ASL-3.1, 3-DIMENSIONAL GEOMETRY

Term Definition / Formula
Distance (3-D) Between P(x1, y1, z1) and Q(x2, y2, z2):
d = √[(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2]
Midpoint (3-D) Midpoint M = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
Sphere Volume: V = (4/3)πr3. Surface area: A = 4πr2.
Hemisphere Volume: V = (1/2)(4/3)πr3 = (2/3)πr3. Curved surface area = 2πr2; total surface area (including base) = 3πr2.
Cone (right) Volume: V = (1/3)πr2h. Surface area: A = πr2 + πr s, where s = slant height = √(r2 + h2).
Pyramid (right) Volume: V = (1/3)·(base area)·h. Lateral area depends on base perimeter and slant height: Alat = (1/2)·(perimeter of base)·s.
Angle between two lines (vectors) If direction vectors u and v, cos θ = (u · v) / (|u||v|).
Angle between line and plane If line direction u and plane normal n: angle φ between line and plane satisfies sin φ = |u · n| / (|u||n|). Equivalently, angle between line and normal is θ where cos θ = |u·n|/(|u||n|), and φ = 90° − θ.

📌 Distance and Midpoint in 3-D

• Distance generalises Pythagoras to three orthogonal directions. Use differences in x, y and z coordinates and take the square root of sum of squares.
• Midpoint is the average of coordinates — useful for bisectors, centres of mass (uniform) and constructing midplanes.

Worked example — distance & midpoint

Find the distance and midpoint between P(−1, 2, 3) and Q(4, −2, 7).

Distance:

  1. Δx = 4 − (−1) = 5; Δy = −2 − 2 = −4; Δz = 7 − 3 = 4.
  2. d = √(52 + (−4)2 + 42) = √(25 + 16 + 16) = √57 ≈ 7.55.

Midpoint:

M = ((−1 + 4)/2, (2 + (−2))/2, (3 + 7)/2) = (3/2, 0, 5).

📌 Volumes & Surface Areas — Key Solids

  • Learn formulae and geometric meaning (why the 1/3 appears for pyramids/cones). • For combined solids, break into parts, compute each volume/area, and add/subtract as required.
  • Important note!: Refer to your formula booklet, find out the formulas that are not included in the formula booklet, memorise them. The IB will ask questions that require you to use formulas they haven’t included!

Worked example — sphere & hemisphere

A sphere of radius 3 cm. Find its volume and surface area. Then find the total surface area of a hemisphere made from the same sphere (including base).

  1. Volume sphere: V = (4/3)πr3 = (4/3)π(3)3 = (4/3)π·27 = 36π cm3 ≈ 113.10 cm3.
  2. Surface area sphere: A = 4πr2 = 4π(9) = 36π cm2 ≈ 113.10 cm2 (interesting: same numeric here because r=3 makes them equal numerically).
  3. Hemisphere curved area = 2πr2 = 18π. Base area = πr2 = 9π. Total hemisphere surface area = 2πr2 + πr2 = 3πr2 = 27π ≈ 84.82 cm2.

Worked example — cone & right pyramid

(a) Right circular cone with base radius r = 4 cm and vertical height h = 3 cm. Find volume and surface area. (b) Regular square pyramid with base side 6 cm and height 4 cm: find volume.

  1. Cone: slant height s = √(r2 + h2) = √(16 + 9) = √25 = 5 cm.
    Volume V = (1/3)πr2h = (1/3)π·16·3 = 16π cm3 ≈ 50.27 cm3.
    Surface area A = πr2 + πr s = π·16 + π·4·5 = 16π + 20π = 36π cm2 ≈ 113.10 cm2.
  2. Pyramid (square base): base area = 6×6 = 36 cm2. Volume V = (1/3)·(base area)·h = (1/3)·36·4 = 48 cm3.

🧮 GDC Tips

  • Use the calculator/matrix solver to compute distances from coordinate differences quickly.
  • For volumes and areas, compute numeric values on the GDC and show algebraic steps in your working.
  • Use vector dot product functions (or compute manually) for angles between vectors; ensure calculator is in degree mode if required.
  • When sketching, use the graphing tool to visualise cross-sections; always include a labelled hand sketch in exam answers.

📐 IA Spotlight

  • Model volumes of composite containers (e.g., water tank made of a cylinder + cone) and compare theoretical capacity with physical measurements.
  • Collect data on irregularly shaped objects and approximate volume via decomposition into known solids; discuss errors.

📌 Combining Solids & Practical Strategy

  • Break complex solids into simpler parts (e.g., cylinder + cone, hemisphere + cylinder).
  • Compute volumes/areas for each part then add or subtract as required.
  • Be careful with units and ensure heights/radii refer to the same axes.
Quick strategy: draw a labelled diagram, mark knowns, identify simple shapes, compute each volume, state final units, and include a short interpretation sentence (e.g., “the container holds X cm3“).

🌐 EE Focus

  • Investigate the derivation of the volume formulas (e.g., why cone/pyramid have 1/3 factor) using calculus methods (integration).
  • Explore geometric optimisation problems: e.g., given surface area, maximise volume (and vice versa).

🔍 TOK Perspective

How do idealised mathematical objects (perfect cones, spheres) relate to messy real objects? Discuss validity of modelling and approximation, the role of measurement error, and the limits of mathematical idealisation.

📌 Angles between Lines and Angle between a Line & a Plane

Use vector methods: represent a line by its direction vector. For two lines with direction vectors u and v, use the dot product to find the angle. For a line and a plane, use the plane’s normal vector n.

Worked example — angle between two lines

Line L1 has direction vector u = <2, 1, −2> and line L2 has v = <1, 2, 1>. Find the acute angle between the lines.

  1. Compute dot product: u·v = 2·1 + 1·2 + (−2)·1 = 2 + 2 − 2 = 2.
  2. Compute magnitudes: |u| = √(4 + 1 + 4) = √9 = 3. |v| = √(1 + 4 + 1) = √6 ≈ 2.449.
  3. cos θ = (u·v)/(|u||v|) = 2 / (3·√6) = 2 / (3√6) ≈ 2 / 7.348 = 0.2722 → θ ≈ cos−1(0.2722) ≈ 74.2°.

Worked example — angle between a line and a plane

Line has direction vector u = <3, 0, 4>. Plane has equation 2x − y + 2z = 5, so normal n = <2, −1, 2>. Find angle φ between the line and the plane.

  1. Compute dot product: u·n = 3·2 + 0·(−1) + 4·2 = 6 + 0 + 8 = 14.
  2. Magnitudes: |u| = √(9 + 0 + 16) = √25 = 5. |n| = √(4 + 1 + 4) = √9 = 3.
  3. Angle θ between u and n: cos θ = |u·n| / (|u||n|) = 14 / (5·3) = 14/15 ≈ 0.9333 → θ ≈ cos−1(0.9333) ≈ 21.0°.
  4. Angle between line and plane φ = 90° − θ ≈ 90° − 21.0° = 69.0° (or use sin φ = |u·n|/(|u||n|) directly because sin φ = cos θ).

❤️ CAS Ideas

  • Organise a community challenge to design and build small scale shelters; compute required material volume/area using decomposition into solids.
  • Lead a workshop teaching 3-D geometry through hands-on models (paper models of cones, pyramids, hemispheres).

📝 Paper Tips

  • Always include a clear labelled diagram with units.
  • State the formula you are using before substituting numbers.
  • Show intermediate arithmetic (e.g., computing slant height) — method marks are awarded for structure.
  • When combining solids, explicitly list parts and state whether you add or subtract volumes/areas.