Squaring walks one way — side → area. The square root walks back — area → side.
Lay tiles on a square floor. If each side runs 5 tiles long, the floor swallows 5 × 5 = 25 tiles — and we write that 52, read "five squared." That is the easy direction. The whole new idea of this lesson is the reverse trip: a friend tells you a square floor holds 25 tiles and asks how long each side is. Undoing a square is called taking a square root, and by the end of this page you'll read the symbol √, know exactly what number it points to, and be able to pin down even the "messy" roots like √20 to a tenth.
The steady color habit for this lesson: a side a is teal, the square a², the area is blue, the square-root operation recovers the teal side from the blue area, perfect squares glow blue, and the estimate bounds that trap a messy root are amber.
6.2.1 Squaring and the area of a square
Why do we say "squared" for the little raised 2? Because of an actual square. Take a side of length a and build a square on it. Slice it into a grid of unit boxes and you get a rows of a boxes each, so the box count — the area — is a × a. We write that product a2 and read it "a squared" or "the square of a." The word and the shape are the same fact.
So squaring a number is exactly "find the area of the square built on it." 32 = 9 because a 3-by-3 square holds 9 unit squares; 72 = 49 because a 7-by-7 square holds 49. The result is an area, and an area is never negative — already a hint about what's coming.
A square of side a = 4 is cut into 4 × 4 = 16 unit squares, so its area is 42 = 16.
Worked example
Read each square out loud, then count the area.
62 = 6 × 6 = 36 ("six squared is thirty-six"). 102 = 10 × 10 = 100 ("ten squared is one hundred").
A square patio with side 12 ft has area 122 = 144 square feet.
Key idea
To square a number is to build the square on it and count its tiles: a2 = a × a = the area. Side in, area out.
🎮 Try itGrow the square, watch the area
Step the side a up and down. The grid fills with unit squares, and the readout shows a2 = area.
Side a3
6.2.2 Taking a square root: from area back to side
Now flip the question. A friend says, "my square rug covers 9 square feet — how long is each side?" You are no longer given the side and asked for the area; you are given the area and asked for the side. You need a number that, when squared, gives back 9. That number is 3, because 32 = 9.
Written as an equation, you are solving x2 = 9 — "what number, squared, makes nine?" This undoing of a square is called taking the square root, and it gets its own symbol, the radical sign√. The little roof over the number is the vinculum; it tells you exactly which area is going under the root. So √9 is read "the square root of nine" and asks: what side gives area nine?
The square root reads the picture backward: from area = 9 it recovers the side = 3. So √9 = 3.
Worked example
Each square root is a "what side?" question. Find the side whose square is the area.
√16 = 4, because 42 = 16. √49 = 7, because 72 = 49. √1 = 1 and √0 = 0 (a side of length 0 makes a square of area 0).
🎮 Try itReverse finder: hunt the side
Pick a target area (a perfect square), then step the side until side2 matches the area. When it locks, the root is revealed.
Target area9
Your side2
6.2.3 Square roots and the principal square root
Here is the subtle part. If we only ask "what squares to 9?", there are two answers, because squaring kills minus signs: 32 = 9 and also (−3)2 = (−3) × (−3) = 9. So the equation x2 = 9 has two solutions, x = +3 and x = −3. Both +3 and −3 are "square roots of nine."
But a side length can't be negative, and we want the symbol √ to name one definite number, not two. So mathematicians agreed: √ always means the non-negative one, called the principal square root. Therefore √9 = 3 exactly — never ±3. If you actually want both solutions of x2 = 9, you write the ± yourself: x = ±√9 = ±3.
Both −3 and +3 square to 9 — but the radical √ points only to the positive one. So √9 = 3.
Watch out
The number-one mix-up in this whole lesson: writing "√9 = ±3." No. The radical alone is always one non-negative number, so √9 = 3. The ± only appears when you solve an equation like x2 = 9, where two sides could give that area.
🎮 Try itTwo roots, one principal
Step the candidate side across negatives and positives. Watch both −n and +n land on the same area — but see where √ points.
Candidate3
6.2.4 When √a exists, and why it's never negative
Because a square root is "the side of a square of area a," two facts fall out for free. First, the number inside the root, the radicand, must satisfy a ≥ 0 — there is no square with a negative area, so there is no real side to find. That means √−4 has no real value: no real number squared gives −4, because squaring any real number lands at 0 or above.
Second, the value that comes out is always ≥ 0 — it's a length. So you will never see a real square root return a negative answer. Put together: √a is defined only when a ≥ 0, and when it is defined, √a ≥ 0. The smallest case is √0 = 0.
Radicand a
Real value?
Why
a > 0
yes, one
a positive side: e.g. √25 = 5
a = 0
yes, zero
√0 = 0
a < 0
none
no real square is negative
Watch out
Keep the minus sign outside straight in your head. −√25 = −5 is perfectly fine — that's "the negative of √25," and the radicand 25 is still positive. The forbidden thing is a negative under the roof, like √−25, which has no real value.
