Square Roots and Radicals for the Pythagorean Theorem

The square root of a number n, written √n, is the non-negative number that gives n when multiplied by itself: √49 = 7 because 7 × 7 = 49. In the Pythagorean theorem the square root is the last step, turning the area c² into the length c. When the result isn't a whole number, you simplify the radical (√50 = 5√2) or round it (≈ 7.07).

What a square root means

Squaring and square-rooting are opposites. Squaring a side gives the area of the square built on it; taking the square root of an area gives back the side. That's exactly why the theorem, which is about areas (a² + b² = c²), needs a square root to give you a length.

Some vocabulary you will see:

• √ is the radical sign, and an expression like √50 or 5√2 is a radical.
• The number under the sign is the radicand (50 in √50).
• A perfect square is a whole number whose square root is also a whole number: 1, 4, 9, 16, 25 and so on.

A square root doesn't have to be a whole number to be a real length. The tilted squares below are drawn on ordinary graph paper, yet their sides are exactly √2 and √5 units long.

Perfect squares from 1 to 30

Knowing these by sight is the single biggest time-saver in Pythagorean problems. When c² turns out to be one of them, you can write the answer down immediately.

Two patterns help you spot perfect squares quickly:

• Last digits. A perfect square always ends in 0, 1, 4, 5, 6 or 9. If a number ends in 2, 3, 7 or 8, its square root is not a whole number, so don't waste time looking for one.
• Squares of numbers ending in 5. Multiply the tens digit by the next number up and write 25 after it: 25² is 2 × 3 = 6, then 25, giving 625. Likewise 15² = 225 and 35² = 1225.

Many of these squares come in matching sets: 9 + 16 = 25, 25 + 144 = 169, 64 + 225 = 289. Those are the Pythagorean triples, the right triangles with whole-number sides.

Estimating a square root

Before you reach for a calculator, you can pin any square root between two whole numbers, and usually get the first decimal place right too.

1. Find the perfect squares on either side. For √50: 49 < 50 < 64, so 7 < √50 < 8.
2. See how far along the gap you are. From 49 to 64 is 15 steps, and 50 is 1 step in. So √50 is about 1 over 15 of the way from 7 to 8.
3. Add that fraction. √50 ≈ 7 + 1 over 15 ≈ 7.07. The true value is 7.0711.

This "fraction of the gap" estimate always comes out slightly low, because squares spread out as numbers grow. For √30 it gives 5 + 5 over 11 ≈ 5.45, while the true value is 5.477.

Sharpen the estimate: divide and average

To get more accuracy by hand, divide the number by your guess and average the guess with the result. This is often called Heron's method or the Babylonian method.

One round already matches the true value, 5.47723…, to three decimal places. Repeat the step with the new guess and the number of correct digits roughly doubles each time.

Simplifying radicals (simplest radical form)

A radical is in simplest radical form when the radicand has no perfect-square factor other than 1. So √50 is not simplified, because 50 = 25 × 2 and 25 is a perfect square, but 5√2 is. The whole method rests on one rule:

Method 1: find the largest perfect-square factor

1. List perfect squares that might divide the radicand: 4, 9, 16, 25, 36, 49, 64, 81, 100…
2. Pick the largest one that divides evenly. For 72, both 4 and 36 work; choose 36.
3. Split and simplify. √72 = √36 · 2 = √36 · √2 = 6√2.

If you pick a smaller square by mistake, you're not wrong, just not finished: √72 = √4 · 18 = 2√18, and √18 still contains 9, so keep going to 2 · 3√2 = 6√2.

Method 2: prime factors in pairs

For larger numbers, break the radicand into prime factors. Every pair of equal primes comes out of the root as a single factor; anything unpaired stays inside.

Radicals that keep turning up

These come up again and again as hypotenuses and legs. The decimals are rounded to three places.

When a side is already a radical

In multi-step problems, the answer to one triangle often becomes a side of the next, and it may be a radical. Squaring undoes the root, which keeps the arithmetic clean:

Exact answers, decimals and rounding

An exact answer keeps the radical (√52, better written as 2√13). A decimal answer is an approximation (≈ 7.21). Use the one the question asks for:

• "Exact", "simplest radical form" or "leave in radical form": give the simplified radical.
• "Nearest tenth" (one decimal place), "nearest hundredth" (two) and so on: give the decimal, rounded as asked, and use the ≈ sign.
• Measurement and word problems: usually a decimal with units, rounded to a sensible precision.
• No instruction: give both, as in 2√13 ≈ 7.21.

To round, look at the digit just after the last place you keep: 5 or more rounds up, 4 or less rounds down. √52 = 7.2111… is 7.2 to the nearest tenth and 7.21 to the nearest hundredth.

Never round in the middle

Rounding an intermediate result and then using it again lets the error grow. Suppose a triangle has legs 1 and 1, and its hypotenuse becomes a leg of a second triangle whose other leg is also 1 (the first steps of the spiral of Theodorus):

Keep the radical (or the full calculator value) until the final step, then round once.

Using a calculator

• Use parentheses around a sum. To find √9 + 16, enter √(9 + 16). Without the parentheses many calculators compute √9 + 16 = 19.
• Know your key order. Most scientific and graphing calculators want √ first, then the number. Some basic calculators want the number first, then √. Test with √25 = 5 if you are unsure.
• Spreadsheets and code use a function: =SQRT(50) in Excel or Google Sheets, Math.sqrt(50) in JavaScript, math.sqrt(50) in Python.
• Check by squaring. Square your answer: if √50 ≈ 7.0711, then 7.0711² ≈ 50.0.

The simplify square root calculator gives both the simplest radical form and the decimal, and the Pythagorean theorem calculator simplifies the root of every answer automatically.

Why the negative root is thrown away

The equation c² = 25 has two solutions, c = 5 and c = −5, because (−5)² = 25 too. The symbol √25, though, means only the non-negative one, 5. In a triangle every side is a positive length, so the negative solution has no meaning and you drop it.

That's specific to lengths. When the unknown is a coordinate rather than a length, both signs can matter. On the circle x² + y² = 25, the points with x = 3 have y = 4 and y = −4 (see the equation of a circle). And a negative number has no real square root at all, which is why a negative result when you find a missing leg signals an impossible triangle.