Study guide
9.2 3D Geometry: Surface Area & Volume of Solids
ACCUPLACER Quantitative Reasoning, Algebra, and Statistics Exam
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1 Chapter 1: Rational Numbers & Operations
1.1 Rational Number Arithmetic & Signed Numbers 1.2 Fractions, Decimals, & Mixed Numbers Operations 1.3 Absolute Value, Number Line, & Estimation
2 Chapter 2: Ratios, Proportions & Rates
2.1 Ratios, Unit Rates, & Dimensional Analysis 2.2 Proportions & Direct/Inverse Variation 2.3 Percent Applications, Markup, Discount, & Simple Interest
3 Chapter 3: Exponents & Scientific Notation
3.1 Integer Exponents & Laws of Exponents 3.2 Scientific Notation & Operations
4 Chapter 4: Algebraic Expressions
4.1 Evaluating & Translating Algebraic Expressions 4.2 Simplifying, Combining Like Terms, & Factoring
5 Chapter 5: Linear Equations & Inequalities
5.1 Solving Single-Variable Linear Equations 5.2 Solving & Graphing Linear Inequalities 5.3 Systems of Linear Equations
6 Chapter 6: Linear Applications & Graphing
6.1 Slope & Rate of Change 6.2 Forms of Linear Equations & Graphing 6.3 Linear Modeling & Inequality Graphs
7 Chapter 7: Probability & Set Theory
7.1 Fundamental Probability Concepts 7.2 Set Theory, Operations, & Venn Diagrams
8 Chapter 8: Descriptive Statistics & Data Displays
8.1 Measures of Center: Mean, Median, Mode, & Weighted Average 8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation 8.3 Data Displays: Histograms, Box Plots, & Scatter Plots
9 Chapter 9: Geometry Concepts & Measurement
9.1 2D Geometry: Perimeter, Area, & Circles 9.2 3D Geometry: Surface Area & Volume of Solids 9.3 Triangles, Angles, & the Pythagorean Theorem
9.2 3D Geometry: Surface Area & Volume of Solids
Key Takeaways Explain This Topic AI
Volume measures three-dimensional internal capacity (V in cubic units), whereas surface area measures total external two-dimensional boundary coverage (SA in square units).
Right rectangular prisms have V = lwh and SA = 2(lw + lh + wh); right circular cylinders have V = πr²h and total SA = 2πr² + 2πrh.
Pointed solids (cones and pyramids) contain exactly one-third the volume of their corresponding prisms/cylinders: Cone V = (1/3)πr²h and Pyramid V = (1/3)Bh; slant height l = √(r² + h²) is required for cone lateral area (LA = πrl).
Spheres have volume V = (4/3)πr³ and surface area SA = 4πr²; a closed solid hemisphere has total surface area SA = 3πr² (2πr² curved dome + πr² circular flat base).
Under 3D linear scaling by factor k, surface area scales by k² and volume (as well as fluid capacity and mass) scales by k³.
Last updated: August 2026
Foundations of Three-Dimensional Geometry
Three-dimensional (3D) geometry on the ACCUPLACER Quantitative Reasoning, Algebra, and Statistics (QAS) test extends planar concepts into three physical dimensions: length, width, and height. Spatial geometric problems evaluate two distinct quantities:
Volume ( V V V ) : The amount of three-dimensional space enclosed within a solid, representing its capacity or interior containment. Volume is measured in cubic units (e.g., in 3 , ft 3 , cm 3 , m 3 \text{in}^3, \text{ft}^3, \text{cm}^3, \text{m}^3 in 3 , ft 3 , cm 3 , m 3 ) or liquid capacity units (gallons, liters, milliliters, where 1 cm 3 = 1 mL 1\text{ cm}^3 = 1\text{ mL} 1 cm 3 = 1 mL and 1 , 000 cm 3 = 1 L 1,000\text{ cm}^3 = 1\text{ L} 1 , 000 cm 3 = 1 L ).
Surface Area ( S A SA S A ) : The total two-dimensional area of all exterior faces and curved surfaces enclosing the 3D solid. Surface area is measured in square units (e.g., in 2 , ft 2 , cm 2 , m 2 \text{in}^2, \text{ft}^2, \text{cm}^2, \text{m}^2 in 2 , ft 2 , cm 2 , m 2 ).
Lateral Surface Area ( L A LA L A ) vs. Total Surface Area ( S A SA S A ) :
Lateral Area ( L A LA L A ) : The area of the surrounding side surfaces of the solid, strictly excluding top and bottom base faces.
