9.2 3D Geometry: Surface Area & Volume of Solids

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 ($V$): The amount of three-dimensional space enclosed within a solid, representing its capacity or interior containment. Volume is measured in cubic units (e.g., $\text{in}^3, \text{ft}^3, \text{cm}^3, \text{m}^3$) or liquid capacity units (gallons, liters, milliliters, where $1\text{ cm}^3 = 1\text{ mL}$ and $1,000\text{ cm}^3 = 1\text{ L}$).
• Surface Area ($SA$): 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., $\text{in}^2, \text{ft}^2, \text{cm}^2, \text{m}^2$).
• Lateral Surface Area ($LA$) vs. Total Surface Area ($SA$):
  • Lateral Area ($LA$): The area of the surrounding side surfaces of the solid, strictly excluding top and bottom base faces.
  • Total Surface Area ($SA$): The complete sum of lateral area plus the area of all bases: $SA = LA + \sum A_{\text{bases}}$.

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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 $B$ separated by perpendicular height $h$:

• Volume: $V = B \cdot h$
• Lateral Area: $LA = P_{\text{base}} \cdot h$ (where $P_{\text{base}}$ is the perimeter of the base polygon)
• Total Surface Area: $SA = LA + 2B = P_{\text{base}} \cdot h + 2B$

2. Right Rectangular Prisms

• Volume: $V = l \cdot w \cdot h$
• Total Surface Area: $SA = 2lw + 2lh + 2wh = 2(lw + lh + wh)$
• Open Box (No Top Lid): $SA_{\text{open}} = lw + 2lh + 2wh$
• Interior 3D Space Diagonal: The longest straight line segment connecting two opposite interior corners: $d = \sqrt{l^2 + w^2 + h^2}$

3. Cubes

A regular rectangular prism where length, width, and height are all equal to side length $s$ ($l = w = h = s$):

• Volume: $V = s^3$
• Total Surface Area: $SA = 6s^2$ (since all $6$ square faces each have area $s^2$)
• Interior 3D Space Diagonal: $d = \sqrt{s^2 + s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3}$

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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 $r$ separated by vertical height $h$:

• Base Area: $B = \pi r^2$
• Volume: $V = B \cdot h = \pi r^2 h$
• Lateral Area ($LA$): Unrolling the curved wall of a cylinder produces a rectangle of length equal to base circumference ($2\pi r$) and height $h$: $LA = 2\pi r h$
• Total Surface Area ($SA$): Lateral area plus the two circular base caps: $SA = 2\pi r^2 + 2\pi rh = 2\pi r (r + h)$
• Open Cylinder (Pipe or Open Tank):
  • Hollow open pipe (no end caps): $SA = 2\pi rh$
  • Open-top container (one circular base): $SA = \pi r^2 + 2\pi rh$

2. Right Circular Cones

A cone has a single circular base of radius $r$ tapering smoothly to an apex point directly above the center at vertical height $h$:

• Volume: Exactly one-third of a cylinder with matching radius and height: $V = \frac{1}{3}\pi r^2 h$
• Slant Height ($l$): The diagonal surface length from the outer circular base rim to the apex, calculated using the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$
• Lateral Surface Area ($LA$): $LA = \pi r l = \pi r \sqrt{r^2 + h^2}$
• Total Surface Area ($SA$): Base circle area plus lateral curved surface: $SA = \pi r^2 + \pi r l = \pi r (r + l)$

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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 $B$ and triangular lateral faces meeting at an apex point at height $h$:

• Volume: $V = \frac{1}{3} B h$
• Square Pyramid ($B = s^2$): $V = \frac{1}{3} s^2 h$
• Slant Height of Triangular Faces ($l_{\text{face}}$): $l_{\text{face}} = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}$
• Total Surface Area: $SA = s^2 + 4 \left(\frac{1}{2} s \cdot l_{\text{face}}\right) = s^2 + 2 s \cdot l_{\text{face}}$

2. Spheres & Hemispheres

A sphere is the set of all points in 3D space equidistant from a center point by radius $r$:

• Volume: $V = \frac{4}{3}\pi r^3$
• Surface Area: $SA = 4\pi r^2$
• Hemisphere (Half-Sphere):
  • Volume: $V = \frac{2}{3}\pi r^3$
  • Curved Dome Surface Area: $SA_{\text{dome}} = 2\pi r^2$
  • Total Surface Area of Closed Solid Hemisphere: Curved dome plus flat circular base: $SA_{\text{total}} = 2\pi r^2 + \pi r^2 = 3\pi r^2$

