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compulsory · CSS study notes

11. Solid Geometry

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 11 (Solid Geometry)

Past Paper Weightage

  • Frequently tested — volume/surface area of cubes, cylinders, spheres, cones recur almost every year. High-priority, formula-driven topic (marked as a focus area).

Key 3D Formulas

Cube (side a):

V=a3S=6a2V = a^3 \qquad S = 6a^2

Cuboid (l, w, h):

V=lwhS=2(lw+wh+hl)V = lwh \qquad S = 2(lw+wh+hl)

Cylinder (r, h):

V=πr2hS=2πr(r+h)V = \pi r^2 h \qquad S = 2\pi r(r+h)

Sphere (r):

V=43πr3S=4πr2V = \frac{4}{3}\pi r^3 \qquad S = 4\pi r^2

Cone (r, h, slant l):

V=13πr2hS=πr(r+l),l=r2+h2V = \frac{1}{3}\pi r^2 h \qquad S = \pi r(r+l), \quad l=\sqrt{r^2+h^2}

Key Concepts

  • Volume = space occupied (cubic units); Surface area = total area of all outer faces (square units)
  • Diagonal of a cube (side a):
d=a3d = a\sqrt{3}

Diagonal of a cuboid:

d=l2+w2+h2d = \sqrt{l^2+w^2+h^2}
  • Curved surface area (CSA) excludes the flat top/bottom (e.g., cylinder CSA = 2πrh); total surface area (TSA) includes them

Worked Example

  • Cylinder with r=7, h=10 (use π≈22/7): Volume = 22/7 × 49 × 10 = 1540 cubic units
  • Sphere with r=3: Volume = (4/3)×π×27 ≈ 113.1 cubic units

Past Paper Style Numericals (Solved)

  1. Find the volume of a cuboid with dimensions 8 cm × 5 cm × 3 cm.

    Answer: 8×5×3 = 120 cm³

  2. A sphere has a radius of 7 cm. Find its total surface area (use π≈22/7).

    Answer: 4×22/7×49 = 616 cm²

  3. A cylindrical tank has radius 3.5 m and height 6 m. Find its volume (use π≈22/7).

    Answer: 22/7×12.25×6 = 231 m³

  4. Find the diagonal of a cube with side 5 cm.

    Answer: a√3 = 5×1.732 ≈ 8.66 cm

  5. A cone has base radius 6 cm and height 8 cm. Find its slant height and volume.

    Answer: slant l=√(36+64)=10 cm; Volume=(1/3)×π×36×8 ≈ 301.6 cm³

MCQ Traps

  • Cone volume is 1/3 of a cylinder with the same base and height — a very common comparison question
  • Don't confuse curved surface area with total surface area for cylinders/cones

Self-Test Questions

  1. Find the volume of a cube with side 6 cm.
  2. Find the total surface area of a cylinder with radius 5 cm and height 12 cm.
  3. A cone and a cylinder have the same base radius and height. What is the ratio of their volumes?

Past-Paper MCQs (FPSC)

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The FPSC MCQ papers you shared (2024 and 2025) had no direct solid-geometry question.

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me.

  • CSS 2025 Q8(b): A solid cuboid has a 10 cm by 6 cm base, height y cm, and total surface area 376 cm². Find the height. (5)
    1. Total surface area = 2(lw + lh + wh) = 2(60 + 10y + 6y)
    2. 120 + 32y = 376 → 32y = 256
    3. y = 8 cm
  • CSS 2024 Q7(c): Surface area and volume of a sphere of radius 7 m. (5)
    1. Use π = 22/7
    2. Surface area = 4πr² = 4 × (22/7) × 49 = 616 m²
    3. Volume = (4/3)πr³ = (4/3) × (22/7) × 343 = 4,312/3 ≈ 1,437.33 m³
  • CSS 2023 Q7(d): The Egyptian pyramid is 146.6 m high with a 230.6 m base length. Find its volume. (5)
    1. Volume = (1/3) × base area × height
    2. Base area = 230.6² = 53,176.36 m²
    3. Volume = (1/3) × 53,176.36 × 146.6 ≈ 2,598,551 m³
  • CSS 2020 Q7(b): A farmer builds a boundary wall around a square farm of area 484 m². The wall is 2 m high on three sides and 3 m on one. Find the total wall area. (5)
    1. Side = √484 = 22 m
    2. Three sides: 3 × 22 × 2 = 132 m²; one side: 22 × 3 = 66 m²
    3. Total = 198 m²

CSS Exam Perspective

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  • Most repeated areas: cuboid, sphere, pyramid and wall-area problems (2020, 2023, 2024, 2025)
  • Tip: write the formula first; use π = 22/7 when the radius is a multiple of 7