Total Surface Area ( S A SA S A ) : The complete sum of lateral area plus the area of all bases: S A = L A + ∑ A bases SA = LA + \sum A_{\text{bases}} S A = L A + ∑ A bases .
Polyhedra: Prisms & Cubes
A polyhedron is a solid bounded by flat polygonal faces. Rectangular Prism: Cube (All edges equal to s):
- +-----------------+ +----------+
- / /| / /|
- / / | / / |
- +-----------------+ | h +----------+ | s
- | | | | | |
- | | | | | |
- | | + | | +
- | | / w | | / s
- | |/ | |/
- +-----------------+ +----------+
- l s
- V = l·w·h, SA = 2(lw+lh+wh) V = s³, SA = 6s²
- Space Diagonal = √(l²+w²+h²) Space Diagonal = s√3
- 1. General Right Prisms
In any right prism with identical parallel bases of area B B B separated by perpendicular height h h h :
Volume : V = B ⋅ h V = B \cdot h V = B ⋅ h
Lateral Area : L A = P base ⋅ h LA = P_{\text{base}} \cdot h L A = P base ⋅ h (where P base P_{\text{base}} P base is the perimeter of the base polygon)
Total Surface Area : S A = L A + 2 B = P base ⋅ h + 2 B SA = LA + 2B = P_{\text{base}} \cdot h + 2B S A = L A + 2 B = P base ⋅ h + 2 B
- 2. Right Rectangular Prisms
- Volume : V = l ⋅ w ⋅ h V = l \cdot w \cdot h V = l ⋅ w ⋅ h
Total Surface Area : S A = 2 l w + 2 l h + 2 w h = 2 ( l w + l h + w h ) SA = 2lw + 2lh + 2wh = 2(lw + lh + wh) S A = 2 lw + 2 l h + 2 w h = 2 ( lw + l h + w h )
Open Box (No Top Lid) : S A open = l w + 2 l h + 2 w h SA_{\text{open}} = lw + 2lh + 2wh S A open = lw + 2 l h + 2 w h
Interior 3D Space Diagonal : The longest straight line segment connecting two opposite interior corners: d = l 2 + w 2 + h 2 d = \sqrt{l^2 + w^2 + h^2} d = l 2 + w 2 + h 2
3. Cubes
A regular rectangular prism where length, width, and height are all equal to side length s s s ( l = w = h = s l = w = h = s l = w = h = 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 (since all 6 6 6 square faces each have area s 2 s^2 s 2 )
Interior 3D Space Diagonal : d = s 2 + s 2 + s 2 = 3 s 2 = s 3 d = \sqrt{s^2 + s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3} d = s 2 + s 2 + s 2 = 3 s 2 = s 3
Solids of Revolution: Cylinders & Cones
Solids of revolution feature circular cross-sections formed by rotating 2D planar shapes around an axis. Right Circular Cylinder: Right Circular Cone:
- .---. /\
- / r \ / \
- | • | / | \ Slant Height l
- +---------+ / |h \ l = √(r²+h²)
- | | / | \
- | | h +----+-----+
- | | ( • r )
- +---------+ '---. .---'
- \ /
- '---' V = (1/3)πr²h
- V = πr²h LA = πrl
- LA = 2πrh, SA = 2πr² + 2πrh SA = πr² + πrl
- 1. Right Circular Cylinders
A cylinder consists of two parallel, congruent circular bases of radius r r r separated by vertical height h h h :
- Base Area : B = π r 2 B = \pi r^2 B = π r 2
- Volume : V = B ⋅ h = π r 2 h V = B \cdot h = \pi r^2 h V = B ⋅ h = π r 2 h
Lateral Area ( L A LA L A ) : Unrolling the curved wall of a cylinder produces a rectangle of length equal to base circumference ( 2 π r 2\pi r 2 π r ) and height h h h : L A = 2 π r h LA = 2\pi r h L A = 2 π r h
