Calculate area of square knowing the diagonal

[Spoo]

I'm stumped

I was watching It's Academic, a local TV quiz show
among area high school teams, and one of the questions
was:

What is the area of this square?
Given: On the monitor, a square with the length of the diagonal shown as 12.

The kid did it in his head in about 5 seconds.

It has something to do with .707 and the square-root of 2.
but I've developed brain-lock and don't get it.


Spoo

[NotLKH]

Since it is a square, then all the sides are equal length.
Lets call the length of a side "s"

The diagonals length follows the right triangle rule:
a² + b² = c², where a and b are the values "s" and c is the value 12:

2*s² = 144

s² = 72

And the area of the square is s², which we see is 72.

Seems too simple, maybe I need more coffee.

[Spoo]

NotLKH

Yes, that was beautiful.

My approach was:

s = .707 x 12
s² = 72 .. try to do that in your head

I'd forgotten that .707 = .5 x sqrt(2), so alternatively,

s = .5 x sqrt(2) x 12
s² = .5² x sqrt(2)² x 12² = .25 x 2 x 144 = .5 x 144 = 72

Spoo

[jemidiah]

I'd guess the way the kid in the quiz show did it was by remembering that, in a 45-45-90 triangle, the diagonal is Sqrt(2) times as long as the base. So, the base is (diagonal) / Sqrt(2), but since you want area you can square this expression out and just get (diagonal ^ 2) / 2 = 144 / 2 = 72. Of course this is basically equivalent to the two ways mentioned, it's just a slightly different perspective. This way doesn't require the Pythagorean Theorem or remembering that 1/Sqrt(2) = Sqrt(2)/2 as the above algebra requires, which is why I'd guess it's how you'd do it very quickly.

If you're curious, I was in some of these type of academic competitions around high school and this question would have been considered easy for our team. Amongst good competitors, in our versions of this quiz show (which were timed and usually simultaneous with all teams getting the same question at the same time) it would have been a race to see who signaled first much more than who could answer. Of course, training increases speed with these type of questions dramatically.

You could ask a very similar question which would illustrate the memorization I mentioned, though it'd be much more complicated to do very quickly. Draw a rectangle with diagonal length 2*Sqrt(Sqrt(3)). Suppose the diagonal's angle with the smaller side is labeled as 60 degrees. What is the area of the rectangle?

In a 30-60-90 triangle, the smallest side is half the length of the diagonal, and the longer side is Sqrt(3) times the length of the smallest side. So, the smallest side here is of length Sqrt(Sqrt(3)). To find the area we just have to square this and multiply by Sqrt(3)--since the longer side is Sqrt(Sqrt(3)) * Sqrt(3) in length. Doing this, the factors cancel beautifully. Squaring the smaller side length gives Sqrt(3), and multiplying by Sqrt(3) gives, in all, simply 3. If you were quick on your 30-60-90 triangles, this reasoning could probably be done in ~5 seconds (varying from person to person significantly).

[Spoo]

Jemi

Nice. The closest I got to the show was being in the audience
when I was in high school, but I did know all 3 guys who were on
the team, and was in many of their same classes.

I did great in algebra, trig and geom, but when I got to calculus in
college, I met my match. For some reason, I couldn't visualize it.

Spoo

[Siddharth Rout]

It's pretty simple...

If you split the square diagonally then it will intersect the angle at 45 degrees

Cos 45 = adjacent / hypotenuse

value of Cos 45 = 0.7071 (and hence the value you were looking for)

so

0.7071 = adjacent / 12

adjacent = 0.7071 * 12

Now you can get the area...

You can use the same calculation using Sin 45

Sin 45 = adjacent / hypotenuse

Note: This is applicable only in case of square. In case of Rectangle it becomes

Sin 45 = Opposite Side / hypotenuse

Value of Sin 45 = 0.7071

0.7071 = adjacent / 12

and hence the same calculations as above...

Hope this helps....

[gmiller]

Hi dear, It's very easy if you know the diagonal of square. consider a is the sides of sqare and d is the diagonal of square. Then
a2 + a2 = d2 and d is 12
area of square will be a2.
a2=d2/2
= 12*12/2
=144/2
=72 Ans.

[soft903]

for a question for kid, let's forget about Pythagoras' theorem or sin cos stuff
just simply little knowledge of square and area of triangle

a diagonal splits square into 2 same size triangle
with base is diagonal and height is half of diagonal
Area of triangle = base * height / 2
Area of square = area of triangle * 2
= base * height / 2 * 2
= diagonal * diagonal / 2

(or consider 2 diagonal split square into 4 triangle with base and height is half of diagonal)

the calculation may take ~2 seconds, so total time depend on how long you realize the triangles