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📐 F4 Mathematics — Three Dimensional Geometry
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📌 Learning Outcomes

By the end of this lesson, you will be able to:

🏗️

State the geometric properties of common 3D solids

📐

Identify projection of a line onto a plane and skew lines

📏

Calculate the length between two points in 3D geometry

🔺

Calculate the angle between two lines

📋

Calculate the angle between a line and a plane

🔶

Calculate the angle between two planes

🏗️ Properties of Common Solids

Dimensions

📦Rectangular Prism
6 faces, 8 vertices, 12 edges
🔺Triangular Prism
5 faces, 6 vertices, 9 edges
🔔Cone
2 faces, 1 edge, 1 vertex
⬛Cube
6 faces, 8 vertices, 12 edges
🥫Cylinder
3 faces, 2 edges, 0 vertices
🔷Rect. Pyramid
5 faces, 5 vertices, 8 edges
🌐Sphere
1 curved face, 0 edges, 0 vertices

↔️ Skew Lines & Projections

Skew Lines

Skew lines are lines that are NOT parallel and do NOT intersect — they exist in different planes. For example, the top-front edge of a cuboid and the bottom-right edge are skew lines.

Skew lines cannot be in the same flat plane.

Projection of a Line onto a Plane

The projection of a line onto a plane is like the "shadow" the line casts on the plane when light shines perpendicular to the plane.

To find the projection: drop perpendiculars from both ends of the line to the plane. The line joining the feet of these perpendiculars is the projection.
Example: In a cuboid ABCDEFGH, the projection of diagonal AG onto the base ABCD is the diagonal AC.

📏 Length in 3D Geometry

Distance Between Two Points in 3D

For two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂):

AB = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

This is an extension of Pythagoras' theorem into 3 dimensions.

Space Diagonal of a Cuboid

For a cuboid with length l, width w, height h:

Space diagonal = √(l² + w² + h²)
Example: A cuboid is 3 cm × 4 cm × 5 cm. Find the length of the space diagonal.
d = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 = 5√2 ≈ 7.07 cm

Face Diagonal

Face diagonal (base) = √(l² + w²)
Example (continued): Face diagonal of base = √(3² + 4²) = √25 = 5 cm

📐 Angle Between a Line and a Plane

Definition

The angle between a line and a plane is the angle between the line and its projection onto the plane. It is the smallest angle between the line and any line in the plane.

Steps to find this angle:

  1. Identify the foot of the perpendicular from a point on the line to the plane
  2. The projection is the line joining this foot to the point where the line meets the plane
  3. Use trigonometry: tan θ = height ÷ projection length

Worked Example

A pole PQ stands vertically at corner Q of a rectangular floor. Q is 10 m from corner A, and the pole is 4 m tall. Find the angle PA makes with the floor.

tan θ = opposite/adjacent = 4/10 = 0.4
θ = tan⁻¹(0.4) = 21.8°

🔶 Angle Between Two Planes

Definition

Any two planes either are parallel or intersect in a straight line. The dihedral angle between two planes is measured by drawing a line on each plane perpendicular to the line of intersection at the same point, then measuring the angle between these two lines.

Worked Example — Tetrahedron

PQRS is a regular tetrahedron of side 4 cm. M is the midpoint of RS.

Step 1: Find PM (in triangle PRS, equilateral with side 4):
PM² = 4² – 2² = 16 – 4 = 12 → PM = 2√3 cm

Step 2: Find QM (triangle QRS is also equilateral):
QM = 2√3 cm → triangle PMQ is isosceles ✓

Step 3: Angle PMQ = angle between planes PSR and QRS
In triangle PMQ: PM = QM = 2√3, PQ = 4
cos(PMQ) = (PM² + QM² – PQ²) / (2·PM·QM)
= (12 + 12 – 16) / (2·12) = 8/24 = 1/3
Angle PMQ = cos⁻¹(1/3) ≈ 70.5°

🏠 Real-Life Example: Cuboid

Full Cuboid Analysis

A room is 6 m long, 4 m wide, and 3 m tall. A rope runs from corner A at the base to corner G diagonally opposite at the top.

a) Length of rope AG:
AG = √(6² + 4² + 3²) = √(36+16+9) = √61 ≈ 7.81 m

b) Projection of AG onto base = AC:
AC = √(6² + 4²) = √52 ≈ 7.21 m

c) Angle AG makes with base floor:
sin θ = 3/7.81 → θ = sin⁻¹(0.384) ≈ 22.6°

Kenyan Context 🇰🇪

These calculations are used in construction of buildings, bridges, and structures like the Nairobi Expressway pillars, roofing trusses, and telecommunications towers in Kenya.

📋 Summary

Distance in 3D: d = √(Δx² + Δy² + Δz²)

Space diagonal of cuboid: d = √(l² + w² + h²)

Angle: line to plane: tan θ = height / projection

Angle between planes: Use perpendiculars to line of intersection + cosine rule

Key Concepts to Remember

✏️ Practice Questions

1. How many edges does a rectangular prism have?
2. A cuboid is 5m × 3m × 4m. Find the space diagonal (1 dp).
3. What is the term for lines that are not parallel and do not intersect?

🎯 Quiz — 3D Geometry

Fill in the Blanks (1 mark each)

1. A cuboid has _____ edges.
2. Lines in different planes that don't meet and aren't parallel are called _____ lines.
3. The projection of a line onto a plane is the _____ of the line on the plane.
4. The angle between a line and a plane uses: tan θ = _____ ÷ projection length.
5. The space diagonal of a cuboid l×w×h = √(l² + w² + _____²).

Short Answer (2 marks each)

6. A cuboid is 8cm × 6cm × 5cm. Find the length of the space diagonal (1 dp).
7. A vertical mast 12m tall stands at corner B of a rectangular field. Corner A is 9m from B. Find the angle the wire from A to top of mast makes with the ground (1 dp).
8. State TWO examples of 3D solids and give the number of faces for each.

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