Same area but perimeter?

Geometry Level 2

If the area of equilateral triangle XYZ is equal to area of rhombus ABCD, in which Angle A = 60°,

Then which will have greater perimeter?

Rhombus Both have same perimeter Triangle Cannot be determined

1 solution

Richard Costen
May 27, 2018

Let the rhombus have side length $a$ . It’s perimeter is $4a$ . Since one angle is $60^\circ$ , divide the rhombus into 2 equilateral triangles. Each has area $\frac12 a^2\sin60=\frac{\sqrt3}{4}a^2$ . Therefore the area of the rhombus & $\triangle XYZ$ is $\frac{\sqrt3}{4}(2a^2)$ . That makes each triangle side $\sqrt{2a^2}=\sqrt2 a$ . The perimeter is $3\sqrt2 a$ . $3\sqrt2>4$ so the triangle has the greater perimeter.

Same area but perimeter?

1 solution

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