Rhombus and Square — Complete Guide (Grade 8)
Big Idea: A rhombus and a square are both special types of parallelograms. A square is actually a special case of a rhombus — one where all angles are right angles. Understanding their properties helps us solve complex geometry problems.
♦ Rhombus
Definition
A rhombus is a parallelogram in which all four sides are equal in length.
Key Properties
- All sides are equal: $AB = BC = CD = DA$
- Opposite angles are equal: $\angle A = \angle C$ and $\angle B = \angle D$
- Adjacent angles are supplementary: $\angle A + \angle B = 180°$
- Diagonals bisect each other at right angles: $AC \perp BD$
- Diagonals bisect the vertex angles
- Opposite sides are parallel
Formulas
| Perimeter | $P = 4a$ where $a$ is the side length |
| Area (base × height) | $A = a \times h$ |
| Area (diagonals) | $$A = \frac{d_1 \times d_2}{2}$$ |
| Side from diagonals | $$a = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}$$ |
◼ Square
Definition
A square is a parallelogram in which all four sides are equal AND all four angles are right angles ($90°$).
Key Properties
- All sides are equal: $AB = BC = CD = DA$
- All angles are $90°$
- Diagonals are equal, bisect each other at right angles
- Each diagonal = $a\sqrt{2}$
- Diagonals bisect vertex angles (each half = $45°$)
- A square is simultaneously a rhombus, a rectangle, and a parallelogram
Formulas
| Perimeter | $P = 4a$ |
| Area | $A = a^2$ |
| Area (diagonal) | $$A = \frac{d^2}{2}$$ |
| Diagonal | $$d = a\sqrt{2}$$ |