A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Geometry and algebra
CSCA Mathematics · Lesson 14 of 16Solid geometry
Géométrie dans l'espace · الهندسة الفضائية
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Solid geometry asks for lengths, areas and volumes in three dimensions. A 3D coordinate system turns most of it into calculation.
Key ideas
- A point in space is (x, y, z); distance = √(Δx² + Δy² + Δz²).
- Cube of edge a: face diagonal a√2, space diagonal a√3.
- Prism and cylinder: V = base area × height.
- Pyramid and cone: V = ⅓ × base area × height.
Formulas to know
- Sphere: V = (4/3)πr³, S = 4πr²
- Cylinder: lateral area = 2πrh
- Cuboid diagonal = √(a² + b² + c²)
Worked example
A cube has edge 2. Find its space diagonal and its volume.
- Diagonal: d = √(2² + 2² + 2²) = √12
- Simplify: √12 = 2√3 ≈ 3.46
- Volume: V = 2³ = 8
Answer: d = 2√3, V = 8
Practice questions
1. Distance between (1, 2, 3) and (4, 6, 3)?
5 ✓ · 7 · √50 · 25
√(9 + 16 + 0) = 5.
2. Volume of a cone with r = 3 and h = 4?
36π · 12π ✓ · 48π · 9π
⅓ · π · 9 · 4 = 12π.
3. Surface area of a sphere with r = 2?
8π · 16π ✓ · 32π/3 · 4π
4π · 2² = 16π.