Total Surface Area ( S A SA S A ) : Lateral area plus the two circular base caps: S A = 2 π r 2 + 2 π r h = 2 π r ( r + h ) SA = 2\pi r^2 + 2\pi rh = 2\pi r (r + h) S A = 2 π r 2 + 2 π r h = 2 π r ( r + h )
Open Cylinder (Pipe or Open Tank) :
- Hollow open pipe (no end caps): S A = 2 π r h SA = 2\pi rh S A = 2 π r h
- Open-top container (one circular base): S A = π r 2 + 2 π r h SA = \pi r^2 + 2\pi rh S A = π r 2 + 2 π r h
- 2. Right Circular Cones
A cone has a single circular base of radius r r r tapering smoothly to an apex point directly above the center at vertical height h h h :
Volume : Exactly one-third of a cylinder with matching radius and height: V = 1 3 π r 2 h V = \frac{1}{3}\pi r^2 h V = 3 1 π r 2 h
Slant Height ( l l l ) : The diagonal surface length from the outer circular base rim to the apex, calculated using the Pythagorean theorem: l = r 2 + h 2 l = \sqrt{r^2 + h^2} l = r 2 + h 2
Lateral Surface Area ( L A LA L A ) : L A = π r l = π r r 2 + h 2 LA = \pi r l = \pi r \sqrt{r^2 + h^2} L A = π r l = π r r 2 + h 2
Total Surface Area ( S A SA S A ) : Base circle area plus lateral curved surface: S A = π r 2 + π r l = π r ( r + l ) SA = \pi r^2 + \pi r l = \pi r (r + l) S A = π r 2 + π r l = π r ( r + l )
Pyramids & Spheres Right Square Pyramid: Sphere:
/\ .---' '---.
- / \ / r \
- / | \ Slant Height l_face | •---------> | Radius r
- / |h \ \ /
- +---+----+ '---. .---'
- / / V = (4/3)πr³
- +--------+ s SA = 4πr²
- s
- V = (1/3)s²h, SA = s² + 2sl_face
- 1. Right Pyramids
A pyramid consists of a polygonal base of area B B B and triangular lateral faces meeting at an apex point at height h h h :
- Volume : V = 1 3 B h V = \frac{1}{3} B h V = 3 1 B h
- Square Pyramid ( B = s 2 B = s^2 B = s 2 ) : V = 1 3 s 2 h V = \frac{1}{3} s^2 h V = 3 1 s 2 h
Slant Height of Triangular Faces ( l face l_{\text{face}} l face ) : l face = h 2 + ( s 2 ) 2 l_{\text{face}} = \sqrt{h^2 + \left(\frac{s}{2}\right)^2} l face = h 2 + ( 2 s ) 2
Total Surface Area : S A = s 2 + 4 ( 1 2 s ⋅ l face ) = s 2 + 2 s ⋅ l face SA = s^2 + 4 \left(\frac{1}{2} s \cdot l_{\text{face}}\right) = s^2 + 2 s \cdot l_{\text{face}} S A = s 2 + 4 ( 2 1 s ⋅ l face ) = s 2 + 2 s ⋅ l face
2. Spheres & Hemispheres
A sphere is the set of all points in 3D space equidistant from a center point by radius r r r :
- Volume : V = 4 3 π r 3 V = \frac{4}{3}\pi r^3 V = 3 4 π r 3
- Surface Area : S A = 4 π r 2 SA = 4\pi r^2 S A = 4 π r 2
Hemisphere (Half-Sphere) :
- Volume : V = 2 3 π r 3 V = \frac{2}{3}\pi r^3 V = 3 2 π r 3
- Curved Dome Surface Area : S A dome = 2 π r 2 SA_{\text{dome}} = 2\pi r^2 S A dome = 2 π r 2
Total Surface Area of Closed Solid Hemisphere : Curved dome plus flat circular base: S A total = 2 π r 2 + π r 2 = 3 π r 2 SA_{\text{total}} = 2\pi r^2 + \pi r^2 = 3\pi r^2 S A total = 2 π r 2 + π r 2 = 3 π r 2
3D Geometry Formula Summary Table
Solid Type Volume Formula ( V V V ) Lateral Area ( L A LA L A ) Total Surface Area ( S A SA S A )
Rectangular Prism V = l w h V = lwh V = lw h L A = 2 l h + 2 w h LA = 2lh + 2wh L A = 2 l h + 2 w h S A = 2 l w + 2 l h + 2 w h SA = 2lw + 2lh + 2wh S A = 2 lw + 2 l h + 2 w h
