How to use the SQRT function

What is the SQRT function?

The SQRT function calculates the positive square root.

1. Introduction

What is a square root?

The square root of x is denoted by √x but it can also be x^1/2 or x^0.5 A square root of a number x is a value that when multiplied by itself equals x.

√x * √x = x

For example

√4 * √4 = 4

It is the reverse operation of squaring a number.

x² = 4
x = √4
x₁ = +2
x₂ = -2

Why are there two solutions to a square root?

There are two solutions to a square root is because when you square a number whether it is positive or negative you get the same result.

For example, (-2)² = 2² = 4.

If you want to find the number that was squared to get 4, you have two possible answers:

√4 = 2 or √4 = -2.

Often we only want the positive solution of a square root, like when we are dealing with length, area, or time.

Is there a solution to a square root of a negative number?

No, there is no real number solution for the square root of a negative number. For example: √(-16) has no solution because if you square a real number you always get a positive result:

(-4)2 = 16
4^2 = 16

So no real number squared can produce a negative result.

However, we can find a complex number solution by introducing the imaginary unit i:

i = √(-1)

Where i is defined such that i² = -1. Using i we can take square roots of negatives:

√(-16) = 4i

Because (4i)(4i) = 16*i² = 16*-1 = -16

Square roots of negative real numbers are undefined, the concept of imaginary numbers and the imaginary unit i allows extending square roots to negative and complex numbers.

Check out the Engineering category for Excel functions dealing with complex numbers involving imaginary units.

What is an even power?

An even power or exponent refers to when a number is raised to an even integer exponent. Even powers have the general form x²ⁿ where n is some integer.
The first even power is the square x². Higher even powers are x², x⁴, x⁶, etc.

What is an even number?

An even number is an integer (whole number) that is divisible by 2 with no remainder.

2. SQRT Function Syntax

SQRT(number)

number

Required. The positive numerical value for which you want the square root.

3. Example

The first example in cell B3 and C3 demonstrates what you get if you try to square root a negative number in Excel. The formula returns the #NUM! error value if a number is negative.

Formula in cell C3:

Use the ABS function to remove the sign if a number is negative.

The second example in cell B4 and C4 calculates the square root of 0 (zero) which is 0 (zero).

The third example in cell B5:C5 calculates the square root of 1 which is equal to 1.

The fourth example in cell B6:C6 calculates the square root of 4 which is equal to 2. 2*2 = 4

The fifth example in cell B7:C7 calculates the square root of 9 which is equal to 3. 3 * 3 = 9

The sixth example in cell B8:C8 calculates the square root of 36 which is equal to 6. 6 * 6 = 36

4. Calculate the hypothenuse in a right triangle

The SQRT function lets you calculate the hypotenuse using the Pythagorean Theorem:
c = √(a² + b²)

Excel formula in cell C5:

The formula in cell C5 calculates the hypotenuse in a right triangle based one side equal to 6 units and the other side equal to 8 units. A right triangle must have one angle equal to 90 degrees.

What is the Pythagorean Theorem?

The Pythagorean Theorem is a mathematical relationship that describes the lengths of the sides of a right triangle.

In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

a² + b² = c²

a and b are the lengths of the legs of the triangle.

c is the length of the hypotenuse.

To calculate the hypothenuse the formula becomes:

c = √(a² + b²)

Explaining formula

Step 1 - Square leg 1

The caret character lets you raise a number to a given power. The caret symbol is only one character compared to the POWER function that also raises a number to a given power.

C2^2

becomes

6^2

and returns 36 (6 * 6 = 36).

Step 2 - Square leg 2

C3^2

becomes

8^2

and returns 64 (8 * 8 = 64).

Step 3 - Add the squared numbers

C2^2+C3^2

becomes

36 + 64 equals 100

Step 4 - Calculate the square root of the sum

SQRT(C2^2+C3^2)

becomes

SQRT(100)

and returns 10.

5. Calculate the legs in a right triangle

The SQRT function lets you calculate the legs of a right triangle using the Pythagorean Theorem:
a = √(c² - b²)
or
b = √(c² - a²)

Excel formula in cell C5:

The formula in C5 calculates the length of leg 1 based on leg 2 equal to 8 units and the hypotenuse equal to 10.

What is the Pythagorean Theorem?

The Pythagorean Theorem is a mathematical relationship that describes the lengths of the sides of a right triangle.

In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

a² + b² = c²

a and b are the lengths of the legs of the triangle.

c is the length of the hypotenuse.

To calculate the leg of a right triangle the formula becomes:

a = √(c² - b²)

or

b = √(c² - a²)

Explaining formula

Step 1 - Square the hypotenuse

The caret character lets you raise a number to a given power. The caret symbol is only one character compared to the POWER function that also raises a number to a given power.

C2^2

becomes

10^2

and returns 100 (10 * 10 = 100).

Step 2 - Square leg 2

C3^2

becomes

8^2

and returns 64 (8 * 8 = 64).

Step 3 - Subtract the squared hypotenuse with the squared leg 2

C2^2-C3^2

becomes

100 - 63 equals 36

Step 4 - Calculate the square root of the sum

SQRT(C2^2+C3^2)

becomes

SQRT(36)

and returns 6.

