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8.3 3D Geometry: Surface Area & Volume of Solids

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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

8.3 3D Geometry: Surface Area & Volume of Solids

Key Takeaways Explain This Topic AI

Volume measures three-dimensional space in cubic units (u³) while surface area measures total boundary area in square units (u²).

The volume of a rectangular prism is V = l · w · h, and its total surface area is SA = 2(lw + lh + wh); for a cube with edge s, V = s³ and SA = 6s².

Cylinder volume is V = πr²h, with total surface area SA = 2πr² + 2πrh; cones and pyramids equal one-third of the volume of their corresponding cylinder/prism (V = ⅓ B h).

Sphere formulas are V = (4/3)πr³ for volume and SA = 4πr² for surface area.

Scaling linear dimensions of a solid by factor k multiplies surface area by k² and volume by k³.

Last updated: July 2026

8.3 3D Geometry: Surface Area & Volume of Solids

Three-dimensional geometry questions on the AFQT test your spatial reasoning and ability to calculate volumetric capacity and total surface area of 3D solids. Understanding the distinction between two-dimensional surface measurements (measured in square units like in 2 \text{in}^2 in 2 or cm 2 \text{cm}^2 cm 2 ) and three-dimensional volume measurements (measured in cubic units like ft 3 \text{ft}^3 ft 3 or m 3 \text{m}^3 m 3 ) is fundamental for accurate problem-solving.

Rectangular Prisms & Cubes

A prism is a 3D polyhedron with two congruent parallel bases connected by rectangular lateral faces. A rectangular prism (cuboid) has six rectangular faces. +-------------------+

  • / /|
  • / / | h (height)
  • +-------------------+ |
  • | | +
  • | | / w (width)
  • | |/
  • +-------------------+
  • l (length)
  • Rectangular Prism Formulas

Volume ( V V V ): The internal space or capacity enclosed by the solid. V = Base Area × Height = l × w × h V = \text{Base Area} \times \text{Height} = l \times w \times h V = Base Area × Height = l × w × h

Total Surface Area ( S A SA S A ): The combined sum of the areas of all six rectangular faces (front/back, top/bottom, left/right). S A = 2 ( l w + l h + w h ) SA = 2(lw + lh + wh) S A = 2 ( lw + l h + w h )

Space Diagonal ( d d d ): The 3D straight-line distance between opposite corners. d = l 2 + w 2 + h 2 d = \sqrt{l^2 + w^2 + h^2} d = l 2 + w 2 + h 2 ​

Cube Formulas

A cube is a special rectangular prism where length, width, and height are all equal to edge length s s s .

  • Volume: V = s 3 V = s^3 V = s 3
  • Total Surface Area: S A = 6 s 2 SA = 6s^2 S A = 6 s 2
  • Space Diagonal: d = s 3 d = s\sqrt{3} d = s 3 ​
  • Right Circular Cylinders & Cones

Circular solids replace polygonal bases with circular bases, requiring the use of π \pi π .

1. Right Circular Cylinder

A cylinder consists of two parallel circular bases of radius r r r connected by a curved lateral surface of height h h h .

  • Base Area ( B B B ): B = π r 2 B = \pi r^2 B = π r 2
  • Volume ( V V V ): Base Area × \times × Height V = π r 2 h V = \pi r^2 h V = π r 2 h

Lateral Surface Area ( L S A LSA L S A ): Unrolled into a rectangle of length equal to circumference ( 2 π r 2\pi r 2 π r ) and height h h h . L S A = 2 π r h LSA = 2\pi r h L S A = 2 π r h

Total Surface Area ( T S A TSA TS A ): Sum of lateral surface area plus both circular bases. T S A = 2 π r 2 + 2 π r h = 2 π r ( r + h ) TSA = 2\pi r^2 + 2\pi r h = 2\pi r (r + h) TS A = 2 π r 2 + 2 π r h = 2 π r ( r + h )

2. Right Circular Cone

A cone has a single circular base of radius r r r tapering to a point (apex) at perpendicular height h h h .

Slant Height ( l l l ): Distance from any point on the base circle edge to the apex. Uses the Pythagorean theorem: l = r 2 + h 2 l = \sqrt{r^2 + h^2} l = r 2 + h 2 ​

Volume ( V V V ): Exactly one-third the volume of a cylinder with the same base and height. V = 1 3 π r 2 h V = \frac{1}{3} \pi r^2 h V = 3 1 ​ π r 2 h

Total Surface Area ( S A SA S A ): Base area plus lateral area. S A = π r 2 + π r l SA = \pi r^2 + \pi r l S A = π r 2 + π r l

  • Pyramids & Spheres
  • 1. Pyramids

A pyramid has a polygonal base (e.g., square or triangle) and triangular faces meeting at an apex.

Volume Formula (Any Pyramid): V = 1 3 B h V = \frac{1}{3} B h V = 3 1 ​ B h where B B B is the area of the base polygon.

Square Pyramid Volume: V = 1 3 s 2 h V = \frac{1}{3} s^2 h V = 3 1 ​ s 2 h .

Total Surface Area: S A = B + Sum of lateral triangular faces SA = B + \text{Sum of lateral triangular faces} S A = B + Sum of lateral triangular faces .

2. Spheres

A sphere is the 3D set of all points equidistant from a central point.