Cube V = s 3 V = s^3 V = s 3 L A = 4 s 2 LA = 4s^2 L A = 4 s 2 S A = 6 s 2 SA = 6s^2 S A = 6 s 2
Right Cylinder V = π r 2 h V = \pi r^2 h V = π r 2 h L A = 2 π r h LA = 2\pi rh L A = 2 π r h S A = 2 π r 2 + 2 π r h SA = 2\pi r^2 + 2\pi rh S A = 2 π r 2 + 2 π r h
Right Cone V = 1 3 π r 2 h V = \frac{1}{3}\pi r^2 h V = 3 1 π r 2 h L A = π r l LA = \pi r l L A = π r l S A = π r 2 + π r l SA = \pi r^2 + \pi r l S A = π r 2 + π r l
Right Pyramid V = 1 3 B h V = \frac{1}{3}Bh V = 3 1 B h L A = ∑ Triangular Faces LA = \sum \text{Triangular Faces} L A = ∑ Triangular Faces S A = B + L A SA = B + LA S A = B + L A
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
Closed Hemisphere V = 2 3 π r 3 V = \frac{2}{3}\pi r^3 V = 3 2 π r 3 L A = 2 π r 2 LA = 2\pi r^2 L A = 2 π r 2 S A = 3 π r 2 SA = 3\pi r^2 S A = 3 π r 2
Composite 3D Solids & Engineering Applications
In real-world applications, physical containers and architectural structures combine multiple geometric solids:
Additive Volume Decomposition : The total volume of a composite solid is the simple sum of its component volumes: V total = V solid 1 + V solid 2 + ⋯ + V solid n V_{\text{total}} = V_{\text{solid 1}} + V_{\text{solid 2}} + \dots + V_{\text{solid } n} V total = V solid 1 + V solid 2 + ⋯ + V solid n
Subtractive Surface Area (Excluding Internal Seams) : When two solids are joined (such as a cylinder topped with a hemisphere to form a grain silo), the shared connecting face becomes internal. Do not include internal touching faces in the total exterior surface area: S A silo = S A cylinder walls + S A hemisphere dome = 2 π r h + 2 π r 2 SA_{\text{silo}} = SA_{\text{cylinder walls}} + SA_{\text{hemisphere dome}} = 2\pi rh + 2\pi r^2 S A silo = S A cylinder walls + S A hemisphere dome = 2 π r h + 2 π r 2
Scaling Dimensionality Laws in Three Dimensions
Understanding how scaling linear dimensions impacts multi-dimensional measurements is crucial for the ACCUPLACER test:
When all linear dimensions of any 3D solid are scaled by a positive factor k k k ( k > 0 k > 0 k > 0 ):
Linear dimensions (length, width, height, radius, perimeter) scale by k 1 = k k^1 = k k 1 = k .
Surface areas (base area, lateral area, total surface area) scale by k 2 k^2 k 2 : S A new = k 2 ⋅ S A original SA_{\text{new}} = k^2 \cdot SA_{\text{original}} S A new = k 2 ⋅ S A original
Volumes, Capacities, and Weights scale by k 3 k^3 k 3 : V new = k 3 ⋅ V original V_{\text{new}} = k^3 \cdot V_{\text{original}} V new = k 3 ⋅ V original 3D Scaling Impact Summary (Linear Scale Factor k = 2):
- Linear Dimensions: 1x ---> 2x (k = 2)
- Surface Area: 1x ---> 4x (k² = 4)
- Volume / Capacity: 1x ---> 8x (k³ = 8)
- Step-by-Step Worked Examples
- Worked Example 1: Conical Hopper Capacity & Lateral Sheet Metal
A conical feed hopper has a top circular base radius of r = 6 ft r = 6\text{ ft} r = 6 ft and a vertical depth of h = 8 ft h = 8\text{ ft} h = 8 ft . Calculate the exact storage volume of the hopper and the exact area of sheet metal required to construct the curved lateral wall (excluding the open circular top).
