# Screen Size Calculations

A friend of mine today asked me the following: *"I am looking to buy a 52 inch
TV with an aspect ratio of 16 by 9. I need to figure out the horizontal and
vertical sizes in inches in order to know if it fits in my entertainment
center. How do I do that?"*

I told him to take a tape measure to each side of the TV and write down the
measurements. That wasn't the answer he was looking for. He is looking at the
TVs online. I told him he would need a little bit of knowledge about the
*Pythagorean Theorem* and a little bit of trigonometry.

### The Setup

This is what my friend is looking at:

When we take all the shiny parts away, we end up with the following setup for our problem:

### What We Know

Given the problem statement, we know that side *c* is 52 inches, angle
*C* is 90 degrees, and that side *b* is 16 units and side *a*
is 9 units, which is screen aspect ratio. We also know that in order to find the
length of sides *a* and *b* in inches we need at least two values:

- The value of one of the sides, which we know
*c*to be 52 inches - The value of one of the angles of the triangle, which we can get to by using the ratio
*16:9*

### Calculating The Angle

We can calculate the angle *A* by using trigonometry functions. We can use the
*Tangent* function on angle *A*, which is to say that the *Tangent* of *A* is
equal to *a* divided by *b*. From the screen ratio we know that *a* and *b* are
*9* and *16* respectively, therefore *Tan(A) = a/b* which is *0.5625* in
radians.

We also know from trigonometry class that the value of angle *A* is the inverse
of the *Tangent* of *A*, or the *ArcTangent* of the *Tangent* of *A*. Therefore,
*ATan(0.5625) = 29.35* in degrees. I cheated and used a calculator to get to the
value of *A*.

Now that we have both the value of side *c* and of angle *A*, we can calculate
the values of the sides *a* and *b*.

### Calculating The Sides

Let us focus on side *a* first. We know that the *Sine* of angle *A* is the
quotient of *a* divided by *c*, or *Sin(A) = a/c*. We can rewrite the equation
like *a = Sin(A) c_ in order to find the value of side a. So _a =
Sin(29.35)52* which is

*25.49*.

Now that we have the value of *a* we can find *b* by using the *Tangent* of *A*
again. When we calculate the angle *A* we used the *Tangent* function which is
equivalent to *a* divided by *b*, or *Tan(A) = a/b*. In order to find the value
of side *b*, we can rewrite this equation to *b = a/Tan(A)*. So *b =
25.49/0.5625* which is *45.32*.

Hence, the value of side *a* is *25.49* inches and the value of side *b* is
*45.32* inches. But are we sure about that?

### Verification

We can verify this results by using the *Pythagorean Theorem*. We know that the
square of the hypotenuse is equal to the sum of the squares of the sides, or
*c ^{2} = a^{2} + b^{2}*. Let us plug in the numbers that
we have for sides

*a*and

*b*to see what our value for

*c*is.

We have *a* and *b* equal to *25.49* and *45.32* respectively. The value of *a*
squared is *649.9228*. The value of *b* squared is *2054.0771*. The sum of the
squares of *a* and *b* is *2704*. The square root of *2704* is *52*.

I used a python session in idle to do all this math and this is what it looks like:

>>> import math >>> A = math.degrees(math.atan(9.0/16.0)) >>> A 29.357753542791276 >>> a = math.sin(math.atan(9.0/16.0)) * 52 >>> a 25.493584460893068 >>> b = a/(9.0/16.0) >>> b 45.321927930476562 >>> c = math.sqrt(a**2 + b**2) >>> c 52.0

Please let me know if you see any flaws in my logic.

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