8.2 Quadrilaterals, Polygons & Circle Geometry

Armed Forces Qualification Test AFQT

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1 Chapter 1: AFQT Structure, Scoring & Enlistment Requirements

1.1 AFQT Overview, Subtest Breakdown & CAT-ASVAB vs. P&P Format 1.2 The AFQT Scoring Formula (2×VE + AR + MK) & Percentile Norming 1.3 Branch Minimum Scores, AFQT Categories & Retake Policies

2 Chapter 2: Arithmetic Reasoning (AR) — Ratios, Rates & Percentages

2.1 Ratios, Proportions & Scale Conversions 2.2 Percentages, Discounts, Interest & Tax Calculations 2.3 Rate, Distance, Time & Speed Word Problems 2.4 Work Rates & Combined Labor Problems

3 Chapter 3: Arithmetic Reasoning (AR) — Mixtures, Sequences & Applied Math

3.1 Mixture, Concentration & Price-Blending Problems 3.2 Sequences, Patterns & Series Calculations 3.3 Averages, Weighted Averages & Statistical Word Problems

4 Chapter 4: Word Knowledge (WK) — Vocabulary & Word Structure

4.1 High-Frequency AFQT Vocabulary & Synonym Recognition 4.2 Latin & Greek Roots, Prefixes & Suffixes for Word Decoding 4.3 Context Clues & Words with Multiple Meanings

5 Chapter 5: Paragraph Comprehension (PC) — Reading Analysis & Inference

5.1 Identifying Main Ideas & Primary Purpose 5.2 Locating Supporting Details & Direct Facts 5.3 Drawing Logical Inferences & Implied Conclusions 5.4 Analyzing Tone, Author Perspective & Contextual Vocabulary

6 Chapter 6: Mathematics Knowledge (MK) — Fundamentals & Linear Algebra

6.1 Arithmetic Operations, Order of Operations & Number Properties 6.2 Exponents, Scientific Notation & Square Roots / Radicals 6.3 Linear Equations, Inequalities & Absolute Value 6.4 Systems of Linear Equations & Word-to-Algebra Translation

7 Chapter 7: Mathematics Knowledge (MK) — Advanced Algebra & Functions

7.1 Polynomial Operations & Factoring Techniques 7.2 Quadratic Equations, Factoring & Quadratic Formula 7.3 Coordinate Geometry: Slope, Distance, Midpoint & Linear Graphs

8 Chapter 8: Mathematics Knowledge (MK) — Geometry & Spatial Math

8.1 Angles, Triangles & Pythagorean Theorem 8.2 Quadrilaterals, Polygons & Circle Geometry 8.3 3D Geometry: Surface Area & Volume of Solids 8.4 Basic Geometric Transformations & Trigonometric Ratios

8.2 Quadrilaterals, Polygons & Circle Geometry

Key Takeaways Explain This Topic AI

The sum of the interior angles of an n-sided polygon is given by (n - 2) × 180°, while the exterior angles of any convex polygon always sum to 360°.

Parallelograms, rectangles, squares, rhombuses, and trapezoids each have specific area formulas: Rectangle (A = lw), Parallelogram (A = bh), Rhombus (A = ½ d₁ d₂), and Trapezoid (A = ½(b₁ + b₂)h).

For circles, circumference is C = 2πr = πd and area is A = πr²; arc length and sector area are proportional fractions (θ / 360°) of full circumference and area.

An inscribed angle in a circle measures exactly half the measure of its intercepted arc or central angle.

A line tangent to a circle is always perpendicular (90°) to the radius drawn to the point of tangency.

Last updated: July 2026

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 × h A = b \times h A = b × h (where h h h is the perpendicular height, not the slanted side length).

Perimeter Formula: P = 2 b + 2 a P = 2b + 2a P = 2 b + 2 a .

2. Rectangle

Properties: A parallelogram with four right angles (90°). Diagonals are equal in length and bisect each other.

Area Formula: A = l × w A = l \times w A = l × w .

Perimeter Formula: P = 2 l + 2 w P = 2l + 2w P = 2 l + 2 w .

