Quadrilaterals • Topic 3 of 5

Squares

A square is a rectangle with all four sides equal — equivalently, a quadrilateral with four right angles and four equal sides. It inherits every rectangle and parallelogram property and adds full symmetry. Its perimeter is 4 × side, its area is side², and its diagonal is side × √2 (from the 45-45-90 triangle the diagonal creates). The diagonals are equal, bisect each other at right angles, and bisect the corner angles. Because everything depends on a single side length, one measurement determines all others. The SAT uses squares for area, perimeter and diagonal questions and inside composite-figure problems.

A square with four equal sides and a diagonal of length s root 2Squaress√2Four equal sides; diagonal = s√2 (45°-45°-90°)

✅ Solved examples

1. A square has side 6. Find its area.
6² = 36.
2. A square has side 6. Find its perimeter.
4 × 6 = 24.
3. A square has side 5. Find its diagonal.
5√2.
4. A square has area 49. Find its side.
√49 = 7.

✏️ Practice — try these, take hints as needed

1. A square has side 9. Find its area.
side².
9².
—
81.
2. A square has side 9. Find its perimeter.
4 × side.
4 × 9.
—
36.
3. A square has side 8. Find its diagonal.
side × √2.
—
—
8√2.
4. A square has area 64. Find its side.
√area.
√64.
—
8.
5. A square has perimeter 20. Find its side.
Perimeter ÷ 4.
20 ÷ 4.
—
5.

📝 Topic test — 8 questions

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