  • Volume ( V V V ): V = 4 3 π r 3 V = \frac{4}{3} \pi r^3 V = 3 4 ​ π r 3
  • Surface Area ( S A SA S A ): S A = 4 π r 2 SA = 4\pi r^2 S A = 4 π r 2

Hemisphere (Half-Sphere):

  • Volume = 2 3 π r 3 = \frac{2}{3}\pi r^3 = 3 2 ​ π r 3
  • Total Surface Area (including flat circular base) = 3 π r 2 = 3\pi r^2 = 3 π r 2
  • Dimensional Scaling & Multi-Dimensional Effects

AFQT Mathematics Knowledge often features conceptual questions testing how scaling linear dimensions affects area and volume. If all linear dimensions (length, width, height, radius) of a solid are multiplied by a scale factor k k k :

Linear Dimensions (edges, perimeters, heights, radii) scale by factor k 1 = k k^1 = k k 1 = k .

Surface Area & Base Area scale by factor k 2 k^2 k 2 .

Volume & Capacity scale by factor k 3 k^3 k 3 .

Scaling Effect Summary Table

Scale Factor ( k k k ) Linear Change Surface Area Multiplier ( k 2 k^2 k 2 ) Volume Multiplier ( k 3 k^3 k 3 )

Doubled ( k = 2 k = 2 k = 2 ) 2 × 2\times 2 × 2 2 = 4 × 2^2 = 4\times 2 2 = 4 × (+300% increase) 2 3 = 8 × 2^3 = 8\times 2 3 = 8 × (+700% increase)

Tripled ( k = 3 k = 3 k = 3 ) 3 × 3\times 3 × 3 2 = 9 × 3^2 = 9\times 3 2 = 9 × 3 3 = 27 × 3^3 = 27\times 3 3 = 27 ×

Quadrupled ( k = 4 k = 4 k = 4 ) 4 × 4\times 4 × 4 2 = 16 × 4^2 = 16\times 4 2 = 16 × 4 3 = 64 × 4^3 = 64\times 4 3 = 64 ×

Halved ( k = 0.5 k = 0.5 k = 0.5 ) 0.5 × 0.5\times 0.5 × 0.25 × 0.25\times 0.25 × (¼) 0.125 × 0.125\times 0.125 × (⅛)

Critical Exam Tip: If asked "If every edge of a cube is doubled, by how much does its volume increase?", remember 2 3 = 8 2^3 = 8 2 3 = 8 . The new volume is 8 times the original volume!

  • AFQT Step-by-Step Problem Walkthroughs
  • Example 1: Total Surface Area of a Rectangular Prism

Problem: A military cargo crate has a length of 6 feet, a width of 4 feet, and a height of 3 feet. What is the total surface area of the crate in square feet?

Identify dimensions: l = 6 l = 6 l = 6 , w = 4 w = 4 w = 4 , h = 3 h = 3 h = 3 .

Apply Surface Area Formula: S A = 2 ( l w + l h + w h ) SA = 2(lw + lh + wh) S A = 2 ( lw + l h + w h ) .

Calculate products:

  • l w = 6 × 4 = 24 lw = 6 \times 4 = 24 lw = 6 × 4 = 24
  • l h = 6 × 3 = 18 lh = 6 \times 3 = 18 l h = 6 × 3 = 18
  • w h = 4 × 3 = 12 wh = 4 \times 3 = 12 w h = 4 × 3 = 12

Sum and multiply by 2: S A = 2 ( 24 + 18 + 12 ) = 2 ( 54 ) = 108 sq ft SA = 2(24 + 18 + 12) = 2(54) = 108\text{ sq ft} S A = 2 ( 24 + 18 + 12 ) = 2 ( 54 ) = 108 sq ft .

Example 2: Volume of a Cylinder

Problem: A fuel cylinder has a base diameter of 10 meters and a height of 8 meters. What is the volume of the tank in terms of π \pi π ?

Find radius: Radius r = Diameter 2 = 10 2 = 5 r = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5 r = 2 Diameter ​ = 2 10 ​ = 5 meters.

Apply Volume Formula: V = π r 2 h V = \pi r^2 h V = π r 2 h .

Substitute values: V = π ( 5 2 ) ( 8 ) = π ( 25 ) ( 8 ) = 200 π m 3 V = \pi (5^2) (8) = \pi (25) (8) = 200\pi\text{ m}^3 V = π ( 5 2 ) ( 8 ) = π ( 25 ) ( 8 ) = 200 π m 3 .

Key Takeaways Summary

Rectangular Prism: V = l w h V = lwh V = lw h ; S A = 2 ( l w + l h + w h ) SA = 2(lw + lh + wh) S A = 2 ( lw + l h + w h ) .

Cube: V = s 3 V = s^3 V = s 3 ; S A = 6 s 2 SA = 6s^2 S A = 6 s 2 .

Cylinder: V = π r 2 h V = \pi r^2 h V = π r 2 h ; T S A = 2 π r 2 + 2 π r h TSA = 2\pi r^2 + 2\pi rh TS A = 2 π r 2 + 2 π r h .

Cone: V = 1 3 π r 2 h V = \frac{1}{3}\pi r^2 h V = 3 1 ​ π r 2 h ; Pyramid: V = 1 3 B h V = \frac{1}{3}Bh V = 3 1 ​ B h .

Sphere: V = 4 3 π r 3 V = \frac{4}{3}\pi r^3 V = 3 4 ​ π r 3 ; S A = 4 π r 2 SA = 4\pi r^2 S A = 4 π r 2 .

Scaling Rules: Linear → k \to k → k , Area → k 2 \to k^2 → k 2 , Volume → k 3 \to k^3 → k 3 .