Calculate Storage Volume ( V V V ) : V = 1 3 π r 2 h = 1 3 π ( 6 2 ) ( 8 ) = 1 3 π ( 36 ) ( 8 ) = 12 π × 8 = 96 π cu ft V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6^2)(8) = \frac{1}{3}\pi (36)(8) = 12\pi \times 8 = 96\pi\text{ cu ft} V = 3 1 π r 2 h = 3 1 π ( 6 2 ) ( 8 ) = 3 1 π ( 36 ) ( 8 ) = 12 π × 8 = 96 π cu ft
Calculate Slant Height ( l l l ) : l = r 2 + h 2 = 6 2 + 8 2 = 36 + 64 = 100 = 10 ft l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ ft} l = r 2 + h 2 = 6 2 + 8 2 = 36 + 64 = 100 = 10 ft
Calculate Lateral Sheet Metal Surface Area ( L A LA L A ) : L A = π r l = π ( 6 ) ( 10 ) = 60 π sq ft LA = \pi r l = \pi (6)(10) = 60\pi\text{ sq ft} L A = π r l = π ( 6 ) ( 10 ) = 60 π sq ft
Worked Example 2: Agricultural Grain Silo (Composite Solid)
An agricultural grain silo consists of a vertical right cylinder of radius r = 3 m r = 3\text{ m} r = 3 m and height h = 10 m h = 10\text{ m} h = 10 m , topped by a solid hemisphere of radius r = 3 m r = 3\text{ m} r = 3 m . Find the total internal capacity volume of the silo.
Calculate Cylinder Volume : V cyl = π r 2 h = π ( 3 2 ) ( 10 ) = 90 π m 3 V_{\text{cyl}} = \pi r^2 h = \pi (3^2)(10) = 90\pi\text{ m}^3 V cyl = π r 2 h = π ( 3 2 ) ( 10 ) = 90 π m 3
Calculate Hemisphere Dome Volume : V hemi = 2 3 π r 3 = 2 3 π ( 3 3 ) = 2 3 π ( 27 ) = 18 π m 3 V_{\text{hemi}} = \frac{2}{3}\pi r^3 = \frac{2}{3}\pi (3^3) = \frac{2}{3}\pi (27) = 18\pi\text{ m}^3 V hemi = 3 2 π r 3 = 3 2 π ( 3 3 ) = 3 2 π ( 27 ) = 18 π m 3
Total Silo Volume : V total = V cyl + V hemi = 90 π + 18 π = 108 π cubic meters V_{\text{total}} = V_{\text{cyl}} + V_{\text{hemi}} = 90\pi + 18\pi = 108\pi\text{ cubic meters} V total = V cyl + V hemi = 90 π + 18 π = 108 π cubic meters
Worked Example 3: 3D Scaling & Casting Mold Mass Ratio
A machine shop manufactures solid brass spherical bearings. A standard bearing with radius r 1 = 2 cm r_1 = 2\text{ cm} r 1 = 2 cm weighs 0.30 kg 0.30\text{ kg} 0.30 kg . If the shop manufactures a large industrial bearing with radius r 2 = 6 cm r_2 = 6\text{ cm} r 2 = 6 cm using the same brass alloy, what is the weight of the large bearing?
Determine Linear Scale Factor ( k k k ) : k = r 2 r 1 = 6 cm 2 cm = 3 k = \frac{r_2}{r_1} = \frac{6\text{ cm}}{2\text{ cm}} = 3 k = r 1 r 2 = 2 cm 6 cm = 3
Apply Cubic Scaling Law to Mass/Volume : Mass Factor = k 3 = 3 3 = 27 \text{Mass Factor} = k^3 = 3^3 = 27 Mass Factor = k 3 = 3 3 = 27
Calculate New Bearing Weight : Weight large = 27 × 0.30 kg = 8.10 kg \text{Weight}_{\text{large}} = 27 \times 0.30\text{ kg} = 8.10\text{ kg} Weight large = 27 × 0.30 kg = 8.10 kg
Common Pitfalls & ACCUPLACER Exam Traps
Confusing Vertical Height ( h h h ) with Slant Height ( l l l ) : Always use vertical height h h h for volume formulas ( V = 1 3 π r 2 h V = \frac{1}{3}\pi r^2 h V = 3 1 π r 2 h ) and slant height l l l for cone/pyramid lateral surface area ( L A = π r l LA = \pi r l L A = π r l ).
Including Internal Boundary Faces in Composite Solids : When calculating surface area for composite objects, never add the shared partition base where two shapes meet.
Omitting the 1 3 \frac{1}{3} 3 1 Factor for Cones and Pyramids : Cones and pyramids hold exactly one-third the volume of cylinders and prisms with identical bases and heights.
Linear vs. Cubic Scaling Confusion : Multiplying linear dimensions by k k k multiplies volume by k 3 k^3 k 3 , not k k k .
Closed vs. Open Hemisphere Surface Area : An open hemispherical bowl has surface area 2 π r 2 2\pi r^2 2 π r 2 , while a solid closed hemisphere with a flat base has total area 3 π r 2 3\pi r^2 3 π r 2 .