8.3 3D Geometry: Surface Area & Volume of Solids

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 $\text{in}^2$ or $\text{cm}^2$) and three-dimensional volume measurements (measured in cubic units like $\text{ft}^3$ or $\text{m}^3$) 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 ($V$): The internal space or capacity enclosed by the solid. $V = \text{Base Area} \times \text{Height} = l \times w \times h$
• Total Surface Area ($SA$): The combined sum of the areas of all six rectangular faces (front/back, top/bottom, left/right). $SA = 2(lw + lh + wh)$
• Space Diagonal ($d$): The 3D straight-line distance between opposite corners. $d = \sqrt{l^2 + w^2 + h^2}$

Cube Formulas

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

• Volume: $V = s^3$
• Total Surface Area: $SA = 6s^2$
• Space Diagonal: $d = s\sqrt{3}$

---

Right Circular Cylinders & Cones

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

1. Right Circular Cylinder

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

• Base Area ($B$): $B = \pi r^2$
• Volume ($V$): Base Area $\times$ Height $V = \pi r^2 h$
• Lateral Surface Area ($LSA$): Unrolled into a rectangle of length equal to circumference ($2\pi r$) and height $h$. $LSA = 2\pi r h$
• Total Surface Area ($TSA$): Sum of lateral surface area plus both circular bases. $TSA = 2\pi r^2 + 2\pi r h = 2\pi r (r + h)$

2. Right Circular Cone

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

• Slant Height ($l$): Distance from any point on the base circle edge to the apex. Uses the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$
• Volume ($V$): Exactly one-third the volume of a cylinder with the same base and height. $V = \frac{1}{3} \pi r^2 h$
• Total Surface Area ($SA$): Base area plus lateral area. $SA = \pi r^2 + \pi r l$

---

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): $V = \frac{1}{3} B h$ where $B$ is the area of the base polygon.
• Square Pyramid Volume: $V = \frac{1}{3} s^2 h$.
• Total Surface Area: $SA = B + \text{Sum of lateral triangular faces}$.

2. Spheres

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

• Volume ($V$): $V = \frac{4}{3} \pi r^3$
• Surface Area ($SA$): $SA = 4\pi r^2$
• Hemisphere (Half-Sphere):
  • Volume $= \frac{2}{3}\pi r^3$
  • Total Surface Area (including flat circular base) $= 3\pi r^2$

---

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 $k$:

1. Linear Dimensions (edges, perimeters, heights, radii) scale by factor $k^1 = k$.
2. Surface Area & Base Area scale by factor $k^2$.
3. Volume & Capacity scale by factor $k^3$.

Scaling Effect Summary Table

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

1. Identify dimensions: $l = 6$, $w = 4$, $h = 3$.
2. Apply Surface Area Formula: $SA = 2(lw + lh + wh)$.
3. Calculate products:
   • $lw = 6 \times 4 = 24$
   • $lh = 6 \times 3 = 18$
   • $wh = 4 \times 3 = 12$
4. Sum and multiply by 2: $SA = 2(24 + 18 + 12) = 2(54) = 108\text{ sq ft}$.

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 $\pi$?

1. Find radius: Radius $r = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5$ meters.
2. Apply Volume Formula: $V = \pi r^2 h$.
3. Substitute values: $V = \pi (5^2) (8) = \pi (25) (8) = 200\pi\text{ m}^3$.

---

Key Takeaways Summary

• Rectangular Prism: $V = lwh$; $SA = 2(lw + lh + wh)$.
• Cube: $V = s^3$; $SA = 6s^2$.
• Cylinder: $V = \pi r^2 h$; $TSA = 2\pi r^2 + 2\pi rh$.
• Cone: $V = \frac{1}{3}\pi r^2 h$; Pyramid: $V = \frac{1}{3}Bh$.
• Sphere: $V = \frac{4}{3}\pi r^3$; $SA = 4\pi r^2$.
• Scaling Rules: Linear $\to k$, Area $\to k^2$, Volume $\to k^3$.