Mensuration के सवालों के लिए UltraCalc ! Abhinay Sharma | Abhinay Maths | SSC CGL

Main ideas / concepts taught

• Pythagoras’ Theorem and “triplets”

  • The video begins with the Pythagorean relation: [ a^2 + b^2 = c^2 ]

  • It introduces Pythagorean triplets:

    • If (a^2, b^2, c^2) satisfy (a^2 + b^2 = c^2), then the sides form a right-angled triangle.
    • Example: 8, 15, 17
  • Key advantage emphasized: once you identify a triplet, you can compute results (especially area) without using longer “general triangle” methods.

• Area of a right triangle using the triplet recognition

  • For a right triangle with legs (a) and (b), the area is: [ \text{Area}=\frac{1}{2}ab ]

  • Example using 8–15–17: [ \text{Area}=\frac{1}{2}\cdot 8 \cdot 15 = 60 ]

  • Contrast: if you don’t recognize the triplet, you may resort to longer approaches like:

    • Semi-perimeter
    • Heron’s formula (involving (s) and terms like (s(s-a)(s-b)(s-c)))

• Diophantine-type cube identity (cube-sum pattern)

  • The video highlights cube relationships of the form: [ a^3+b^3=c^3 \quad \text{(with known examples)} ]

  • Memorization examples provided:

    • [ 3^3 + 4^3 + 5^3 = 6^3 ]

    • [ 1^3 + 6^3 + 8^3 = 9^3 ]

  • Exam usefulness: when questions involve melting/combining cubes (equalizing volumes), you can substitute these identities rather than repeatedly calculating cube values.

• Scaling rule for Pythagorean triplets

  • If ((a,b,c)) is a Pythagorean triplet, then ((ka, kb, kc)) is also a triplet.
  • Example from 3–4–5:
    • Multiply by 2: 6–8–10
    • Multiply by 3: 9–12–15

• Volume equalization method for “melting” spheres/cubes

  • When solids are melted into one, use equal total volume to find the new radius/side.

  • Spheres example: [ \frac{4}{3}\pi(2^3 + 12^3 + 16^3) = \frac{4}{3}\pi r^3 ] Cancel (\frac{4}{3}\pi): [ r^3 = 2^3 + 12^3 + 16^3 ]

  • Using the referenced cube identity substitution, the result gives:

    • (r = 18)

• Surface area / ratio questions solved using side scaling

  • After finding the new side length of the combined cube, the video uses surface-area reasoning.

  • Example pattern (illustrative):

    • If the side becomes (9), surface area is (6a^2), and ratios like “(1/4) of surface area” reduce to simpler arithmetic: [ \frac{1}{4}\cdot 6a^2 = \frac{3}{2}a^2 ]

  • Multiple exam-oriented examples are mentioned (DP Constable, Delhi Police exam, RRB NTPC 2025, CPO/CGL/CSSC-type), all reinforcing: recognize the cube identity or triplet scaling → compute quickly.

• How “melting cubes with side lengths” uses cube identities

  • The described method:

    • Identify the given cube side lengths as matching one of the base identities:
      • (3,4,5 \rightarrow) new side (=6)
      • (1,6,8 \rightarrow) new side (=9)
    • Apply scaling:
      • If the base identity is scaled by factor (k), then: [ (3k)^3 + (4k)^3 + (5k)^3 = (6k)^3 ] [ (k)^3 + (6k)^3 + (8k)^3 = (9k)^3 ]

  • Examples referenced:

    • Melting cubes of sides 1, 6, 8 gives side 9
    • Melting sides that are scaled (e.g., 2, 12, 16) gives side 18
    • Melting 3, 4, 5 leads to side 6, then questions about radius/surface area are answered using the updated solid dimensions

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Method / “instruction-like” bullet points extracted

A) For right-triangle area using Pythagorean triplet

• Check whether the given triangle sides form a Pythagorean triplet (e.g., (8,15,17)).

• If yes:

  • Identify the legs (a) and (b) (the non-hypotenuse sides).
  • Compute area directly: [ \text{Area}=\frac{1}{2}ab ]

• Avoid longer steps (semi-perimeter + Heron’s formula).

B) For finding new side/radius after “melting” cubes or spheres (volume equalization)

• Set total volume before = volume after.

• For cubes: [ a^3+b^3+c^3 = r_{\text{new}}^3 ] where (r_{\text{new}}) is the cube side.

• Use memorized cube identities:

  • [ 3^3+4^3+5^3=6^3 ]

  • [ 1^3+6^3+8^3=9^3 ]

• If numbers are scaled by factor (k), apply scaling:

  • Replace (3,4,5) with (3k,4k,5k) → result becomes (6k)
  • Replace (1,6,8) with (k,6k,8k) → result becomes (9k)
• After obtaining the new side length (or radius for spheres), use standard geometry formulas (surface area/volume) as needed.

C) For “triplet scaling” (Pythagoras triplets)

• If ((a,b,c)) is a Pythagorean triplet, then ((2a,2b,2c)), ((3a,3b,3c)), etc., are also triplets.
• Example specifically mentioned:
  • From (3,4,5) to (6,8,10) by multiplying by 2.

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Speakers / sources featured

• Abhinay Sharma (also referenced as Abhinay Maths in the video title)