🎮 Try itDomain checker
Slide the radicand a. While a ≥ 0 the root is a real length; push a below zero and the radical lights up "no real value."
Radicand a
6.2.5 Perfect squares and estimating roots
Some areas "come out perfectly." The numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … are exactly the areas of whole-number squares, so their roots are whole numbers. We call them perfect squares. Memorizing the first dozen makes square roots feel instant: √64 = 8, √81 = 9.
But most numbers fall between perfect squares — and their roots are messy (they're irrational, the surprise of the next lesson). You can still pin them down. Take √20. Since 16 < 20 < 25, taking roots gives √16 < √20 < √25, that is 4 < √20 < 5. So the side is between 4 and 5. To refine, test tenths: 4.42 = 19.36 (too small) and 4.52 = 20.25 (too big), so 4.4 < √20 < 4.5. Squeeze again and you get √20 ≈ 4.47.
√20 sits between the perfect squares: 4 = √16 and 5 = √25. The amber bounds tighten toward 4.47.
Worked example
Estimate √50 to one decimal place.
Bound it.49 < 50 < 64, so 7 < √50 < 8. It's just barely past 49, so try numbers near 7. Refine by tenths.7.12 = 50.41 (a touch too big); 7.02 = 49 (too small). So 7.0 < √50 < 7.1, and since 50 is very close to 50.41, √50 ≈ 7.1 (more precisely 7.07).
🎮 Try itSqueeze a root
Pick a number n. The bar traps √n between consecutive perfect squares; then nudge the tenths slider to tighten the bound.
n20
Tenths guess
★ The big ideas, in one breath
Squaring builds the square on a side and counts its tiles, so a2 is an area and is never negative; the square root√ walks that trip backward, recovering the side from the area. Two numbers square to 9, namely +3 and −3, but the radical points only at the non-negative principal root, so √9 = 3. A real √a needs a ≥ 0 and returns something ≥ 0. The perfect squares come out whole; everything trapped between them you pin down with amber bounds and refine by tenths.
Coming up next — 6.3
If a square turns a side into an area, a cube turns an edge into a volume — and we'll undo it with the cube root3√. There's one delicious difference: cubes keep the sign, so 3√−8 = −2 is perfectly real.
✎ Exercises 6.2
Work each one out first, then open the answer to check your thinking.
Find 82, and say it in words.
Show answer
82 = 8 × 8 = 64. In words: "eight squared is sixty-four" — a square of side 8 holds 64 unit squares.
A square garden has side 11 m. What is its area?
Show answer
Area = 112 = 11 × 11 = 121 square meters.
Evaluate √36.
Show answer
√36 = 6, because 62 = 36. The radical gives the non-negative side whose square is 36.
A square rug covers 144 square feet. How long is each side?
Show answer
The side is √144 = 12 ft, because 122 = 144.
True or false: √25 = ±5.
Show answer
False. The radical alone is the principal (non-negative) root, so √25 = 5 only. Both ±5 square to 25, but the symbol points to just the positive one.
Solve x2 = 49.
Show answer
x2 = 49 has two solutions: x = +7 and x = −7, since both square to 49. Write x = ±√49 = ±7. (The equation gives two; the radical itself is just +7.)
Does √−16 have a real value? Compare it to −√16.
Show answer
√−16 has no real value — no real number squared gives −16. But −√16 is fine: it means "the negative of √16," which is −4. The minus is outside the roof, so the radicand 16 is still positive.
Between which two consecutive whole numbers does √30 lie?
Show answer
The perfect squares around 30 are 25 and 36: 25 < 30 < 36. Taking roots, 5 < √30 < 6. (It's closer to the middle: √30 ≈ 5.48.)
Estimate √40 to one decimal place.
Show answer
36 < 40 < 49, so 6 < √40 < 7. Test tenths: 6.32 = 39.69 (too small), 6.42 = 40.96 (too big). So 6.3 < √40 < 6.4, and since 40 is near 39.69, √40 ≈ 6.3 (truly 6.32).
A square has area 169 square inches. Find its side, then its perimeter.
Show answer
Side = √169 = 13 in, since 132 = 169. Perimeter = 4 × side = 4 × 13 = 52 in. (Find the side first by undoing the square; the perimeter is then just four equal sides.)
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson serves 8.EE.A.2 — using square-root symbols, evaluating √ of small perfect squares, and knowing that √2 is irrational — with grade-6 area (6.G) as the concrete anchor and 8.NS.A.2 for approximating irrational roots by trapping them between rationals. The #1 misconception is writing "√9 = ±3," conflating the radical (a single non-negative number) with the two solutions of x2 = 9. The antidote: tie the radical to a side length, which can't be negative — the symbol always names the principal (non-negative) root, and the ± appears only when the student solves an equation, where the negative side is also a valid algebraic solution.