3. Rhombus

Properties: A parallelogram with four sides of equal length. Opposite angles are equal, and diagonals intersect at perpendicular ( 90 ∘ 90^\circ 9 0 ∘ ) right angles.

Area Formulas: A = b × h A = b \times h A = b × h OR A = 1 2 d 1 d 2 A = \frac{1}{2} d_1 d_2 A = 2 1 ​ d 1 ​ d 2 ​ (where d 1 d_1 d 1 ​ and d 2 d_2 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 A = s^2 A = s 2 or A = 1 2 d 2 A = \frac{1}{2} d^2 A = 2 1 ​ d 2 .

Perimeter Formula: P = 4 s P = 4s P = 4 s .

Diagonal Length: d = s 2 d = s\sqrt{2} d = s 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 b_1 b 1 ​ and b 2 b_2 b 2 ​ ). The non-parallel sides are legs.

Isosceles Trapezoid: Legs are equal in length, and base angles are equal.

Area Formula: A = 1 2 ( b 1 + b 2 ) h = ( b 1 + b 2 2 ) h A = \frac{1}{2} (b_1 + b_2) h = \left(\frac{b_1 + b_2}{2}\right) h A = 2 1 ​ ( b 1 ​ + b 2 ​ ) h = ( 2 b 1 ​ + b 2 ​ ​ ) h . Note: b 1 + b 2 2 \frac{b_1 + b_2}{2} 2 b 1 ​ + b 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 n n sides, the sum of the interior angles S S S is: S = ( n − 2 ) × 180 ∘ S = (n - 2) \times 180^\circ S = ( n − 2 ) × 18 0 ∘

Single Interior Angle of a Regular Polygon

Because all n n n angles are equal in a regular polygon, each interior angle I I I measures: I = ( n − 2 ) × 180 ∘ n I = \frac{(n - 2) \times 180^\circ}{n} I = n ( n − 2 ) × 18 0 ∘ ​

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 = 360 ∘ n E = \frac{360^\circ}{n} E = n 36 0 ∘ ​ .

Supplementary Rule: Interior Angle + Exterior Angle = 180 ∘ \text{Interior Angle} + \text{Exterior Angle} = 180^\circ Interior Angle + Exterior Angle = 18 0 ∘ .

Polygon Reference Table

Polygon Name Sides ( n n n ) Interior Angle Sum Single Interior Angle (Regular) Single Exterior Angle (Regular)

• Triangle 3 180° 60° 120°
• Quadrilateral 4 360° 90° 90°
• Pentagon 5 540° 108° 72°
• Hexagon 6 720° 120° 60°
• Octagon 8 1080° 135° 45°
• Decagon 10 1440° 144° 36°
• Dodecagon 12 1800° 150° 30°
• 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 r r ): Distance from center to any point on the boundary.

Diameter ( d d d ): Straight line distance across the circle passing through the center. d = 2 r d = 2r d = 2 r .

Circumference ( C C C ): The perimeter of the circle. C = 2 π r = π d C = 2\pi r = \pi d C = 2 π r = π d

Area ( A A A ): The measure of the surface enclosed by the circle. A = π r 2 A = \pi r^2 A = π r 2 (On AFQT tests, use π ≈ 3.14 \pi \approx 3.14 π ≈ 3.14 or 22 7 \frac{22}{7} 7 22 ​ unless left in terms of π \pi π ).

Scaling Proportions in Circles

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

The diameter and circumference multiply by k k k .

The area multiplies by k 2 k^2 k 2 . Example: Doubling the radius ( k = 2 k = 2 k = 2 ) quadruples the area ( 2 2 = 4 2^2 = 4 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 ( θ \theta θ ): The angle formed at the center of the circle.

