HighSchoolQ

📐 Geometry · Similarity and Right Triangles

The Pythagorean Theorem

Use a² + b² = c² to find a missing side of a right triangle, test whether a triangle is right, spot Pythagorean triples, and solve word problems.

Grades 9, 10⏱ 18 min✔ AI-checkedCCSS 8.G.B.6CCSS 8.G.B.7
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-08.

  • 8.G.B.6: Explain a proof of the Pythagorean Theorem and its converse.
  • 8.G.B.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

By the end you can…

  • Name the legs and the hypotenuse of a right triangle, and use a2 + b2 = c2 to find a missing side, giving an exact answer (a simplified radical) and a rounded decimal with ≈.
  • Explain why the theorem is true, using the areas of squares on the sides and the two ways of filling a square with four copies of the triangle.
  • Use the converse to decide whether three side lengths make a right triangle, and recognize Pythagorean triples and their multiples.
  • Solve word problems about ladders, diagonals (including the diagonal of a box) and shortcuts, and avoid the common mistakes: a wrong hypotenuse, adding instead of squaring, skipping the square root, rounding too early, and using the theorem on a triangle that is not right.

Lesson (school style)

A right triangle has one angle of exactly 90°, like the corner of a sheet of paper. The sides of every right triangle are linked by one famous rule, the Pythagorean theorem. If you know any two sides, the rule gives you the third. It tells you how long a ladder must be, how big a TV screen is, and how much walking you save by cutting across a field. It also works backward: it can tell you whether a corner is truly square. This lesson shows the rule, why it is true, how to use it, and where it stops working.

The parts of a right triangle

The two sides that meet at the right angle are the legs. The side across from the right angle, the one that does not touch it, is the hypotenuse. To find it, stand in the right-angle corner and look straight across. The side you are facing is the hypotenuse.

The Pythagorean theorem. If a right triangle has legs of length a and b and a hypotenuse of length c, then

a2 + b2 = c2.

In words: (leg)² + (leg)² = (hypotenuse)². The letters are only names. What matters is that the hypotenuse stands alone on its own side of the equation. If a book calls the hypotenuse z, the rule reads x2 + y2 = z2.

Because c2 = a2 + b2 is larger than a2 and larger than b2 (side lengths are positive), the hypotenuse is longer than each leg. It is the longest side of a right triangle. Keep all lengths in the same unit, and the answer comes out in that unit too.

Why it works: squares on the sides

Draw a square on each side of a right triangle, using that side as one edge of the square. The rule says the areas of the two small squares add up to the area of the big one. In the first picture the legs are 3 and 4 and the hypotenuse is 5, so the areas are 32 = 9, 42 = 16 and 52 = 25, and 9 + 16 = 25.

One triangle is only an example. Here is why it works for every right triangle. Make four copies of a right triangle with legs a and b and hypotenuse c. Fit them inside a big square of side a + b in two different ways (the second picture):

  • Arrangement 1. Put a triangle in each corner so that their hypotenuses form a tilted square in the middle. Its side is c, so its area is c2.
  • Arrangement 2. Pair the triangles into two rectangles. That leaves one square of side a and one square of side b, with areas a2 and b2.

Both arrangements fill the same big square with the same four triangles. Each triangle has area ab/2, so the four have area 4 × ab/2 = 2ab. The big square's area is therefore 2ab + c2 in Arrangement 1 and 2ab + a2 + b2 in Arrangement 2. Take away the same 2ab and what is left must match: c2 = a2 + b2.

Why is the tilted shape really a square? Its four sides all have length c. At each corner of the tilted shape, three angles sit along a straight line (180°): an acute angle from one triangle, the corner of the shape, and the other acute angle from the neighboring triangle. Because all four triangles are copies, those two acute angles are the two different acute angles of the same right triangle. The two acute angles of a right triangle add to 90°, because the three angles of any triangle add to 180° and one of them is 90°. So each corner of the tilted shape measures 180° − 90° = 90°. A shape with four equal sides and four right angles is a square.

