Chapter 5. Geometry and Measurement
Basic figures. Points, lines, rays, segments and angles. Angles are measured in degrees (a full turn is 360°). Complementary angles sum to 90°; supplementary angles sum to 180°. The angles of a triangle sum to 180° [5][9].
Pythagorean theorem. In a right triangle with legs a and b and hypotenuse c: a² + b² = c². Example: the 3–4–5 triangle.
Congruence and similarity. Congruent figures have the same size and shape; similar figures have the same shape and proportional sides. If the scale factor is k, lengths scale by k, areas by k², and volumes by k³.
Area and perimeter
| Figure | Area | Perimeter / Circumference |
| Rectangle (l, w) | lw | 2l + 2w |
| Triangle (base b, height h) | ½ bh | sum of sides |
| Parallelogram | bh | sum of sides |
| Trapezoid (parallel sides a, b) | ½ (a + b)h | sum of sides |
| Circle (radius r) | πr² | 2πr |
Volume and surface area
| Solid | Volume | Surface area |
| Cube (side s) | s³ | 6s² |
| Rectangular prism | lwh | 2(lw + lh + wh) |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | ⅓ πr²h | πr² + πrℓ (ℓ = slant height) |
| Sphere | 4/3 πr³ | 4πr² |
Coordinate geometry. Distance between (x₁, y₁) and (x₂, y₂): √((x₂ − x₁)² + (y₂ − y₁)²). Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2). A circle of centre (h, k) and radius r: (x − h)² + (y − k)² = r² [4].
Axiomatic geometry. Euclid’s Elements built geometry from definitions, postulates and proofs and remains a model of mathematical reasoning [10].
Review: Find the area of a circle with radius 5 and the volume of a sphere with radius 3. (Answer: 25π ≈ 78.5 and 36π ≈ 113.1.)