7.2 Quadrilaterals, Polygons & Perimeter/Area

In the Quantitative Reasoning section of the Saudi Arabia Qiyas General Aptitude Test (GAT), quadrilaterals and polygons represent a substantial portion of geometry questions. These problems test your ability to recall formulas, recognize properties of specific shapes, and solve compound figures where multiple shapes overlap. Mastering the relationships between perimeter, side lengths, diagonals, and area is essential for finishing the exam on time.

---

Properties of Special Quadrilaterals

A quadrilateral is any four-sided polygon. In GAT, you will primarily deal with special quadrilaterals, each possessing distinct properties that unlock shortcuts during calculations.

1. Parallelogram

A parallelogram is a quadrilateral where opposite sides are parallel.

• Properties: Opposite sides are equal, opposite interior angles are equal, and consecutive interior angles are supplementary (sum to $180^\circ$). Diagonals bisect each other.
• Area Formula: $\text{Area} = \text{base} \times \text{height}$ (where height is the perpendicular distance between the bases, not the slant height).

2. Rectangle

A rectangle is a parallelogram with four right angles.

• Properties: Inherits all parallelogram properties, all four interior angles are $90^\circ$, and diagonals are equal in length ($AC = BD$).
• Formulas: $\text{Perimeter} = 2(l + w)$; $\text{Area} = l \times w$; $\text{Diagonal} = \sqrt{l^2 + w^2}$.

3. Rhombus

A rhombus is a parallelogram with four equal sides.

• Properties: Inherits all parallelogram properties, all four sides are equal, and diagonals are perpendicular bisectors of each other. Diagonals also bisect the vertex angles.
• Formulas: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$. The perpendicular diagonals divide the rhombus into four identical right triangles with legs $\frac{d_1}{2}$ and $\frac{d_2}{2}$, allowing you to find the side length using the Pythagorean theorem.

4. Square

A square is a regular quadrilateral with four equal sides and four right angles. It combines all properties of rectangles and rhombuses.

• Formulas: $\text{Perimeter} = 4s$; $\text{Area} = s^2 = \frac{d^2}{2}$; $\text{Diagonal} = s\sqrt{2}$. The diagonal area formula is a major time-saver on the GAT.

5. Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel sides (called bases).

• Properties: The midsegment connects the midpoints of the legs, and its length is the average of the bases: $\text{midsegment} = \frac{b_1 + b_2}{2}$.
• Area Formula: $\text{Area} = \frac{b_1 + b_2}{2} \times h = \text{midsegment} \times h$.

---

Interior and Exterior Angles of Polygons

A polygon is a closed 2D shape with straight sides. A regular polygon has equal sides and equal interior angles.

Angle and Diagonal Formulas:

1. Sum of Interior Angles: For any $n$-sided polygon: $S = (n - 2) \times 180^\circ$.
2. Individual Interior Angle (Regular Polygon Only): $I = \frac{(n - 2) \times 180^\circ}{n}$.
3. Sum of Exterior Angles: The sum of the exterior angles of any convex polygon is always exactly $360^\circ$.
4. Individual Exterior Angle (Regular Polygon Only): $E = \frac{360^\circ}{n}$.
5. Number of Diagonals: The number of unique diagonals in an $n$-gon is: $\text{Diagonals} = \frac{n(n - 3)}{2}$.

Let us review the properties of common regular polygons:

• Pentagon (5 sides): Sum of interior angles is $540^\circ$; regular interior angle is $108^\circ$; regular exterior angle is $72^\circ$; has 5 diagonals.
• Hexagon (6 sides): Sum of interior angles is $720^\circ$; regular interior angle is $120^\circ$; regular exterior angle is $60^\circ$; has 9 diagonals.
• Octagon (8 sides): Sum of interior angles is $1080^\circ$; regular interior angle is $135^\circ$; regular exterior angle is $45^\circ$; has 20 diagonals.
• Decagon (10 sides): Sum of interior angles is $1440^\circ$; regular interior angle is $144^\circ$; regular exterior angle is $36^\circ$; has 35 diagonals.

---

Special Focus: The Regular Hexagon

Regular hexagons are highly popular on the GAT. You must know these properties:

• A regular hexagon can be divided into 6 congruent equilateral triangles by drawing diagonals connecting opposite vertices.
• Area of Regular Hexagon: Since it is made of 6 equilateral triangles, its area is: $\text{Area} = \frac{3s^2\sqrt{3}}{2}$.
• Diagonals: The longest diagonals connect opposite vertices and have length $2s$. The shorter diagonals (connecting alternate vertices) have length $s\sqrt{3}$.

---

Perimeter and Area Optimization and Scaling

• Scaling Rule: If the linear dimensions (side lengths) of a polygon are multiplied by a scale factor $k$, the perimeter is multiplied by $k$, and the area is multiplied by $k^2$.
• Maximized Area for Fixed Perimeter: Among all quadrilaterals with a fixed perimeter, a square has the maximum area. Among all rectangles with a perimeter of 24, the square with side 6 has an area of 36, while a rectangle of dimensions $5 \times 7$ has an area of 35.
• Minimized Perimeter for Fixed Area: Among all quadrilaterals with a fixed area, a square has the minimum perimeter.

---

Qiyas GAT Common Traps and Exam Strategies

• Slant Height Trap: In parallelograms and trapezoids, never use the slant side length as the height when calculating area. You must find or calculate the perpendicular height, often using a right triangle formed inside the shape.
• Unshaded Area Problems: GAT frequently asks you to find the area of a shaded region by subtracting the area of a smaller polygon from a larger one (e.g., a square minus a rectangle, or a regular hexagon minus a triangle).
• Hexagon Diagonal Shortcuts: If a GAT comparison asks you to evaluate the longest diagonal of a regular hexagon with side length 5, it is simply $2 imes 5 = 10$. No complex calculations are necessary.