If the height of the cone is twice of the radius of its base circle then find the ratio of the area of base with total surface area:

Given:

Height of cone = 2 × Radius of base circle.

Radius of base circle = r.

Total Surface Area (TSA) of cone = Area of base + Curved Surface Area.

Formula Used:

Area of base = πr².

Curved Surface Area = πr × l, where l is the slant height.

Slant height (l) = √(r² + h²).

Total Surface Area = πr² + πr × l.

Ratio = Area of base / Total Surface Area.

Calculation:

Height (h) = 2r.

Slant height (l) = √(r² + (2r)²)

⇒ l = √(r² + 4r²)

⇒ l = √(5r²)

⇒ l = r√5.

Curved Surface Area = πr × r√5

⇒ Curved Surface Area = πr²√5.

Total Surface Area = πr² + πr²√5.

⇒ Total Surface Area = πr²(1 + √5).

Area of base = πr².

Ratio = Area of base / Total Surface Area

⇒ Ratio = πr² / [πr²(1 + √5)]

⇒ Ratio = 1 / (1 + √5).