The same idea works with algebra: the big square has area (a + b)2 = a2 + 2ab + b2, and Arrangement 1 says that area is 2ab + c2. Subtracting 2ab gives a2 + b2 = c2. The picture needs positive a and b, and it works for any right triangle, not only 3, 4, 5.

Finding the hypotenuse

When both legs are known, the hypotenuse is the unknown side. Follow four steps:

  1. Spot the right angle and check that the unknown side is the one across from it.
  2. Square each leg.
  3. Combine by adding the two squares. The sum is c2.
  4. Root: take the positive square root. The side length is c = √(a2 + b2).

For legs 3 and 4: 32 + 42 = 9 + 16 = 25, so c = √25 = 5.

Step 4 is the one people forget. The sum of the squares is c2, not c. Then check that your answer is longer than each leg and shorter than the sum of the legs: walking along both legs is always farther than going straight across.

Finding a missing leg

Sometimes you know the hypotenuse and one leg, and the unknown is the other leg. The equation is the same, but now a leg is the unknown. Start from a2 + b2 = c2, where c is the hypotenuse you know and b is the leg you know, and subtract b2 from both sides:

a2 = c2 − b2, so a = √(c2 − b2).

In words: big square minus small square. The hypotenuse's square is the total, and the two leg squares are its parts, so a missing part is the total minus the known part. The classic mistake is adding out of habit. A quick check: a leg must be shorter than the hypotenuse. If your "leg" came out longer, you added when you should have subtracted.

The mix-up usually starts with labels. Do not call the unknown side c automatically. Ask which side is across from the right angle, and put that side alone on its own side of the equation.

Exact answers and decimals

Most right triangles do not have whole-number sides. When c2 = 65, the side length is √65, the positive number whose square is 65. For a length we use only the positive root. There are two good ways to give an answer:

  • Exact: a simplified radical such as √65 or 6√5. Nothing has been rounded.
  • Approximate: a decimal such as 8.06. Write ≈ (approximately equal), never =, because the decimal digits of √65 never end.

Simplifying a radical. Find the largest perfect square that divides the number under the root, and pull its root out: √180 = √(36 × 5) = 6√5. This uses the fact that √(mn) = √m × √n when m and n are 0 or more. A radical is simplified when no perfect square other than 1 divides the number under the root. Handy perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100.

Round only at the end. Keep the exact radical (or at least four decimal places) until the last step. Suppose the legs of one right triangle are 5 and 8, and its hypotenuse is a leg of a second right triangle whose other leg is 6. The first hypotenuse is √89 ≈ 9.43. Squaring a square root undoes it, so the second hypotenuse is exactly √(89 + 36) = √125 = 5√5 ≈ 11.18. If you round the first hypotenuse to 9.4 and square that, you get 88.36 instead of 89, and the second hypotenuse comes out as √124.36 ≈ 11.15. That is not 11.18.

Sanity check. Find the two whole numbers the root sits between. Since 82 = 64 and 92 = 81, the number √65 is between 8 and 9.

The converse: is it a right triangle?

The theorem starts with a right triangle and tells you about its sides. Its converse goes the other way: start with three side lengths and ask whether they make a right triangle. (A converse is not automatically true, but this one is.)

Converse of the Pythagorean theorem. If the side lengths of a triangle satisfy a2 + b2 = c2, where c is the longest side, then the triangle is a right triangle, and the right angle is across from the side of length c.

Why it is true: draw a right triangle with legs a and b. By the theorem, its hypotenuse h has h2 = a2 + b2 = c2, so h = c. That triangle and the triangle you were given have the same three side lengths, and two triangles with the same three side lengths have the same shape and size (the SSS idea from the lesson on triangle congruence). So your triangle has a right angle too.

To test three sides:

  1. Find the longest side. It plays the role of c, whatever order the numbers were listed in.
  2. Square the two shorter sides and add them.
  3. Square the longest side.
  4. Compare. If the two results are exactly equal, the triangle is a right triangle. If they are not equal, it is not a right triangle.

