Pythagoras Theorem Calculator

Pythagoras Theorem Guide

What should I know about Pythagoras Theorem?

In a right triangle: a² + b² = c². Where c is the hypotenuse (longest side, opposite the right angle). Finding the hypotenuse: c = √(a² + b²). Finding a shorter side: a = √(c² − b²). The 3-4-5 triangle: 3² + 4² = 9 + 16 = 25 = 5². This is the simplest Pythagorean triple. Other triples: 5-12-13, 8-15-17, 7-24-25. Distance formula in coordinate geometry: distance between (x₁,y₁) and (x₂,y₂) = √((x₂−x₁)² + (y₂−y₁)²) is Pythagoras in 2D. 3D distance: d = √(a² + b² + c²).

What do I need to know about Trigonometry in Right Triangles?

SOH CAH TOA: sin(θ) = Opposite/Hypotenuse. cos(θ) = Adjacent/Hypotenuse. tan(θ) = Opposite/Adjacent. Finding angles: angle A = arcsin(a/c) = arccos(b/c) = arctan(a/b). All angles of a right triangle: one is 90°. The other two sum to 90° (complementary angles). If angle A = 30°: angle B = 60°. Special triangles: 30-60-90 (sides in ratio 1:√3:2). 45-45-90 (sides in ratio 1:1:√2). These give exact values: sin(30°) = 0.5, cos(60°) = 0.5, tan(45°) = 1.

What do I need to know about Pythagoras in 3D?

Length of space diagonal of a cuboid (box): d = √(l² + w² + h²). This is applying Pythagoras twice: first to the base (√(l² + w²)), then using that diagonal and the height. Example: box 3m × 4m × 5m. Base diagonal = √(9+16) = 5m. Space diagonal = √(25+25) = √50 = 7.07m. Maximum length of item fitting in the box: 7.07m. Construction uses: checking square corners: a 3-4-5 rope gives a perfect right angle. Surveying: horizontal distances calculated from slant measurements. Navigation: calculating the shortest straight-line distance between two points given their coordinates in three dimensions, essential for both marine and aviation route planning.

What should I know about Pythagorean Triples?

Pythagorean triples are integer solutions to a² + b² = c². Generator formula: a = m² − n², b = 2mn, c = m² + n² for any positive integers m > n. For m=2, n=1: a=3, b=4, c=5. For m=3, n=2: a=5, b=12, c=13. For m=4, n=3: a=7, b=24, c=25. For m=5, n=2: a=21, b=20, c=29. Infinite families: multiplying any triple by a constant gives another triple. 3-4-5, 6-8-10, 9-12-15... Fermat's Last Theorem (proved 1994 by Andrew Wiles): no integer solutions exist for aⁿ + bⁿ = cⁿ when n > 2. The Pythagorean triple pattern continues indefinitely, with infinitely many such whole-number combinations existing, generated systematically using formulas involving pairs of integers.

What do I need to know about the Converse of Pythagoras' Theorem?

If a triangle's three sides satisfy a² + b² = c² (where c is the longest side), the triangle is guaranteed to be right-angled — this converse is just as useful as the theorem itself, because it lets you test whether a triangle is right-angled purely from its side lengths, without measuring any angle. Builders use this constantly: the '3-4-5 rule' checks a corner is square by measuring 3 units along one wall, 4 along the other, and confirming the diagonal between those points is exactly 5 (since 3²+4²=9+16=25=5²). If the measured diagonal doesn't match, the corner isn't a true right angle. This converse also explains why Pythagorean triples (3-4-5, 5-12-13, 8-15-17, and their multiples) are so useful on site: any multiple of a known triple (6-8-10, 9-12-15) also forms a right angle, giving builders flexible whole-number measurements to check squareness at different scales.

What should I know about the Distance Formula as Pythagoras in Disguise?

