8.2 Quadrilaterals, Polygons & Circle Geometry

8.2 Quadrilaterals, Polygons & Circle Geometry

Two-dimensional geometry questions on the AFQT Mathematics Knowledge subtest require a solid command of formulas for perimeters, areas, angle sums, and circle properties. Questions often combine algebraic expressions with geometric shapes or ask you to calculate the area of shaded regions. This section provides a thorough reference for all required 2D geometric shapes.

---

Classification & Properties of Quadrilaterals

A quadrilateral is a four-sided polygon. The sum of the interior angles of any quadrilateral is always 360°.

                  Quadrilaterals (360°)
                       /	     \
             Trapezoid	     Parallelogram
					    /             \
				    Rectangle         Rhombus
					    \             /
						    Square

1. Parallelogram

• Properties: Opposite sides are parallel and equal in length. Opposite angles are equal, and consecutive angles are supplementary (sum to 180°). Diagonals bisect each other.
• Area Formula: $A = b \times h$ (where $h$ is the perpendicular height, not the slanted side length).
• Perimeter Formula: $P = 2b + 2a$.

2. Rectangle

• Properties: A parallelogram with four right angles (90°). Diagonals are equal in length and bisect each other.
• Area Formula: $A = l \times w$.
• Perimeter Formula: $P = 2l + 2w$.

3. Rhombus

• Properties: A parallelogram with four sides of equal length. Opposite angles are equal, and diagonals intersect at perpendicular ($90^\circ$) right angles.
• Area Formulas: $A = b \times h$ OR $A = \frac{1}{2} d_1 d_2$ (where $d_1$ and $d_2$ are diagonal lengths).

4. Square

• Properties: A regular quadrilateral with four equal sides and four 90° right angles. Combines all properties of rectangles and rhombuses.
• Area Formula: $A = s^2$ or $A = \frac{1}{2} d^2$.
• Perimeter Formula: $P = 4s$.
• Diagonal Length: $d = s\sqrt{2}$ (derived from the 45°-45°-90° triangle ratio).

5. Trapezoid

• Properties: A quadrilateral with exactly one pair of parallel sides (called bases $b_1$ and $b_2$). The non-parallel sides are legs.
• Isosceles Trapezoid: Legs are equal in length, and base angles are equal.
• Area Formula: $A = \frac{1}{2} (b_1 + b_2) h = \left(\frac{b_1 + b_2}{2}\right) h$. Note: $\frac{b_1 + b_2}{2}$ is the average base length, also known as the midsegment or median.

---

Interior & Exterior Angles of Polygons

A polygon is a closed 2D shape with straight line segments as sides. A regular polygon has all sides equal in length (equilateral) and all interior angles equal in measure (equiangular).

Interior Angle Sum Formula

For any convex polygon with $n$ sides, the sum of the interior angles $S$ is:

$$S = (n - 2) \times 180^\circ$$

Single Interior Angle of a Regular Polygon

Because all $n$ angles are equal in a regular polygon, each interior angle $I$ measures:

$$I = \frac{(n - 2) \times 180^\circ}{n}$$

Exterior Angle Rules

• An exterior angle is formed by extending one side of a polygon past the vertex.
• Sum of Exterior Angles: The exterior angles of any convex polygon always sum to 360°, regardless of the number of sides.
• Single Exterior Angle of a Regular Polygon: $E = \frac{360^\circ}{n}$.
• Supplementary Rule: $\text{Interior Angle} + \text{Exterior Angle} = 180^\circ$.

Polygon Reference Table

---

Circle Geometry: Perimeter, Area, Arcs & Sectors

A circle is the set of all points in a plane equidistant from a fixed center point.

Core Definitions

• Radius ($r$): Distance from center to any point on the boundary.
• Diameter ($d$): Straight line distance across the circle passing through the center. $d = 2r$.
• Circumference ($C$): The perimeter of the circle. $$C = 2\pi r = \pi d$$
• Area ($A$): The measure of the surface enclosed by the circle. $$A = \pi r^2$$ (On AFQT tests, use $\pi \approx 3.14$ or $\frac{22}{7}$ unless left in terms of $\pi$).