Any three positive numbers that pass the test do form a triangle, because c2 = a2 + b2 is smaller than (a + b)2, which gives c < a + b. A near miss is still a miss. Builders use this idea to check that a corner is square.

Pythagorean triples

A Pythagorean triple is a set of three positive whole numbers a, b, c with a2 + b2 = c2. A whole-number right triangle is handy because no square root is left over. The three most common triples:

  • 3, 4, 5, since 9 + 16 = 25
  • 5, 12, 13, since 25 + 144 = 169
  • 8, 15, 17, since 64 + 225 = 289

Multiples. Multiply all three numbers of a triple by the same positive whole number k and you get another triple, because (ka)2 + (kb)2 = k2(a2 + b2) = k2c2 = (kc)2. Doubling 3, 4, 5 gives 6, 8, 10. So if two sides of a right triangle are the same multiple of two numbers from one triple, the third side is that multiple of the third number. This can save a calculation.

Two cautions. First, the numbers must sit in the right places: legs 5 and 12 have hypotenuse 13, but legs 5 and 13 give √194, which is not a whole number. Second, most right triangles are not triples at all. Legs 1 and 2 give a hypotenuse of √5. Triples are the easy-to-spot exceptions.

Word problems

Many word problems hide a right triangle. Use this routine:

  1. Draw a picture and mark the right angle.
  2. Label the sides with the numbers you know and a letter for the unknown. Decide which side is the hypotenuse: the one across from the right angle.
  3. Write the equation and solve it.
  4. Check the answer against the picture and give the units.

Pictures that come up again and again:

  • A ladder on a wall. The wall is vertical and the ground is level, so they meet at a right angle. The ladder is the hypotenuse, the height it reaches is one leg, and the distance from the wall to the foot of the ladder is the other leg.
  • The diagonal of a rectangle or a screen. The sides of a rectangle meet at right angles, so a diagonal is the hypotenuse of a triangle whose legs are the width and the height. TVs and phones are named by this diagonal.
  • The shortest path. Walking along two sides of a rectangle covers a + b. Cutting straight across covers the hypotenuse c, which is shorter, because c2 = a2 + b2 is less than (a + b)2 = a2 + 2ab + b2.
  • The diagonal of a box. Use the theorem twice. For a box 3 by 4 by 12, the diagonal of the 3-by-4 floor is √(32 + 42) = 5. That floor diagonal and the 12-unit vertical edge at the end of that floor diagonal are the legs of a second right triangle, because the edge stands straight up from the floor. Its hypotenuse is the diagonal of the box: √(52 + 122) = 13.

Watch what each number measures. The length of the ladder is the hypotenuse, but the height of its top is a leg.

Where the theorem stops

The theorem is only for right triangles. In a triangle with no right angle, a2 + b2 = c2 is not true: an equilateral triangle with all sides 2 has a2 + b2 = 8 but c2 = 4. Other triangles need other tools. Later lessons cover special right triangles and the ratios called sine, cosine and tangent. On a coordinate grid, the same idea gives the distance formula, which appears in the lesson on midpoint and distance.

Key words

Right triangle
A triangle with one angle of exactly 90°.
Leg
One of the two sides of a right triangle that meet at the right angle.
Hypotenuse
The side across from the right angle. It is the longest side of a right triangle.
Pythagorean theorem
In a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2.
Converse
A statement with its “if” and “then” parts swapped. The converse of the Pythagorean theorem says that if a2 + b2 = c2, with c the longest side, then the triangle is a right triangle.
Pythagorean triple
Three positive whole numbers a, b, c with a2 + b2 = c2, such as 3, 4, 5.
Simplified radical
A square root with no perfect square other than 1 dividing the number under the root. 6√5 is simplified; √180 is not.
Perfect square
The square of a whole number, such as 49, which is 72.

Worked examples

Example 1. Find the hypotenuse of a right triangle with legs 4 and 7. Give the exact answer and a decimal rounded to hundredths.

