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 c² = a² + b². 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.