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3D Geometry Formula Summary Table

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Composite 3D Solids & Engineering Applications

In real-world applications, physical containers and architectural structures combine multiple geometric solids:

1. Additive Volume Decomposition: The total volume of a composite solid is the simple sum of its component volumes: $V_{\text{total}} = V_{\text{solid 1}} + V_{\text{solid 2}} + \dots + V_{\text{solid } n}$
2. 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: $SA_{\text{silo}} = SA_{\text{cylinder walls}} + SA_{\text{hemisphere dome}} = 2\pi rh + 2\pi r^2$

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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 $k$ ($k > 0$):

• Linear dimensions (length, width, height, radius, perimeter) scale by $k^1 = k$.
• Surface areas (base area, lateral area, total surface area) scale by $k^2$: $SA_{\text{new}} = k^2 \cdot SA_{\text{original}}$
• Volumes, Capacities, and Weights scale by $k^3$: $V_{\text{new}} = k^3 \cdot V_{\text{original}}$

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)

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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 $r = 6\text{ ft}$ and a vertical depth of $h = 8\text{ ft}$. 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).

1. Calculate Storage Volume ($V$): $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6^2)(8) = \frac{1}{3}\pi (36)(8) = 12\pi \times 8 = 96\pi\text{ cu ft}$
2. Calculate Slant Height ($l$): $l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ ft}$
3. Calculate Lateral Sheet Metal Surface Area ($LA$): $LA = \pi r l = \pi (6)(10) = 60\pi\text{ sq ft}$

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Worked Example 2: Agricultural Grain Silo (Composite Solid)

An agricultural grain silo consists of a vertical right cylinder of radius $r = 3\text{ m}$ and height $h = 10\text{ m}$, topped by a solid hemisphere of radius $r = 3\text{ m}$. Find the total internal capacity volume of the silo.

1. Calculate Cylinder Volume: $V_{\text{cyl}} = \pi r^2 h = \pi (3^2)(10) = 90\pi\text{ m}^3$
2. Calculate Hemisphere Dome Volume: $V_{\text{hemi}} = \frac{2}{3}\pi r^3 = \frac{2}{3}\pi (3^3) = \frac{2}{3}\pi (27) = 18\pi\text{ m}^3$
3. Total Silo Volume: $V_{\text{total}} = V_{\text{cyl}} + V_{\text{hemi}} = 90\pi + 18\pi = 108\pi\text{ cubic meters}$

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Worked Example 3: 3D Scaling & Casting Mold Mass Ratio

A machine shop manufactures solid brass spherical bearings. A standard bearing with radius $r_1 = 2\text{ cm}$ weighs $0.30\text{ kg}$. If the shop manufactures a large industrial bearing with radius $r_2 = 6\text{ cm}$ using the same brass alloy, what is the weight of the large bearing?

1. Determine Linear Scale Factor ($k$): $k = \frac{r_2}{r_1} = \frac{6\text{ cm}}{2\text{ cm}} = 3$
2. Apply Cubic Scaling Law to Mass/Volume: $\text{Mass Factor} = k^3 = 3^3 = 27$
3. Calculate New Bearing Weight: $\text{Weight}_{\text{large}} = 27 \times 0.30\text{ kg} = 8.10\text{ kg}$

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Common Pitfalls & ACCUPLACER Exam Traps

1. Confusing Vertical Height ($h$) with Slant Height ($l$): Always use vertical height $h$ for volume formulas ($V = \frac{1}{3}\pi r^2 h$) and slant height $l$ for cone/pyramid lateral surface area ($LA = \pi r l$).
2. Including Internal Boundary Faces in Composite Solids: When calculating surface area for composite objects, never add the shared partition base where two shapes meet.
3. Omitting the $\frac{1}{3}$ Factor for Cones and Pyramids: Cones and pyramids hold exactly one-third the volume of cylinders and prisms with identical bases and heights.
4. Linear vs. Cubic Scaling Confusion: Multiplying linear dimensions by $k$ multiplies volume by $k^3$, not $k$.
5. Closed vs. Open Hemisphere Surface Area: An open hemispherical bowl has surface area $2\pi r^2$, while a solid closed hemisphere with a flat base has total area $3\pi r^2$.