  1. The legs meet at the right angle, so the unknown side across from it is the hypotenuse, c.
  2. Write the theorem with the numbers: 42 + 72 = c2.
  3. Square each leg and add: 16 + 49 = 65, so c2 = 65.
  4. Take the positive square root: c = √65. Since 65 = 5 × 13 has no perfect-square factor, √65 is already simplified.
  5. Approximate: 82 = 64 and 92 = 81, so √65 is a little more than 8. A calculator gives √65 ≈ 8.06.
  6. Check: 8.06 is longer than each leg (4 and 7) and shorter than 4 + 7 = 11. Also 8.062 ≈ 64.96, close to 65.

Answer: Answer: c = √65 ≈ 8.06

Example 2. Find the hypotenuse of a right triangle with legs 6 and 12. Simplify the radical.

  1. The unknown side is across from the right angle, so it is the hypotenuse: 62 + 122 = c2.
  2. 36 + 144 = 180, so c2 = 180.
  3. Find the largest perfect-square factor of 180. Since 180 = 36 × 5 and 36 is a perfect square, use 36.
  4. c = √180 = √36 × √5 = 6√5.
  5. Decimal: √5 ≈ 2.2361, so 6√5 ≈ 13.416, which rounds to 13.42. (Rounding √5 to 2.2 first would give 13.2, which is off.)
  6. Check: 13.42 is longer than 12 and shorter than 6 + 12 = 18.

Answer: Answer: c = 6√5 ≈ 13.42

Example 3. A right triangle has hypotenuse 11 and one leg 5. Find the other leg.

  1. The hypotenuse is given: it is 11. So the equation is 52 + b2 = 112, with the hypotenuse alone on its side.
  2. A missing leg means big square minus small square: b2 = 112 − 52 = 121 − 25 = 96.
  3. Find the largest perfect-square factor of 96. Since 96 = 16 × 6, b = √96 = √16 × √6 = 4√6.
  4. Decimal: √6 ≈ 2.4495, so 4√6 ≈ 9.798, which rounds to 9.80.
  5. Check: a leg must be shorter than the hypotenuse, and 9.80 < 11. Also 52 + 96 = 25 + 96 = 121 = 112.

Answer: Answer: b = 4√6 ≈ 9.80

Example 4. Find the hypotenuse of a right triangle with legs 10 and 24 using a triple, then confirm it.

  1. Look for a pattern. 10 = 2 × 5 and 24 = 2 × 12, so the legs are the 5, 12, 13 triple doubled.
  2. The hypotenuse should then be 2 × 13 = 26.
  3. Confirm with the theorem: 102 + 242 = 100 + 576 = 676, and 262 = 676.
  4. The shortcut works only when the legs really are a multiple of two numbers in a triple. For legs 10 and 23, there is no pattern: 102 + 232 = 629, which is not a perfect square, so you must use a root.

Answer: Answer: c = 26

Example 5. A triangle has side lengths 29, 21 and 20. Is it a right triangle?

  1. Use the converse. The longest side, 29, plays the role of c.
  2. Square the two shorter sides and add: 202 + 212 = 400 + 441 = 841.
  3. Square the longest side: 292 = 841.
  4. The two results are equal, so the triangle is a right triangle. Its right angle is across from the side of length 29.

Answer: Answer: Yes. It is a right triangle, with the right angle across from the side of length 29.

Example 6. Is a triangle with side lengths 10, 11 and 15 a right triangle?

  1. The longest side, 15, plays the role of c.
  2. Add the squares of the two shorter sides: 102 + 112 = 100 + 121 = 221.
  3. Square the longest side: 152 = 225.
  4. Since 221 ≠ 225, the triangle is not a right triangle, even though the numbers are close.
  5. If you had put 11 in the c spot, you would compare 102 + 152 = 325 with 112 = 121, which tests the wrong thing. The longest side always goes alone.

Answer: Answer: No. 102 + 112 = 221, but 152 = 225.

