1. 3D Shapes Overview

Tips:

Check number of faces, edges, and vertices for tricky questions.

Know which faces are counted for TSA and which are joined for combined solids.

2. Square-Based Pyramid

Key Terms:

a = base side, h = vertical height, l = slant height, e = lateral edge

Base diagonal = a√2, Half diagonal = a√2 / 2

Relations:

l² = h² + (a/2)²

e² = h² + (a√2 /2)²

Formulas:

LSA = 2 a l

TSA = a² + 2 a l = a(a + 2 l)

Volume = 1/3 a² h

Important Questions:

Q1: a = 12 cm, l = 10 cm → TSA, h, V

TSA = 12(12 + 20) = 384 cm²

h = √(10² − 6²) = 8 cm

Volume = 1/3 × 144 × 8 = 384 cm³

Q2: TSA = 144 cm², l = 5 → a

a(a + 10) = 144 → a² + 10a − 144 = 0 → a = 8 cm

Q3: Volume = 384 cm³, a = 12 → h, l

h = 384×3 /144 = 8 cm

l = √(8² + 6²) = 10 cm

3. Cone

Key Terms: r = radius, h = height, l = slant height

Relations: l² = h² + r²

Formulas:

CSA = π r l

TSA = π r(r + l)

Volume = 1/3 π r² h

Important Questions:

Q4: r = 7, h = 24 → l, CSA, TSA, Volume

l = √(24² + 7²) = 25 cm

CSA = 3.14 × 7 × 25 ≈ 549.5 cm²

TSA = 3.14 × 7 × 32 ≈ 703 cm²

V = 1/3 × 3.14 × 49 × 24 ≈ 1231.7 cm³

Q5: TSA = 2816, r + l = 64 → r, l

r(64) = 896 → r = 14, l = 50

Q6: r:h = 5:12, V = 314.16 → r, h, l

r = 5, h = 12, l = 13

4. Cylinder

Formulas:

CSA = 2 π r h

TSA = 2 π r(h + r)

Volume = π r² h

Important Questions:

Q7: r = 7, h = 20 → CSA, TSA, V

CSA = 2 × 3.14 × 7 × 20 ≈ 879.2

TSA = 879.2 + 2 × 3.14 × 49 ≈ 1186.92

V = 3.14 × 49 × 20 ≈ 3077.2

Q8: V = 1540, r = 7 → h, TSA

h ≈ 10 cm, TSA ≈ 747.32

5. Combined Solids

Rules:

Volume → add all parts

TSA → exclude joined faces

Painting → only visible surfaces

Formulas:

Q9: Pencil r = 2, cylinder h = 10, cone h = 5

Cone slant: l = √(2² +5²) ≈ 5.385

Cylinder CSA = 125.6, Cone CSA ≈ 33.8, Base = 12.56

TSA ≈ 171.96, Volume ≈ 146.53

Q10: Ice-cream cone r = 7, cone h = 12, hemisphere r = 7

l = √(12² +7²) ≈ 13.89

Cone CSA ≈ 304.4, Hemisphere ≈ 307.72

TSA ≈ 612.12, Volume ≈ 1334.13

Q11: Two cones joined, r = 3, h₁ = 8, h₂ = 12

l₁ = 8.54, l₂ = 12.37

LSA = π r (l₁ + l₂) ≈ 197

V = 1/3 π r² (8 +12) ≈ 188.4

6. Sphere & Hemisphere

Sphere:

CSA = 4 π r²

Volume = 4/3 π r³

Hemisphere:

CSA = 2 π r², TSA = 3 π r²

Volume = 2/3 π r³

Tip: Hemisphere’s curved surface is half of a sphere; TSA includes base.

7. Real-Life Applications

Water tanks → V × 1000 = litres

Fencing → perimeter × rounds

Painting → visible area only

Bricks/Tiles → Total area ÷ area per brick

Golden Rules:

Pyramid → 1/3 a² h, TSA = a(a+2l)

Cone → l = √(h²+r²), TSA = π r(r+l)

Cylinder → CSA = 2 π r h, TSA = 2 π r(h+r)

Combined solids → add volumes, subtract internal faces

Always solve stepwise: find dimensions first, then area/volume

Important questions for practice

1. Square-Based Pyramid

Q1: Base side a = 12 cm, slant height l = 10 cm. Find TSA, Volume, and height h.

Solution:
TSA:
TSA = a(a + 2l)
TSA = 12(12 + 20)
TSA = 12 × 32
TSA = 384 cm²

Height:
l² = h² + (a/2)²
10² = h² + 6²
100 = h² + 36
h² = 64
h = 8 cm

Volume:
V = 1/3 × a² × h
V = 1/3 × 144 × 8
V = 384 cm³

Q2: TSA = 144 cm², slant height l = 5 cm. Find base side a.

