3D Geometry & Volume | Math Learning Guide

📋 Prerequisites

3D geometry extends 2D ideas into space — volume, surface area, and spatial relationships. The good news: most competition 3D problems reduce to 2D problems if you take the right cross-section. Find the right slice, draw it flat, and use everything you already know about triangles and circles.

📚 Key Concepts

Volume Formulas

Rectangular prism (box): $ V = lwh $ (length $\times$ width $\times$ height)
Cube: $ V = s^3 $
Cylinder: $ V = \pi r^2 h $ (base area $\times$ height)
Prism (any base): $ V = B \cdot h $ where $B$ = area of base, $h$ = height (distance between bases)
Pyramid: $ V = \frac{1}{3} B h $ (1/3 of the corresponding prism)
Cone: $ V = \frac{1}{3} \pi r^2 h $ (1/3 of the corresponding cylinder)
Sphere: $ V = \frac{4}{3} \pi r^3 $

Surface Area Formulas

Rectangular prism: $ SA = 2(lw + lh + wh) $ (6 faces, 3 pairs)
Cube: $ SA = 6s^2 $
Cylinder: $ SA = 2\pi r^2 + 2\pi r h $ (two circles + lateral surface)
Sphere: $ SA = 4\pi r^2 $
Cone: $ SA = \pi r^2 + \pi r l $ where $l$ = slant height

Space Diagonal of a Box

For a rectangular prism with dimensions $ l, w, h $, the distance between opposite corners (space diagonal):

\[ d = \sqrt{l}}^2 + w^2 + h^2} \]

This is the 3D Pythagorean theorem — apply it twice: first find a face diagonal, then use that as one leg with the height.

Cube special case: $ d = s\sqrt{3} $

Cross-Section Method

Many 3D problems become 2D problems if you take the right cross-sectional slice:

$Slicing a cylinder vertically \to rectangle$
$Slicing a cone through its apex and diameter \to isosceles triangle$
$Slicing a sphere through its center \to circle (great circle)$
$Slicing a pyramid parallel to base \to smaller similar base shape$

Strategy: identify the plane that contains the information you need, draw the cross-section, and solve in 2D.

Similar Solids

If two solids are similar with scale factor $k$ (ratio of corresponding linear dimensions):

Surface area ratio: $ k^2 $
Volume ratio: $ k^3 $

$Linear scale factor k \to area scales by k^{2}, volume by k^{3}. This is the 3D version of the similarity area ratio from 2D.$

Cavalieri's Principle

If two solids have the same height and the same cross-sectional area at every level, they have the same volume.

Intuition: same height, same "width" $at every height \to same total volume, regardless of shape.$

Used to derive volume formulas and solve tricky volume problems where direct calculation is hard.

Cone & Pyramid: 1/3 Factor

Why 1/3? A cone/pyramid of the same base and height as a cylinder/prism has exactly 1/3 the volume.

Three pyramids of equal volume make up the corresponding prism. You can think of it as the 3D analog of the triangle area factor of 1/2.

⚠️ Common Mistake: Don't confuse lateral (curved) surface area with total surface area. A cylinder's lateral surface area is $2\pi rh$; total surface area adds the two circular bases ($2\pi r^2$). Also: volume of cone/pyramid needs the 1/3 — it's easy to forget. And for similar solids: area ratio is $k^2$, volume is $k^3$ — make sure you use the right exponent for what's being asked.

💡 Key Insight: Almost every competition 3D problem can be reduced to 2D by taking a strategic cross-section. When you see a sphere in a cube, or a cone with given height and radius, ask: what plane cuts through the important points? Draw that slice flat. The space diagonal formula is just Pythagoras applied twice — don't memorize it separately, derive it by finding a face diagonal first, then the space diagonal.

✏️ Example Problems

📝 Example 1 (Volume of a cylinder)

A cylinder has radius 3 and height 7. What is its volume?
(Give just the coefficient of $\pi$.)

Volume of cylinder: $ V = \pi r^2 h $

Step 1: Identify values: r = 3, h = 7

Step 2: Compute base area first.

Base area = $ \pi r^2 = \pi (3)^2 = 9\pi $

Step 3: Multiply by height.

\[ V = 9\pi \times 7 = 63\pi \]

The coefficient is 63.

Answer: 63

📝 Example 2 (Space diagonal of a cube)

A cube has edge length 5. What is the length of its space diagonal
(from one corner to the opposite corner through the interior)?
Express your answer in simplest radical form: $a\sqrt{b}$. What is $a + b$?

Step 1: Find the diagonal of one face (2D Pythagoras).

Face diagonal = $ \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} $

Step 2: Use the face diagonal and an edge to find the space diagonal (another Pythagoras).

\[ \text{Space diagonal} = \sqrt{(5\sqrt{2})^2 + 5^2} \]

$ = \sqrt{50 + 25} = \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} $

Direct formula: For cube with side s: space diagonal = $ s\sqrt{3} $

$ 5\sqrt{3} $, so a = 5, b = 3

a + b = 5 + 3 = 8

Answer: 8

📝 Example 3 (Similar solids — volume ratio)

Two similar spheres have radii in ratio 2:3.
If the smaller sphere has volume 32, what is the volume of the larger sphere?

Similar solids: $linear scale factor k \to volume scales by$ $k^3$

Step 1: Linear ratio (small : large) = 2 : 3

Scale factor from small to large: $ k = \frac{3}{2} $

Step 2: Volume ratio = $k^3$.

\[ \text{Volume ratio} = \left(\frac{3}{2}\right)^3 = \frac{27}{8} \]

Step 3: $Larger volume = smaller volume \times volume ratio.$

$ V_{large} = 32 \times \frac{27}{8} = 4 \times 27 = 108 $

Check with sphere formula:

Small: $ \frac{4}{3}\pi(2)^3 = \frac{32}{3}\pi = 32 $ (in our units)

Large: $ \frac{4}{3}\pi(3)^3 = \frac{108}{3}\pi = 108\pi/\text{units} = 108 $. ✓

Answer: 108