LogIn
লগ-ইন

Right Circular Cone

1. Definition and Basic Elements ( সংজ্ঞা ও মৌলিক উপাদান)

A right circular cone (লম্ব বৃত্তাকার শঙ্কু) is a three‑dimensional figure with a circular base and a vertex (শৃঙ্গ) directly above the center of the base. Key elements include:

  • Radius (r, ব্যাসার্ধ): Distance from the center of the base to any point on the circumference. 
  • Height (h, উচ্চতা): Perpendicular distance from the base to the vertex. 
  • Slant Height (l, ঢালু উচ্চতা): The length of the line from the base’s circumference to the vertex on the side, satisfying l=r2–h2

These parameters—r, h, and l—form the foundation for understanding surface areas and volume.

2.Slant Height (ঢালু উচ্চতা):

The slant height (l) is calculated using the Pythagorean theorem because the radius, slant height, and height form a right triangle:

l=r2–h2

 

Example:
 If a cone has radius r = 3 cm and height h = 4 cm, then

l= 32–22 = 9+16 = 25 =5cm

  1. Surface Areas (পৃষ্ঠফল)

a) Curved Surface Area (CSA)

The curved surface area (পরিবাহী ক্ষেত্রফল) of a right circular cone is:

CSA=πrl

Using the prior example: CSA=π×3×5=15π cm2

  1. b) Total Surface Area (TSA)

Total surface area (মোট পৃষ্ঠফল) equals the curved surface area plus the base area:

TSA=πrl+πr2=πr(l+r)

 For our example:

TSA=π×3×(5+3)=3π×8=24π cm2

4.Volume ( আয়তন)

The volume (ঘনত্ব) of a right circular cone is given by:

Volume= 13πr2h

Example: With r = 3 cm and h = 4 cm:

Volume= 13π×32×4=13​π×9×4=12π cm3

5. Example & Application (উদাহরণ ও প্রয়োগ)

Example Problem:
 A right circular cone has height h = 6 cm and slant height l = 10 cm. Find its radius, surface areas, and volume.

Solution:

  1. Compute the radius using r= = 102–62= 100-36 = 64 =8 cm.
  2. CSA: πrl=π×8×10=80π cm2.
  3. TSA: πr(l+r)=π×8×(10+8)=8π×18=144π cm2
  4. Volume: 13πr2h = 13π×82×6=13π×64×6=128π cm3.

 

Scroll to Top
লগ-ইন