Here is a line segment on a grid. How can we determine the length of this line segment?

By drawing some circles, we can tell that it’s longer than 2 units, but shorter than 3 units.

Two circles that have the same center are drawn on a square grid with radii 2 and 3. A line segment slanted upward and to the left is drawn such that the bottom endpoint is the center of the two circles and is 1 unit down and 2 units right of the top endpoint of the line segment.

To find an exact value for the length of the segment, we can build a square on it, using the segment as one of the sides of the square.

The area of this square is 5 square units. That means the exact value of the length of its side is units.

Notice that 5 is greater than 4, but less than 9. That means that is greater than 2, but less than 3. This makes sense because we already saw that the length of the segment is between 2 and 3 units.

We can approximate the value of a square root by observing the whole numbers around it and remembering the relationship between square roots and squares. Here are some examples:

• is a little more than 8 because is a little more than , and .
• is a little less than 9 because is a little less than , and .
• is between 8 and 9 (it’s 8 point something) because 75 is between 64 and 81.
• is approximately 8.67 because .

A number line with the numbers 8 through 9, in increments of zero point 1, are indicated. The square root of 64 is indicated at 8. The square root of 65 is indicated between 8 and 8 point 1, where the square root of 65 is closer to 8 point 1. The square root of 75 is indicated between 8 point 6 and 8 point 7, the square root of 75 is closer to 8 point 7. The square root of 80 is indicated between 8 point 9 and 9, where the square root of 80 is closer to 8 point 9. The square root of 81 is indicated at 9.

If we want to find the square root of a number between two whole numbers, we can work in the other direction. For example, since and , then we know that (to pick one possibility) is between 22 and 23. Many calculators have a square root command, which makes it simple to find an approximate value of a square root.