Scaling Proportions in Circles

If the radius of a circle is multiplied by a scale factor $k$:

• The diameter and circumference multiply by $k$.
• The area multiplies by $k^2$. Example: Doubling the radius ($k = 2$) quadruples the area ($2^2 = 4$).

Arc Length & Sector Area

An arc is a portion of the circle's circumference. A sector is a pie-shaped portion of the circle's area enclosed by two radii and an arc.

• Central Angle ($\theta$): The angle formed at the center of the circle.
• Arc Length Formula: $$L = \left(\frac{\theta}{360^\circ}\right) \times 2\pi r$$
• Sector Area Formula: $$A_{\text{sector}} = \left(\frac{\theta}{360^\circ}\right) \times \pi r^2$$

---

Key Circle Theorems & Angle Relationships

1. Inscribed Angle Theorem: An inscribed angle has its vertex on the circle boundary. The measure of an inscribed angle is exactly half the measure of its intercepted central angle or arc. $$\theta_{\text{inscribed}} = \frac{1}{2} \theta_{\text{arc}}$$
2. Angle Inscribed in a Semicircle: Any angle inscribed in a semicircle (intercepting a diameter of 180°) is always a 90° right angle.
3. Tangent Line Rule: A line tangent to a circle touches the circle at exactly one point and is perpendicular ($90^\circ$) to the radius drawn to that point of tangency.

                  Circle Angle Theorems

      Central Angle: = Arc         Inscribed Angle: = 1/2 Arc
         /---------\                   /---------\
        /   (C)     \                 /    .      \
       /   /   \     \               /    / \      \
      |   /  θ  \     |             |    / θ \      |
      |  O-------*    |             |   *-----*     |
       \  Arc = θ    /               \  Arc = 2θ   /
        \-----------/                 \-----------/

---

AFQT Step-by-Step Problem Walkthroughs

Example 1: Area of a Trapezoid

Problem: A trapezoid has parallel bases measuring 12 inches and 18 inches. Its height is 8 inches, and its legs each measure 10 inches. What is the area of the trapezoid?

1. Identify relevant values: $b_1 = 12$, $b_2 = 18$, height $h = 8$. (Ignore leg lengths of 10, as they are not used in the area formula).
2. Apply Trapezoid Area Formula: $A = \frac{1}{2}(b_1 + b_2)h$.
3. Calculate: $A = \frac{1}{2}(12 + 18) \times 8 = \frac{1}{2}(30) \times 8 = 15 \times 8 = 120\text{ sq in}$.

Example 2: Sector Area of a Circle

Problem: A circle has a radius of 6 cm. What is the area of a sector with a central angle of 60°?

1. Find total area: $A_{\text{total}} = \pi r^2 = \pi (6^2) = 36\pi\text{ cm}^2$.
2. Determine fraction of circle: $\frac{\theta}{360^\circ} = \frac{60^\circ}{360^\circ} = \frac{1}{6}$.
3. Calculate sector area: $A_{\text{sector}} = \frac{1}{6} \times 36\pi = 6\pi\text{ cm}^2$.
4. Decimal conversion (if requested): $6 \times 3.14 = 18.84\text{ cm}^2$.

---

Key Takeaways Summary

• Quadrilateral Area Formulas: Rectangle ($lw$), Parallelogram ($bh$), Rhombus ($\frac{1}{2}d_1 d_2$), Trapezoid ($\frac{1}{2}(b_1+b_2)h$).
• Polygon Interior Angles: Sum $= (n - 2) \times 180^\circ$; Single regular angle $= \frac{(n - 2) \times 180^\circ}{n}$.
• Polygon Exterior Angles: Always sum to 360°; Single regular angle $= \frac{360^\circ}{n}$.
• Circles: $C = 2\pi r = \pi d$; $A = \pi r^2$.
• Arc Length & Sector Area: Multiply full $C$ or $A$ by fraction $\frac{\theta}{360^\circ}$.
• Inscribed Angle: Equals half of intercepted arc.