How Do You Mathematically Determine the Dimensions of a 50-Inch TV?

To calculate the screen dimensions of a 50-inch TV, you must know the aspect ratio of the screen — the most common being 4:3 for conventional TV screens and 16:9 for HDTV. Plugging these ratios into the Pythagorean theorem, a 50-inch conventional TV screen is 50 x 0.8 = 40 inches wide and 50 x 0.6 = 30 inches high. A 50-inch HDTV is 50 x 0.87 = 43.6 inches high and 50 x 0.49 = 24.5 high.

TV sets are generally measured in inches diagonally across the screen. The Pythagorean theorem is useful when calculating screen dimensions as it describes the relationship between the diagonal and the two sides of a right triangle. It states that the square of a hypotenuse of a right triangle, or the diagonal, is equal to the sum of the squares of its two sides, or its height and width.

Knowing the length of the diagonal is not enough, however, as there are many height-width measurements that satisfy the theorem. You need to have the aspect ratio, or the ratio of the width to the height of an image or screen to narrow the results to a single height-width pair.

Most TV screens today have aspect ratios of 4:3 for conventional screens and 16:9 for HDTV. Plugging each of these into the Pythagorean theorem and simplifying yields two sets of simple equations. For conventional TV screens, the width is the diagonal times 0.8 and the height is the diagonal times 0.6. For HDTV screens, the width is the diagonal times 0.87 and the height is the diagonal times 0.49. Therefore, a 50-inch conventional TV is 40 inches wide and 30 inches tall. A 50-inch HDTV is 43.6 inches wide and 24.5 inches tall.