Solution:
TSA = a(a + 2l)
144 = a(a + 10)
a² + 10a − 144 = 0

Solve quadratic:
a = 8 cm (positive root)

Q3: Volume = 384 cm³, base a = 12 cm. Find height h and slant height l.

Solution:
Volume formula:
V = 1/3 × a² × h
384 = 1/3 × 144 × h
h = 8 cm

Slant height:
l² = h² + (a/2)²
l² = 8² + 6²
l² = 64 + 36
l = 10 cm

2. Cone

Q4: Radius r = 7 cm, height h = 24 cm. Find slant height, CSA, TSA, Volume.

Solution:
Slant height:
l = √(h² + r²)
l = √(24² + 7²)
l = √625
l = 25 cm

CSA:
CSA = π r l
CSA = 3.14 × 7 × 25
CSA ≈ 549.5 cm²

TSA:
TSA = π r (r + l)
TSA = 3.14 × 7 × 32
TSA ≈ 703 cm²

Volume:
V = 1/3 π r² h
V = 1/3 × 3.14 × 49 × 24
V ≈ 1231.7 cm³

Q5: TSA = 2816 cm², r + l = 64 cm. Find r and l.

Solution:
TSA = π r (r + l)
3.14 r × 64 = 2816
r × 64 = 896
r = 14 cm

l = 64 − r
l = 50 cm

Q6: Ratio r:h = 5:12, Volume = 314.16 cm³. Find r, h, l.

Solution:
Let r = 5x, h = 12x

Volume:
V = 1/3 π r² h
314.16 = 1/3 × 3.14 × (25x²) × 12x
314.16 ≈ 314 x³
x ≈ 1

r = 5 cm, h = 12 cm

Slant height:
l = √(r² + h²)
l = √(25 + 144)
l = √169
l = 13 cm

3. Cylinder

Q7: Cylinder r = 7 cm, h = 20 cm. Find CSA, TSA, Volume.

Solution:
CSA = 2 π r h
CSA = 2 × 3.14 × 7 × 20
CSA ≈ 879.2 cm²

TSA = CSA + 2 π r²
TSA = 879.2 + 2 × 3.14 × 49
TSA ≈ 1186.92 cm²

Volume:
V = π r² h
V = 3.14 × 49 × 20
V ≈ 3077.2 cm³

Q8: Volume = 1540 cm³, r = 7 cm. Find height h and TSA.

Solution:
Volume formula:
V = π r² h
1540 = 3.14 × 49 × h
h ≈ 10 cm

CSA = 2 π r h
CSA = 2 × 3.14 × 7 × 10
CSA ≈ 439.6 cm²

TSA = CSA + 2 π r²
TSA = 439.6 + 2 × 3.14 × 49
TSA ≈ 747.32 cm²

4. Combined Solids

Q9: Pencil: Cylinder r = 2 cm, h = 10 cm; Cone height = 5 cm. Find TSA, Volume.

Solution:
Cone slant height:
l = √(r² + h²)
l = √(4 + 25)
l ≈ 5.385 cm

Cylinder CSA:
2 π r h = 2 × 3.14 × 2 × 10
≈ 125.6 cm²

Cone CSA:
π r l = 3.14 × 2 × 5.385
≈ 33.8 cm²

Base area:
π r² = 3.14 × 4
≈ 12.56 cm²

TSA = 125.6 + 33.8 + 12.56
≈ 171.96 cm²

Volume:
Cylinder = π r² h = 3.14 × 4 × 10 ≈ 125.6 cm³
Cone = 1/3 π r² h = 1/3 × 3.14 × 4 × 5 ≈ 20.93 cm³
Total Volume ≈ 146.53 cm³

Q10: Ice-cream cone: r = 7 cm, cone h = 12 cm, hemisphere r = 7 cm. Find TSA, Volume.

Solution:
Cone slant height:
l = √(h² + r²) = √(144 + 49) ≈ 13.89 cm

Cone CSA = π r l = 3.14 × 7 × 13.89 ≈ 304.4 cm²
Hemisphere surface = 2 π r² = 2 × 3.14 × 49 ≈ 307.72 cm²

TSA = 304.4 + 307.72 ≈ 612.12 cm²

Volume:
Cone = 1/3 π r² h = 1/3 × 3.14 × 49 × 12 ≈ 615.36 cm³
Hemisphere = 2/3 π r³ = 2/3 × 3.14 × 343 ≈ 718.77 cm³
Total Volume ≈ 1334.13 cm³

Q11: Two cones joined at base, r = 3 cm, heights = 8 cm and 12 cm. Find Volume and TSA.

Solution:
Volume = 1/3 π r² (h1 + h2)
= 1/3 × 3.14 × 9 × 20
≈ 188.4 cm³

Slant heights:
l1 = √(3² + 8²) ≈ 8.54 cm
l2 = √(3² + 12²) ≈ 12.37 cm

LSA = π r (l1 + l2)
= 3.14 × 3 × (8.54 + 12.37)
≈ 197 cm²

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https://besidedegree.com/exam/s/academic