Overview

Quadrilateral properties reduce area, angle, and diagonal problems to standard formulas.

Key Ideas

• Parallelogram: opposite sides parallel and equal; area $= ab\sin\theta$.
• Rectangle: all angles $90^\circ$; diagonals equal.
• Rhombus: all sides equal; diagonals perpendicular and bisect.
• Trapezoid: one pair of parallel sides; area $= \frac{1}{2}(b_1+b_2)h$.
• Square: rectangle + rhombus; diagonals perpendicular and equal.

Core Skills

Identify the Shape

Check parallel sides, right angles, and equal sides to classify the quadrilateral. The correct formula depends on this step.

Use Diagonal Properties

Diagonals in rectangles are equal; in rhombi they are perpendicular and bisect each other; in squares they do both.

Area via Bases and Heights

For parallelograms and trapezoids, area comes from a base and a perpendicular height, not a slanted side.

Worked Example

A trapezoid has bases $10$ and $6$ and height $4$. Find its area.

$A = \frac{1}{2}(10+6)\cdot 4 = 32$.

More Examples

Example 1: Parallelogram Area

A parallelogram has side lengths $7$ and $9$ with included angle $30^\circ$. Find its area.

$A = ab\sin\theta = 7\cdot 9\cdot \sin 30^\circ = 31.5$.

Example 2: Rectangle Diagonal

A rectangle has side lengths $5$ and $12$. Find the diagonal length.

Use the Pythagorean theorem: $\sqrt{5^2+12^2} = 13$.

Example 3: Rhombus Diagonals

A rhombus has diagonals $10$ and $6$. Find its area.

$A = \frac{1}{2}d_1 d_2 = 30$.

Strategy Checklist

• Classify the quadrilateral first.
• Use diagonal properties for fast length or angle facts.
• Use perpendicular heights, not slanted edges, for area.

Common Pitfalls

• Confusing trapezoid height with slanted leg length.
• Assuming diagonals are perpendicular in every quadrilateral.
• Using parallelogram area formula in a trapezoid.

Practice Problems