformations, design, and measurement.
Key Facts
- All four sides of a square are equal: AB = BC = CD = DA = s.
- All four interior angles are right angles: 90° each.
- Perimeter of a square: P = 4s.
- Area of a square: A = s^2.
- Diagonal of a square: d = s√2.
- The diagonals of a square are equal, bisect each other, are perpendicular, and bisect the angles.
Vocabulary
- Square
- A square is a quadrilateral with four equal sides and four right angles.
- Side length
- The side length is the distance along one edge of the square, usually represented by s.
- Diagonal
- A diagonal is a line segment connecting two opposite vertices of a polygon.
- Line of symmetry
- A line of symmetry divides a shape into two matching mirror-image halves.
- Rotational symmetry
- Rotational symmetry means a shape matches itself after being turned around its center by certain angles.
Common Mistakes to Avoid
- Using A = 4s for area is wrong because 4s gives the perimeter, not the space inside the square.
- Using d = 2s for the diagonal is wrong because the diagonal is found with the Pythagorean theorem, so d = s√2.
- Forgetting that all angles are 90° is wrong because a four-sided shape with equal sides is not always a square unless the angles are right angles.
- Counting only two lines of symmetry is wrong because a square has four lines of symmetry: two through midpoints of opposite sides and two along the diagonals.
Practice Questions
- 1 A square has side length 7 cm. Find its perimeter and area.
- 2 A square has side length 10 m. Find the exact length of its diagonal and give a decimal approximation using √2 ≈ 1.414.
- 3 Explain why every square is a rectangle and a rhombus, but not every rectangle or rhombus is a square.