6. Function not working

The SQRT function returns

• #NUM! error value if the number argument is negative.
• #VALUE! error if you use a non-numeric input value.
• #NAME? error if you misspell the function name.
• propagates errors, meaning that if the input contains an error (e.g., #VALUE!, #REF!), the function will return the same error.

6.1 Troubleshooting the error value

When you encounter an error value in a cell a warning symbol appears, displayed in the image above. Press with mouse on it to see a pop-up menu that lets you get more information about the error.

1. The first line describes the error if you press with left mouse button on it.
2. The second line opens a pane that explains the error in greater detail.
3. The third line takes you to the "Evaluate Formula" tool, a dialog box appears allowing you to examine the formula in greater detail.
4. This line lets you ignore the error value meaning the warning icon disappears, however, the error is still in the cell.
5. The fifth line lets you edit the formula in the Formula bar.
6. The sixth line opens the Excel settings so you can adjust the Error Checking Options.

Here are a few of the most common Excel errors you may encounter.

#NULL error - This error occurs most often if you by mistake use a space character in a formula where it shouldn't be. Excel interprets a space character as an intersection operator. If the ranges don't intersect an #NULL error is returned. The #NULL! error occurs when a formula attempts to calculate the intersection of two ranges that do not actually intersect. This can happen when the wrong range operator is used in the formula, or when the intersection operator (represented by a space character) is used between two ranges that do not overlap. To fix this error double check that the ranges referenced in the formula that use the intersection operator actually have cells in common.

#SPILL error - The #SPILL! error occurs only in version Excel 365 and is caused by a dynamic array being to large, meaning there are cells below and/or to the right that are not empty. This prevents the dynamic array formula expanding into new empty cells.

#DIV/0 error - This error happens if you try to divide a number by 0 (zero) or a value that equates to zero which is not possible mathematically.

#VALUE error - The #VALUE error occurs when a formula has a value that is of the wrong data type. Such as text where a number is expected or when dates are evaluated as text.

#REF error - The #REF error happens when a cell reference is invalid. This can happen if a cell is deleted that is referenced by a formula.

#NAME error - The #NAME error happens if you misspelled a function or a named range.

#NUM error - The #NUM error shows up when you try to use invalid numeric values in formulas, like square root of a negative number.

#N/A error - The #N/A error happens when a value is not available for a formula or found in a given cell range, for example in the VLOOKUP or MATCH functions.

#GETTING_DATA error - The #GETTING_DATA error shows while external sources are loading, this can indicate a delay in fetching the data or that the external source is unavailable right now.

6.2 The formula returns an unexpected value

To understand why a formula returns an unexpected value we need to examine the calculations steps in detail. Luckily, Excel has a tool that is really handy in these situations. Here is how to troubleshoot a formula:

1. Select the cell containing the formula you want to examine in detail.
2. Go to tab “Formulas” on the ribbon.
3. Press with left mouse button on "Evaluate Formula" button. A dialog box appears.
   The formula appears in a white field inside the dialog box. Underlined expressions are calculations being processed in the next step. The italicized expression is the most recent result. The buttons at the bottom of the dialog box allows you to evaluate the formula in smaller calculations which you control.
4. Press with left mouse button on the "Evaluate" button located at the bottom of the dialog box to process the underlined expression.
5. Repeat pressing the "Evaluate" button until you have seen all calculations step by step. This allows you to examine the formula in greater detail and hopefully find the culprit.
6. Press "Close" button to dismiss the dialog box.

There is also another way to debug formulas using the function key F9. F9 is especially useful if you have a feeling that a specific part of the formula is the issue, this makes it faster than the "Evaluate Formula" tool since you don't need to go through all calculations to find the issue..

1. Enter Edit mode: Double-press with left mouse button on the cell or press F2 to enter Edit mode for the formula.
2. Select part of the formula: Highlight the specific part of the formula you want to evaluate. You can select and evaluate any part of the formula that could work as a standalone formula.
3. Press F9: This will calculate and display the result of just that selected portion.
4. Evaluate step-by-step: You can select and evaluate different parts of the formula to see intermediate results.
5. Check for errors: This allows you to pinpoint which part of a complex formula may be causing an error.

The image above shows cell reference B3 converted to hard-coded value using the F9 key. The SQRT function requires numerical values larger then or equal to 0 (zero) which is not the case in this example. We have found what is wrong with the formula.

Tips!

• View actual values: Selecting a cell reference and pressing F9 will show the actual values in those cells.
• Exit safely: Press Esc to exit Edit mode without changing the formula. Don't press Enter, as that would replace the formula part with the calculated value.
• Full recalculation: Pressing F9 outside of Edit mode will recalculate all formulas in the workbook.

Remember to be careful not to accidentally overwrite parts of your formula when using F9. Always exit with Esc rather than Enter to preserve the original formula. However, if you make a mistake overwriting the formula it is not the end of the world. You can “undo” the action by pressing keyboard shortcut keys CTRL + z or pressing the “Undo” button

6.3 Other errors

Floating-point arithmetic may give inaccurate results in Excel - Article

Floating-point errors are usually very small, often beyond the 15th decimal place, and in most cases don't affect calculations significantly.

Functions in 'Math and trigonometry' category

The SQRT function function is one of 62 functions in the 'Math and trigonometry' category.

Excel function categories

Excel categories

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