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Surface Areas & Volumes

Cuboid and Cube

What are 3D Solids?

Three-dimensional (3D) solids have length, breadth (width), and height. They occupy space.

Definitions:

  • Surface Area: The total area of all faces of a 3D solid (square units)
  • Lateral Surface Area (LSA): Area of only the side faces (excluding top and bottom)
  • Total Surface Area (TSA): Area of all faces including top and bottom
  • Volume: The amount of space inside a 3D solid (cubic units)

Cube (all sides equal = a):

MeasurementFormula
Total Surface Area (TSA)\(6a^2\)
Volume\(a^3\)

Cuboid (length = l, breadth = b, height = h):

MeasurementFormula
Total Surface Area (TSA)\(2(lb + bh + hl)\)
Volume\(l \times b \times h\)
3D Surface Area FormulasCuboidTSA=2(lb+bh+hl)LSA=2h(l+b)CubeTSA=6a²LSA=4a²CylinderTSA=2πr(r+h)CSA=2πrhConeTSA=πr(r+l)CSA=πrlSphereSA=4πr²HemisphereTSA=3πr²CSA=2πr²l = slant height of cone = √(r²+h²)LSA = Lateral; CSA = Curved Surface Area (exclude flat faces)
Example 1: Cube of edge 5 cm. Volume?
5³ = 125 cm³.
Example 2: Cuboid 4 × 3 × 2. Volume?
4 × 3 × 2 = 24 cubic units.
Quick recap
  • Cuboid: TSA = 2(lb + bh + hl), V = lbh.
  • Cube: SA = 6a², V = a³.
✓ Quick check
The lateral surface area of a cube is 256 m². The volume of the cube is:
LSA of cube = 4a² = 256. Therefore, a² = 64 → a = 8 m. Volume = a³ = 8³ = 512 m³.
The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.
Initial radius r₁ = 7 cm. Final radius r₂ = 14 cm. Surface area = 4πr². Ratio = 4πr₁² / 4πr₂² = (r₁/r₂)² = (7/14)² = (1/2)² = 1:4.

Cylinder and Cone

What are Cylinder and Cone?

  • Cylinder: A 3D solid with two parallel circular bases connected by a curved surface
  • Cone: A 3D solid with a circular base and a curved surface tapering to a point (vertex)

Right Circular Cylinder (radius = r, height = h):

MeasurementFormula
Total Surface Area (TSA)\(2\pi r(r + h)\)
Volume\(\pi r^2 h\)

Right Circular Cone (radius = r, height = h, slant height = l):

MeasurementFormula
Curved Surface Area (CSA)\(\pi r l\)
Total Surface Area (TSA)\(\pi r(r + l)\)
Volume\(\frac{1}{3} \pi r^2 h\)

Important Notes:

  • π ≈ 3.14 or 22/7
  • CSA is also called lateral surface area
  • For cone, l (slant height) is NOT the same as h (vertical height)
Volume FormulasCuboidl × b × hCubea³Cylinderπr²hCone(1/3)πr²hSphere(4/3)πr³Hemisphere(2/3)πr³Unit Conversions:1 m³ = 1,000,000 cm³1 litre = 1000 cm³ = 1 dm³1 m³ = 1000 litresCone volume = (1/3) × Cylinder volume (same base & height)Sphere volume = (2/3) × Cylinder volume (same radius & height=2r)
Example 1: Cylinder r = 7, h = 10 (π = 22/7). CSA?
2 × 22/7 × 7 × 10 = 440 sq units.
Example 2: Cone r = 3, h = 4. Slant height?
√(3² + 4²) = √25 = 5.
Quick recap
  • Cylinder: CSA = 2πrh, V = πr²h.
  • Cone: l = √(r² + h²), CSA = πrl, V = ⅓πr²h.
✓ Quick check
The surface area of a sphere of radius 10.5 cm is (take π = 22/7):
10.5 cm
Surface area of sphere = 4πr² = 4 × (22/7) × 10.5 × 10.5 = 4 × (22/7) × (21/2) × (21/2) = 22 × 3 × 21 = 1386 cm².
The curved surface area of a right circular cone of height 15 cm and base diameter 16 cm is:
15 cm8 cm
Diameter = 16 → r = 8 cm. Height h = 15 cm. Slant height l = √(8² + 15²) = √(64 + 225) = √289 = 17 cm. CSA = πrl = π(8)(17) = 136π cm².

Sphere and Hemisphere

A sphere of radius r has surface area 4πr² and volume (4/3)πr². A hemisphere has curved surface area 2πr², total surface area 3πr² (curved plus the flat circle), and volume (2/3)πr².

Combined solids (a cone on a hemisphere, a capsule, etc.) are handled by adding the relevant surface areas or volumes.

Example 1: Surface area of a sphere of radius 7 (π = 22/7)?
4 × 22/7 × 7² = 616 sq units.
Example 2: Why is a hemisphere's TSA 3πr²?
Curved 2πr² plus the flat circular face πr² gives 3πr².
Quick recap
  • Sphere: SA = 4πr², V = (4/3)πr³.
  • Hemisphere: TSA = 3πr², V = (2/3)πr³.
✓ Quick check
A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.
Volume of earth dug out = Volume of cylindrical well = πr²h = (22/7) × (3.5)² × 20 = (22/7) × 12.25 × 20 = 770 m³. Volume of platform = l × b × h = 22 × 14 × h = 308h. 308h = 770 → h = 770 / 308 = 2.5 m.
Find the length of canvas 2 m in width required to make a conical tent 12 m in diameter and 63 m in slant height.
Diameter = 12 m, so r = 6 m. l = 63 m. CSA = πrl = (22/7) × 6 × 63 = 22 × 6 × 9 = 1188 m². Length × Width = Area. Length × 2 = 1188 → Length = 594 m.
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