The formula for the distance between two coordinate points, d = √[(x₂−x₁)² + (y₂−y₁)²], is literally Pythagoras' theorem applied to a right triangle formed by the horizontal and vertical gaps between the points. For points (1,1) and (4,5): horizontal gap = 3, vertical gap = 4, these form the two legs of a right triangle, and the distance (hypotenuse) = √(3²+4²) = √25 = 5. This is why Pythagoras' theorem underlies almost every distance calculation in coordinate geometry, computer graphics, navigation, and physics (calculating displacement magnitude): any straight-line distance between two points on a flat plane decomposes into a horizontal and vertical component, and those two components always relate to the direct distance via the Pythagorean relationship, regardless of which direction the line actually runs.

What should I know about Checking Whether a Triangle Is Right-Angled from Its Sides?

Given only three side lengths, you can determine whether they form a right triangle without measuring any angle, simply by checking whether the Pythagorean relationship holds: if a² + b² = c² for the longest side c, the triangle is right-angled. If instead a² + b² > c², the triangle is acute (all angles under 90°); if a² + b² < c², it's obtuse (one angle over 90°). This converse test is routinely used in construction and surveying to confirm a corner is truly square using nothing but a tape measure — the classic "3-4-5" method marks 3 units along one wall and 4 along the other, and if the diagonal between those marks measures exactly 5, the corner is a true right angle, since 3² + 4² = 9 + 16 = 25 = 5².

Worked Real-World Examples: Ladder, TV Screen, and Navigation

Three classic real-world uses of Pythagoras' theorem show how the same simple formula applies to completely different situations. Ladder against a wall: a 5m ladder leans against a vertical wall with its foot 3m from the base of the wall — how high up the wall does it reach? The ladder is the hypotenuse, the ground distance and the wall height are the two legs, so height = √(5² − 3²) = √(25 − 9) = √16 = 4m. TV screen diagonal: TVs are sized by their diagonal measurement, and a 55-inch TV with a standard 16:9 aspect ratio has a width and height that satisfy width² + height² = 55². Solving the ratio gives a width of 55 × 16 ÷ √(16² + 9²) ≈ 47.9 inches and a height of 55 × 9 ÷ √(16² + 9²) ≈ 27.0 inches — checking the answer, √(47.9² + 27.0²) = √(2294 + 729) = √3023 ≈ 55.0, confirming the diagonal. Navigation: a boat sails 30 km due east, then 40 km due north — how far is it from its starting point in a straight line? The east and north legs form a right angle, so the direct distance is √(30² + 40²) = √(900 + 1600) = √2500 = 50 km, which is exactly the familiar 3-4-5 triple scaled up by a factor of 10.

Common Pythagorean Triples Reference Table

A Pythagorean triple is a set of three positive whole numbers that satisfy a² + b² = c² exactly, with no rounding needed. A handful of these come up repeatedly in maths, construction, and navigation because they give clean, exact right angles from simple tape-measure distances.

Every triple in the table (and infinitely many more) can be generated from any pair of positive integers m > n using a = m² − n², b = 2mn, c = m² + n², and multiplying any known triple by a whole number produces another valid triple (3-4-5 scaled by 10 gives 30-40-50, which is exactly the navigation example above).

Why Pythagoras' Theorem Works: A Short Proof Sketch

One of the simplest proofs of a² + b² = c² is a rearrangement argument that needs no algebra beyond areas. Take a square with side length (a + b), and place four identical right-angled triangles (with legs a and b, hypotenuse c) inside it, one in each corner, arranged so their hypotenuses form a smaller, tilted square in the middle. The big square's total area is (a + b)², which expands to a² + 2ab + b². That same big square is also exactly covered by the four triangles (each with area ½ab, so 4 × ½ab = 2ab in total) plus the tilted inner square (with area c², since its own sides are the length c). Setting the two descriptions of the same area equal: a² + 2ab + b² = 2ab + c². Subtracting 2ab from both sides leaves a² + b² = c² — the theorem falls directly out of the fact that the same physical square has been measured two different ways. This is one of several hundred known proofs (including one attributed to US President James Garfield), but the rearrangement version is widely considered the most visually intuitive.