Question

There are two concentric hexagons. Each of the sides of both the hexagons is parallel. Side of internal hexagon is 8cm and distance between two parallel lines is 2√3. Find the area between both the hexagons.

Answer 1

Gɪᴠᴇɴ :-

• There are two concentric hexagons.
• Each of the sides of both the hexagons is parallel.
• Side of internal hexagon is 8cm .
• Distance between two parallel lines is 2√3.

Tᴏ Fɪɴᴅ :-

• Find the area between both the hexagons. ?

Fᴏʀᴍᴜʟᴀ ᴜsᴇᴅ :-

• Height of Equaliteral ∆ = (√3 * side of ∆) / 2 .
• Area of Hexagon = 6 Equaliteral Area = 6 * (√3/4) * (side)² .

Sᴏʟᴜᴛɪᴏɴ :-

(Refer To Image First. )

→ Side of internal Hexagon = 8cm.

→ Height of Internal Hexagon = (8 * √3) / 2 = 4√3cm.

So,

→ Height of External Hexagon = Height of Internal Hexagon + Distance between two parallel lines = 4√3 + 2√3 = 6√3cm.

Therefore,

→ Side of External Hexagon = Height * (2/√3) = 6√3 * (2/√3) = 12cm.

_____________

So,

→ Shaded Area = Area of External Hexagon - Area of Internal Hexagon

→ Shaded Area = [ 6 * (√3/4) * (12)² ] - [ 6 * (√3/4) * (8)² ]

→ Shaded Area = 6 * (√3/4) * [ 12² - 8² ]

→ Shaded Area = (3√3/2) * [ 144 - 64 ]

→ Shaded Area = (3√3/2) * 80

→ Shaded Area = 3√3 * 40

→ Shaded Area = 120√3cm². (Ans.)

Hence, The Area Between Both The Hexagons is 120√3cm².