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Solid Geometry

Solid Geometry

Space lines and planes, and surface area / volume of prisms, pyramids, cylinders, cones and spheres.

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1

lessons in this chapter

0

previous exam questions

3 hr

suggested study time

Hard

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Last reviewed Sep 2026 · GetCSCA editors

1. Solid Geometry

Study order

Work the lessons top to bottom — each ends with practice, a timed set and a 5-stage drill. Then prove it in a full 48Q mock.

Solid Geometry

Lesson 1 · target ~75s per question

Positions of lines and planes in space, projections, and volume/surface-area formulas for common solids.

Why CSCA tests this

Solid geometry converts CSCA 3D items into flat ones: classifying line–plane positions and cutting cross-sections through the axis exposes the operative 2D figure.

Key points — memorize

  • Classify space relations first: two lines are parallel, intersecting, or skew; a line is parallel to, contained in, or intersecting a plane.
  • Memorize the volume ladder: prisms/cylinders V = Bh, pyramids/cones V = Bh/3, sphere V = 4πr³/3 with surface 4πr².
  • Cut cross-sections through the axis or key vertices to reduce cones, cylinders, and pyramids to triangles and rectangles.
  • A line is perpendicular to a plane if it is perpendicular to two intersecting lines in that plane — the standard perpendicularity test.
  • Distinguish lateral area (sides only) from total surface (plus bases), and keep area vs volume units separate.

Formula sheet

  • Prism/cylinder: V = Bh
  • Pyramid/cone: V = Bh/3
  • Sphere: V = 4πr³/3; S = 4πr²
  • Cylinder lateral area: 2πrh; cone lateral area: πrl
  • a² + b² + c² = space diagonal² of a box
  • Pyramid/cone total surface = base area + lateral area; prism/cylinder total = 2B + lateral
  • Cone slant height: l = √(r² + h²)
  • Cross-section through the axis reduces cone/cylinder to an isosceles triangle/rectangle

Classic traps

  • Forgetting the /3 on pyramid/cone volumes, or applying it to prisms/cylinders.
  • Using vertical height h where slant height l = √(r² + h²) is needed for cone lateral area πrl.
  • Confusing skew lines with parallel ones — skew lines never meet and are not parallel.
  • Reporting area units for volume (or vice versa) after multi-step computation.

Exam tactic

Draw the axial cross-section immediately for any cone/cylinder/sphere item — the 3D formula choice becomes a 2D Pythagoras or area read-off.

Worked examples

  1. Example 1: Find the volume of a cone with r = 3 and h = 4.

    1. V = πr²h/3 = π · 9 · 4/3 = 12π.

    Answer: 12π

  2. Example 2: A box is 3 × 4 × 12. Find its space diagonal.

    1. d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169.

    Answer: 13

FoundationPractice →BasicPractice →CSCA StandardPractice →Timed set · ~75s/Q →

Previous exam questions

Real CSCA-style questions tagged to Solid Geometry — answer right here.

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Related guides

1 lessons · one chapter

Finished Solid Geometry? Prove it.

Run a timed set, clear every mistake, then take a full 48Q mock.

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