Side 3 - Pythagorean Theorem Calculator

    Calculate the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²).

    One side of a right triangle measures 3. What is the hypotenuse? The Pythagorean theorem says a squared plus b squared equals c squared. The classic 3-4-5 triangle: 9 + 16 = 25. Enter the second side or the hypotenuse and the calculator will find the missing dimension. It also checks whether a triangle is right-angled.

    Parameters

    Enter data for calculations

    First leg of the right triangle (leave empty if solving for this side)

    Second leg of the right triangle (leave empty if solving for this side)

    Hypotenuse - the longest side, opposite the right angle (leave empty if solving for this side)

    What Does the Pythagorean Theorem Calculator Do?

    Enter two sides of a right triangle - the calculator instantly finds the third. Formula: a2 + b2 = c2, where a and b are the legs, c is the hypotenuse (longest side). Fill in two fields, leave the third empty.

    How to Use - Step by Step

    1
    Decide what you are looking for
    Know a and b? The calculator will find c (hypotenuse). Know a and c (or b and c)? It will find the missing leg.
    2
    Enter two known sides
    Type values into fields a, b or c. Leave the third field empty - the calculator automatically detects which side to compute.
    3
    Read the result
    The calculator returns the missing side rounded to several decimal places. Remember: c (hypotenuse) is ALWAYS the longest side.

    Formula and Math

    a2 + b2 = c2
    Finding c: c = sqrt(a2 + b2). E.g. a = 3, b = 4 => c = sqrt(9 + 16) = sqrt(25) = 5
    Finding a: a = sqrt(c2 - b2). E.g. c = 13, b = 5 => a = sqrt(169 - 25) = sqrt(144) = 12
    Finding b: b = sqrt(c2 - a2). E.g. c = 10, a = 6 => b = sqrt(100 - 36) = sqrt(64) = 8

    Common Pythagorean Triples

    Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. Worth memorizing - they appear in exams and standardized tests.

    a b c Verification
    3 4 5 9 + 16 = 25
    5 12 13 25 + 144 = 169
    6 8 10 36 + 64 = 100
    7 24 25 49 + 576 = 625
    8 15 17 64 + 225 = 289
    9 40 41 81 + 1600 = 1681

    Common Mistakes

    Mixing up c with a or b. The hypotenuse c is always the LONGEST side and sits opposite the right angle. If you enter c smaller than a or b, the formula gives a negative number under the square root - the calculator will show an error.
    Using on non-right triangles. The Pythagorean theorem only works for triangles with a 90-degree angle. For other triangles, use the Law of Cosines.
    Forgetting to square. The formula is a2 + b2 = c2, not a + b = c. For a = 3, b = 4: the result is sqrt(9 + 16) = 5, not 3 + 4 = 7.

    Practical Examples

    Example 1: A ladder 5 m long leans against a wall. The base is 3 m from the wall. How high does the ladder reach?
    b = sqrt(52 - 32) = sqrt(25 - 9) = sqrt(16) = 4 m
    Example 2: A TV screen is 48 inches wide and 27 inches tall. What is the diagonal?
    c = sqrt(482 + 272) = sqrt(2304 + 729) = sqrt(3033) = 55.07 inches
    Example 3: A room is 4 m by 3 m. What is the diagonal distance corner to corner?
    c = sqrt(42 + 32) = sqrt(16 + 9) = sqrt(25) = 5 m
    Example 4: A ramp rises 1.2 m over a horizontal distance of 5 m. How long is the ramp?
    c = sqrt(52 + 1.22) = sqrt(25 + 1.44) = sqrt(26.44) = 5.14 m
    Example 5: Two ships leave port at the same time. One goes 8 km north, another 6 km east. Distance between them?
    c = sqrt(82 + 62) = sqrt(64 + 36) = sqrt(100) = 10 km

    FAQ - Frequently Asked Questions

    What is a Pythagorean triple?
    A set of three positive integers (a, b, c) satisfying a2 + b2 = c2. The most common is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple - e.g. (6, 8, 10) = 2 x (3, 4, 5).
    Does the theorem work for non-right triangles?
    No. The Pythagorean theorem only applies to right triangles (with a 90-degree angle). For other triangles use the Law of Cosines: c2 = a2 + b2 - 2ab cos(C).
    Can I use decimal numbers?
    Yes. The calculator handles decimals with precision up to several decimal places. Enter values like 3.5 and 4.2 - the result will be computed correctly.
    How do I check if a triangle is right-angled?
    Enter all three sides. If a2 + b2 = c2 (where c is the longest side), the triangle has a right angle. The calculator shows the verification formula automatically.
    What real-world uses does the Pythagorean theorem have?
    Construction (checking right angles with a 3-4-5 triangle), navigation (distance between two points), architecture (roof slopes), screen sizes (diagonal measurement), GPS calculations, and surveying.
    Why does the calculator show an error when c is smaller than a or b?
    Because the hypotenuse must be the longest side. If c < a, then c2 - a2 is negative, and you cannot take the square root of a negative number (in real numbers). Check which side is the hypotenuse.

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    See also

    Pythagorean Theorem with Side 4 - Find the HypotenusePythagorean Theorem with Side 5 - Find the HypotenusePythagorean Theorem with Side 6 - Find the HypotenusePythagorean Theorem with Side 7 - Find the HypotenusePythagorean Theorem with Side 8 - Find the HypotenusePythagorean Theorem with Side 10 - Find the HypotenusePythagorean Theorem with Side 12 - Find the HypotenusePythagorean Theorem with Side 15 - Find the HypotenusePythagorean Theorem with Side 20 - Find the HypotenusePythagorean Theorem with Side 25 - Find the Hypotenuse

    Calculator verified by the LiczGrupa.pl team

    Content, formulas and results have been reviewed for accuracy and relevance by our team of specialists.

    Krystian Szyszka

    Reviewed by: Krystian Szyszka

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