Surface area of inclined XY plane at 45 degrees

[marciokoko]

Homework Statement

A regular 2 dimensional square has a surface area b*h. But if we rotate the plane on a z axis by 45 degrees (and are forced to extend the length of let’s say, the base, then I guess what happens is that the square now becomes a rectangle with longer sides for b.

Is there a way to calculate how much longer the sides got given the plane must still reside within the original square dimensions?

I’m trying to find out how much more surface area you get by growing crops on a flat square versus that now elongated rectangle obtained by rotating the original square 45 degrees on that z axis.

Not sure if I’m being clear enough. Lemme know.

Relevant Equations

Base x height and possibly 1:1:sqrt 2

Is it just sqrt2?

 

[Chestermiller]

Homework Statement: A regular 2 dimensional square has a surface area b*h. But if we rotate the plane on a z axis by 45 degrees (and are forced to extend the length of let’s say, the base, then I guess what happens is that the square now becomes a rectangle with longer sides for b.

Is there a way to calculate how much longer the sides got given the plane must still reside within the original square dimensions?

I’m trying to find out how much more surface area you get by growing crops on a flat square versus that now elongated rectangle obtained by rotating the original square 45 degrees on that z axis.

Not sure if I’m being clear enough. Lemme know.
Relevant Equations: Base x height and possibly 1:1:sqrt 2

Is it just sqrt2?

Are you familiar with the Pythagorean theorem and the definition of the sine of an angle?

 

[kuruman]

When you look at the square of side $a$ before you rotate it, you see an area equal to the base times the height, $Area=a\times a =a^2$. When you rotate about the base, the base is still $a$, but the height gets shorter, call it $h$. The new area is $Area = a\times h$ (see figure on the right.)

Can you find the relation between $a$ and #h##? See @Chestermiller's post.

 

[WWGD]

Homework Statement: A regular 2 dimensional square has a surface area b*h. But if we rotate the plane on a z axis by 45 degrees (and are forced to extend the length of let’s say, the base, then I guess what happens is that the square now becomes a rectangle with longer sides for b.

Is there a way to calculate how much longer the sides got given the plane must still reside within the original square dimensions?

I’m trying to find out how much more surface area you get by growing crops on a flat square versus that now elongated rectangle obtained by rotating the original square 45 degrees on that z axis.

Not sure if I’m being clear enough. Lemme know.
Relevant Equations: Base x height and possibly 1:1:sqrt 2

Is it just sqrt2?

You may be able to do this analytically by using coordinates, with your square having vertex set {$(0,0), (0,a), (a,0), (a,a)$}, then applying a rotation matrix to the vertices, then extending. You'll end up with a figure made of rectangles and triangles who's area you can easily find, or, you may easily integrate some simple linear maps.

 

[haruspex]

how much more surface area you get by growing crops on a flat square versus that now elongated rectangle obtained by rotating the original square 45 degrees on that z axis.

What about insolation?