+0

# Geometry

0
40
1

Find the ratio of the area of a regular hexagon with sides of 1 unit to the area of an equilateral triangle with sides of 2 units. (Nots: "regular" means that all of the sides and angles are equal)

Jun 13, 2021

### 1+0 Answers

#1
+505
+2

ABCDEF is a regular hexagon and PQR is an equilateral triangle.

In ABCDEF,

area(△AOB) = area(△BOC) = area(△COD) = area(△DOE) = area(△EOF) = area(△FOA)

⇒area(ABCDEF) = area(△AOB + △BOC + △COD + △DOE + △EOF + △FOA)

$$= {\sqrt3 \over 4}+ {\sqrt3 \over 4}+ {\sqrt3 \over 4}+ {\sqrt3 \over 4}+ {\sqrt3 \over 4}+ {\sqrt3 \over 4}$$

$$=6× {\sqrt3 \over 4}$$

$$={3\sqrt3 \over 2}$$ sq. units

⇒area(△PQR) $$={\sqrt3 \over 4}×4$$

$$=\sqrt3$$ sq. units

$${area(ABCDEF)\over area(△PQR)} = {3\over 2} = 3:2$$

Jun 14, 2021