Comparing Areas ( 1 )

Geometry Level 2

Examine the figure and decide which colored area is the largest among others, or are they all the same?

Orange Blue Green All areas are the same.

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3 solutions

Mahdi Raza
May 28, 2020

Thank you for sharing your solution.

Hana Wehbi - 1 year ago

Each triangle is congruent! Each triangle is congruent!

From the figure, we see that for 1 1 green triangle, there is 1 1 blue triangle and 1 1 orange triangle.

Hence, All the areas are same.

Thank you for sharing your solution.

Hana Wehbi - 1 year ago

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You're welcome!

Vinayak Srivastava - 1 year ago
Hana Wehbi
May 28, 2020

Area of one blue equilateral triangle is : 3 4 ( 1 2 ) \frac{\sqrt{3}}{4}(1^2)\implies Area of 6 6 blue triangles is: 3 3 2 \frac{3\sqrt{3}}{2}

Area of the regular hexagon of side 1 1 : 3 3 2 \frac{3\sqrt{3}}{2}

Area of one isosceles Green triangle of non-base side 1 1 is : 3 2 sin 12 0 × 1 2 × 1 2 = 3 4 \underbrace{\frac{\sqrt{3}}{2}}_{\sin 120^\circ}\times\frac{1}{2}\times 1^2 =\frac{\sqrt{3}}{4} \implies Area of 6 6 Green is : 3 3 2 \frac{3\sqrt{3}}{2}

All Areas are Equal \boxed {\text{ All Areas are Equal} }

Formulas used: ( s = 1 ) (s=1)

Area of an equilateral Triangle: 3 4 s 2 \frac{\sqrt{3}}{4}s^2

Area of an Isosceles Triangle: 1 2 s 2 sin ( 12 0 ) \frac{1}{2} s^2 \sin (120^\circ)

Area of a regular hexagon: 3 3 2 s 2 \frac{3\sqrt{3}}{2}s^2

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