Arc Length Formula: L = ( θ 360 ∘ ) × 2 π r L = \left(\frac{\theta}{360^\circ}\right) \times 2\pi r L = ( 36 0 ∘ θ ​ ) × 2 π r

Sector Area Formula: A sector = ( θ 360 ∘ ) × π r 2 A_{\text{sector}} = \left(\frac{\theta}{360^\circ}\right) \times \pi r^2 A sector ​ = ( 36 0 ∘ θ ​ ) × π r 2

Key Circle Theorems & Angle Relationships

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. θ inscribed = 1 2 θ arc \theta_{\text{inscribed}} = \frac{1}{2} \theta_{\text{arc}} θ inscribed ​ = 2 1 ​ θ arc ​

Angle Inscribed in a Semicircle: Any angle inscribed in a semicircle (intercepting a diameter of 180°) is always a 90° right angle .

Tangent Line Rule: A line tangent to a circle touches the circle at exactly one point and is perpendicular ( 90 ∘ 90^\circ 9 0 ∘ ) 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?

Identify relevant values: b 1 = 12 b_1 = 12 b 1 ​ = 12 , b 2 = 18 b_2 = 18 b 2 ​ = 18 , height h = 8 h = 8 h = 8 . (Ignore leg lengths of 10, as they are not used in the area formula).

Apply Trapezoid Area Formula: A = 1 2 ( b 1 + b 2 ) h A = \frac{1}{2}(b_1 + b_2)h A = 2 1 ​ ( b 1 ​ + b 2 ​ ) h .

Calculate: A = 1 2 ( 12 + 18 ) × 8 = 1 2 ( 30 ) × 8 = 15 × 8 = 120 sq in A = \frac{1}{2}(12 + 18) \times 8 = \frac{1}{2}(30) \times 8 = 15 \times 8 = 120\text{ sq in} A = 2 1 ​ ( 12 + 18 ) × 8 = 2 1 ​ ( 30 ) × 8 = 15 × 8 = 120 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°?

Find total area: A total = π r 2 = π ( 6 2 ) = 36 π cm 2 A_{\text{total}} = \pi r^2 = \pi (6^2) = 36\pi\text{ cm}^2 A total ​ = π r 2 = π ( 6 2 ) = 36 π cm 2 .

Determine fraction of circle: θ 360 ∘ = 60 ∘ 360 ∘ = 1 6 \frac{\theta}{360^\circ} = \frac{60^\circ}{360^\circ} = \frac{1}{6} 36 0 ∘ θ ​ = 36 0 ∘ 6 0 ∘ ​ = 6 1 ​ .

Calculate sector area: A sector = 1 6 × 36 π = 6 π cm 2 A_{\text{sector}} = \frac{1}{6} \times 36\pi = 6\pi\text{ cm}^2 A sector ​ = 6 1 ​ × 36 π = 6 π cm 2 .

Decimal conversion (if requested): 6 × 3.14 = 18.84 cm 2 6 \times 3.14 = 18.84\text{ cm}^2 6 × 3.14 = 18.84 cm 2 .

Key Takeaways Summary

Quadrilateral Area Formulas: Rectangle ( l w lw lw ), Parallelogram ( b h bh bh ), Rhombus ( 1 2 d 1 d 2 \frac{1}{2}d_1 d_2 2 1 ​ d 1 ​ d 2 ​ ), Trapezoid ( 1 2 ( b 1 + b 2 ) h \frac{1}{2}(b_1+b_2)h 2 1 ​ ( b 1 ​ + b 2 ​ ) h ).

Polygon Interior Angles: Sum = ( n − 2 ) × 180 ∘ = (n - 2) \times 180^\circ = ( n − 2 ) × 18 0 ∘ ; Single regular angle = ( n − 2 ) × 180 ∘ n = \frac{(n - 2) \times 180^\circ}{n} = n ( n − 2 ) × 18 0 ∘ ​ .

Polygon Exterior Angles: Always sum to 360° ; Single regular angle = 360 ∘ n = \frac{360^\circ}{n} = n 36 0 ∘ ​ .

Circles: C = 2 π r = π d C = 2\pi r = \pi d C = 2 π r = π d ; A = π r 2 A = \pi r^2 A = π r 2 .

Arc Length & Sector Area: Multiply full C C C or A A A by fraction θ 360 ∘ \frac{\theta}{360^\circ} 36 0 ∘ θ ​ .

Inscribed Angle: Equals half of intercepted arc.