Math & Everyday Calculations
Pythagorean Theorem Formula: Hypotenuse, Missing Leg & 3D Box Diagonal
Calculate right-triangle hypotenuse c = SQRT(a^2 + b^2), solve for an unknown leg b = SQRT(c^2 - a^2) with triangle-inequality validation, and find 3D shipping-crate space diagonal d = SQRT(L^2 + W^2 + H^2).
Check roof rafters, TV screen diagonals, foundation squareness, and 3D shipping crates!
Carpenters checking 3-4-5 foundation corners, solar installers sizing sloped roof rafters from horizontal run and vertical rise, and packaging engineers checking whether a long ski pole fits diagonally inside a rectangular L x W x H carton all rely on Pythagoras's Theorem: c = SQRT((a ^ 2) + (b ^ 2))!
Spreadsheets also make the two everyday extensions effortless: first, when you know the diagonal hypotenuse c (say a 13.0 ft ladder) and wall base offset a (5.0 ft), the vertical reach leg is b = SQRT((c ^ 2) - (a ^ 2)) = 12.0 ft—guarded by IF(c <= a, ...) so impossible geometry never triggers #NUM!; second, in three dimensions, applying Pythagoras twice across floor diagonal SQRT(L^2 + W^2) and height H gives the 3D interior space diagonal d_3D = SQRT((L ^ 2) + (W ^ 2) + (H ^ 2))!
| Geometric Calculation | Dimensions Entered | Formula Expression | Exact Result |
|---|---|---|---|
| 2D Hypotenuse (3-4-5 Check) | Leg a = 3, Leg b = 4 | SQRT(3^2 + 4^2) | 5.00 |
| 2D Roof Rafter (Run 12, Rise 5) | Leg a = 12, Leg b = 5 | SQRT(12^2 + 5^2) | 13.00 |
| Missing Leg from Hypotenuse | c = 13, Known a = 5 | SQRT(13^2 - 5^2) | 12.00 |
| 3D Carton Space Diagonal | L = 12, W = 4, H = 3 | SQRT(12^2 + 4^2 + 3^2) | 13.00 (Pythagorean quadruple!) |
2D Right-Triangle Hypotenuse c = SQRT(a^2 + b^2)
Validates positive legs B2 and B3, then computes the Euclidean diagonal.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Triangle Legs", ROUND(SQRT((B2 ^ 2) + (B3 ^ 2)), 4))3D Rectangular Box Interior Space Diagonal d = SQRT(L^2 + W^2 + H^2)
Computes maximum corner-to-opposite-corner rod length inside an L x W x H box (B2, B3, B4).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Box Dimensions", ROUND(SQRT((B2 ^ 2) + (B3 ^ 2) + (B4 ^ 2)), 4))Solve Unknown Leg b = SQRT(c^2 - a^2) Given Hypotenuse B5 & Leg B2
Guards against impossible geometry where hypotenuse B5 <= leg B2 (avoiding SQRT of a negative number).
=IF(OR(B2 <= 0, B5 <= B2), "Hypotenuse Must Exceed Leg", ROUND(SQRT((B5 ^ 2) - (B2 ^ 2)), 4))Try the example
Use the original values or change them to test your understanding.