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Question Number 66852 by John Kaloki Musau last updated on 20/Aug/19

Commented by John Kaloki Musau last updated on 20/Aug/19

The cross-section of a head of a bolt is the form of a regular hexagon as shown in the figure below. Determine the area of the cross-section.

Commented by John Kaloki Musau last updated on 20/Aug/19

The answer is 86.62cm^2   please show me how to do it

$${The}\:{answer}\:{is}\:\mathrm{86}.\mathrm{62}{cm}^{\mathrm{2}} \\ $$$$\boldsymbol{{please}}\:\mathrm{show}\:\mathrm{me}\:\mathrm{how}\:\mathrm{to}\:\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{it}} \\ $$

Commented by Prithwish sen last updated on 20/Aug/19

In a regular hexagon there are  6 equilateral  triangles  of height 5 cm  ∴ each side = 5×(2/(√3)) cm  ∴ the total area = 6×((√3)/4) ×(((10)/(√3)))^2 sq. cm  = 50(√3) sq.cm.=86.6 sq.cm

$$\mathrm{In}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{hexagon}\:\mathrm{there}\:\mathrm{are}\:\:\mathrm{6}\:\mathrm{equilateral} \\ $$$$\mathrm{triangles}\:\:\mathrm{of}\:\mathrm{height}\:\mathrm{5}\:\mathrm{cm} \\ $$$$\therefore\:\mathrm{each}\:\mathrm{side}\:=\:\mathrm{5}×\frac{\mathrm{2}}{\sqrt{\mathrm{3}}}\:\mathrm{cm} \\ $$$$\therefore\:\mathrm{the}\:\mathrm{total}\:\mathrm{area}\:=\:\mathrm{6}×\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\:×\left(\frac{\mathrm{10}}{\sqrt{\mathrm{3}}}\right)^{\mathrm{2}} \mathrm{sq}.\:\mathrm{cm} \\ $$$$=\:\mathrm{50}\sqrt{\mathrm{3}}\:\mathrm{sq}.\mathrm{cm}.=\mathrm{86}.\mathrm{6}\:\mathrm{sq}.\mathrm{cm} \\ $$

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