The Pythagorean Theorem in Real Life: The 3-4-5 Rule and Screen Sizes
October 21, 2026
In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². This turns up constantly in practical work, from checking that a foundation corner is square to working out how big a screen really is. A triangle with sides 3, 4, and 5 is the classic example because 9 + 16 = 25.
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Open the Geometry Calculator →Squaring a Corner With 3-4-5
Builders use the 3-4-5 rule to check that a corner is exactly 90°. Measure 3 units along one side from the corner and 4 along the other, then check that the distance between the two marks is exactly 5. If it is, the corner is square.
For larger layouts, scale up for accuracy: 6-8-10, 9-12-15, or 12-16-20 feet all work. A bigger triangle makes small errors easier to see.
- Mark 9 feet along one edge from the corner.
- Mark 12 feet along the other edge.
- Measure the diagonal between the marks. It should be exactly 15 feet.
- Adjust one edge until it reads 15.
Finding a Diagonal
A 12 by 16 foot room has a diagonal of √(12² + 16²) = √(144 + 256) = √400 = 20 feet. A rectangle is square if its two diagonals are equal, which is another way to check a frame or a slab layout.
To fit a long item, like a 10-foot pipe in a box 6 feet by 7 feet by 5 feet, the longest straight line through the box is the 3D diagonal, √(6² + 7² + 5²) = √110 ≈ 10.5 feet, so it will just fit.
From TV Diagonal to Width and Height
TV sizes are quoted as the diagonal. For a 16:9 screen, the width is 16 units, the height is 9, and the diagonal is √(16² + 9²) = √337 ≈ 18.36 units. So for a 55-inch TV, one unit is 55 ÷ 18.36 ≈ 3.0 inches. The width is about 16 × 2.995 ≈ 47.9 inches and the height about 27.0 inches.
Use the same method to decide whether a screen will fit your cabinet. Add the bezel and stand to those figures.
| TV size (16:9) | Width | Height |
|---|---|---|
| 43 in | 37.5 in | 21.1 in |
| 55 in | 47.9 in | 27.0 in |
| 65 in | 56.7 in | 31.9 in |
Finding a Missing Leg
Rearranging the formula gives a = √(c² − b²). A 13-foot ladder with its base 5 feet from a wall reaches √(169 − 25) = √144 = 12 feet up. Safety guidelines often recommend a base distance of about one quarter of the working height, so check the ladder's rating.
Frequently Asked Questions
What is the 3-4-5 rule?
A right triangle with sides in the ratio 3:4:5 has a 90° angle, so measuring 3, 4, and 5 units verifies a square corner.
How do I find the diagonal of a rectangle?
Use the Pythagorean theorem: diagonal = √(length² + width²).
Does the theorem work in 3D?
Yes. The space diagonal of a box is √(length² + width² + height²).
How do I find a TV's width from its diagonal size?
For a 16:9 screen, multiply the diagonal by 16 ÷ √337 (about 0.872) to get the width, and by 9 ÷ √337 (about 0.490) for the height.