C Math sqrt() Function

What You’ll Learn

The sqrt() function computes the square root of a non-negative number—the value that, when multiplied by itself, gives the original. It is one of the most-used functions in geometry, physics, and graphics.

Definition and Usage

The square root of x is the non-negative number y such that y × y = x. In C, sqrt(25.0) returns 5.0 because 5 × 5 = 25.

The domain is x ≥ 0. Passing a negative value returns NaN (Not a Number). Zero is valid: sqrt(0) = 0.

📝 Syntax

Standard C declaration:

double sqrt(double x);

Related variants

float sqrtf(float x);             /* <math.h> */
long double sqrtl(long double x); /* <math.h> */

Parameters

• x — non-negative value; domain is x ≥ 0.

Return Value

• Non-negative square root of x as double.
• sqrt(25.0) returns 5.0.
• sqrt(0) returns 0.0.
• sqrt(negative) returns NaN.

Headers and linking

• #include <math.h>
• Compile: gcc sqrt.c -std=c11 -o sqrt -lm
• Check invalid results with isnan() from <math.h>.

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Square roots appear everywhere from Pythagorean triangles to normalizing vectors.

#include <math.h>
#include <stdio.h>

int main(void) {
    double number = 25.0;
    double result = sqrt(number);

    printf("Square root of %.0f is %.0f\n", number, result);

    return 0;
}

Square root of 25 is 5

#include <math.h>
#include <stdio.h>

int main(void) {
    double x = 2.0;
    double root = sqrt(x);

    printf("sqrt(%.0f) = %.6f\n", x, root);
    printf("Check: %.6f * %.6f = %.6f\n", root, root, root * root);

    return 0;
}

sqrt(2) = 1.414214
Check: 1.414214 * 1.414214 = 2.000000

#include <math.h>
#include <stdio.h>

int main(void) {
    double a = 3.0;
    double b = 4.0;
    double c = sqrt(a * a + b * b);

    printf("Sides: %.0f and %.0f\n", a, b);
    printf("Hypotenuse: %.0f\n", c);

    return 0;
}

Sides: 3 and 4
Hypotenuse: 5

#include <math.h>
#include <stdio.h>

int main(void) {
    double x = 49.0;

    printf("x = %.0f\n", x);
    printf("sqrt(x)      = %.1f\n", sqrt(x));
    printf("pow(x, 0.5)  = %.1f\n", pow(x, 0.5));

    return 0;
}

x = 49
sqrt(x)      = 7.0
pow(x, 0.5)  = 7.0

#include <math.h>
#include <stdio.h>

int main(void) {
    double values[] = { 16.0, -4.0, 0.0 };
    size_t i;

    for (i = 0; i < 3; i++) {
        double x = values[i];
        if (x < 0.0) {
            printf("sqrt(%.0f): skipped (negative)\n", x);
            continue;
        }
        printf("sqrt(%.0f) = %.1f\n", x, sqrt(x));
    }

    return 0;
}

sqrt(16) = 4.0
sqrt(-4): skipped (negative)
sqrt(0) = 0.0

🚀 Common Use Cases

• Geometry — side length from area, Pythagorean theorem.
• Graphics — vector length, normalizing direction vectors.
• Physics — speed from kinetic energy, RMS calculations.
• Statistics — standard deviation uses square roots.
• Signal processing — magnitude of complex values.

📝 Notes

• Domain: x ≥ 0. Negative input → NaN.
• Returns the principal (non-negative) root only.
• sqrt(0) = 0; perfect squares yield exact integers in output.
• Prefer sqrt over pow(x, 0.5).
• For distances, prefer hypot(dx, dy) over manual squaring.
• Link with -lm on GCC/Clang Unix-like builds.

⚡ Optimization

sqrt() is often implemented in hardware and is fast. Avoid replacing it with Newton’s method unless profiling shows it matters. For comparing distances without the actual length, compare squared distances (dx*dx + dy*dy) to skip sqrt entirely in sorting or collision broad-phase checks.

💡 Best Practices