# Thread: Area of projection on a plane?

1. ## Area of projection on a plane?

what is the area of the projection on a plane of an equilateral triangle 10 cm on a side, if one side is parallel to the plane and the other sides form an angle of 30° with the plane?

*** the answer to this problem is approximately 35.35 sq. cm but i need the DIAGRAM/FIGURE of this problem and please explain the steps to solve this problem. thanks!!!

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3. Originally Posted by fishlord40
what is the area of the projection on a plane of an equilateral triangle 10 cm on a side, if one side is parallel to the plane and the other sides form an angle of 30° with the plane?

*** the answer to this problem is approximately 35.35 sq. cm but i need the DIAGRAM/FIGURE of this problem and please explain the steps to solve this problem. thanks!!!
Let e denote the side of the isoceles triangel which is the projection of the equilateral triangle. Let h denot the height in the isoceles triangel.

1. $\displaystyle e = 10\cdot \cos(30^\circ) = 5 \cdot \sqrt{3}$

2. To calculate the height h use Pythagorean theorem:

$\displaystyle h^2+5^2 = e^2~\implies~h^2 = (5 \cdot \sqrt{3})^2-5^2 = 50$

Thus $\displaystyle h = 5 \cdot \sqrt{2}$

3. The area of the projection is

$\displaystyle A = \frac 12 \cdot 10 \cdot 5 \cdot \sqrt{2} \approx 35.355339...$