Example 7. A 15-foot ladder leans against a vertical wall. Its foot is 4 feet from the wall on level ground. How high up the wall does the ladder reach? Round to hundredths.

  1. The wall and the ground meet at a right angle. The ladder is across from it, so the ladder is the hypotenuse: c = 15. The 4 feet along the ground is one leg, and the height h is the other leg.
  2. Write the equation with the hypotenuse alone: 42 + h2 = 152.
  3. A missing leg means subtract: h2 = 225 − 16 = 209.
  4. h = √209. Since 209 = 11 × 19 has no perfect-square factor, this is simplified.
  5. 142 = 196 and 152 = 225, so √209 is between 14 and 15. A calculator gives √209 ≈ 14.457, so h ≈ 14.46 feet.
  6. Check: the height 14.46 is less than the ladder length 15, and 42 + 14.462 ≈ 16 + 209.09 = 225.09, which is close to 225.

Answer: Answer: h = √209 ≈ 14.46 feet

Example 8. A TV screen is 48 inches wide and 27 inches tall. TVs are named by their diagonal. Find the diagonal.

  1. The width and height meet at a corner of the screen, which is a right angle, so they are the legs. The diagonal is the hypotenuse d.
  2. 482 + 272 = d2.
  3. 2304 + 729 = 3033, so d2 = 3033.
  4. Simplify: 3033 = 9 × 337, and 337 has no perfect-square factor, so d = √3033 = 3√337.
  5. Decimal: 552 = 3025, so d is just over 55. A calculator gives d ≈ 55.07 inches.
  6. Check: 55.07 is longer than 48 and shorter than 48 + 27 = 75. A store would call this a 55-inch TV.

Answer: Answer: d = 3√337 ≈ 55.07 inches

Example 9. A park is a rectangle 50 meters wide and 70 meters long. Maya walks from one corner to the opposite corner along the diagonal. How much shorter is that than walking along two sides?

  1. Along two sides: 50 + 70 = 120 meters.
  2. The diagonal is the hypotenuse of a right triangle with legs 50 and 70: d2 = 502 + 702 = 2500 + 4900 = 7400.
  3. 7400 = 100 × 74, so d = √7400 = 10√74. Since 74 = 2 × 37, the radical is simplified.
  4. Keep extra decimals until the end: √74 ≈ 8.6023, so d ≈ 10 × 8.6023 = 86.023, which is 86.02 meters.
  5. The saving is 120 − 86.02 = 33.98 meters.
  6. Check: 86.02 is longer than the 70-meter side and shorter than 120 meters.

Answer: Answer: The diagonal is 10√74 ≈ 86.02 meters, about 33.98 meters shorter than walking along two sides.

Example 10. A right triangle has legs 9 and x. Its hypotenuse is 1 unit longer than the leg x, so it is x + 1. Find x.

  1. The hypotenuse is alone on one side: 92 + x2 = (x + 1)2.
  2. Expand the right side. Square both terms and keep the middle term: (x + 1)2 = x2 + 2x + 1. It is not x2 + 1.
  3. 81 + x2 = x2 + 2x + 1.
  4. Subtract x2 from both sides: 81 = 2x + 1.
  5. Subtract 1 from both sides: 80 = 2x. Divide by 2: x = 40.
  6. The sides are 9, 40 and 41. Check: 81 + 1600 = 1681 = 412. (This is another Pythagorean triple.)

Answer: Answer: x = 40, so the sides are 9, 40 and 41.

In plain English

A right triangle has one square corner, like the corner of a sheet of paper.

The two sides that touch the square corner are called legs. The side across from the corner is called the hypotenuse. It is always the longest side.

Here is the big idea. Draw a square on each side of the triangle. Add the areas of the two small squares. You get the area of the big square.

Try it on a triangle with legs 3 and 4. The small squares have areas 9 and 16. The big square has area 25, because 9 + 16 = 25. So the long side is 5, since 5 × 5 = 25.

In math symbols, the rule is a2 + b2 = c2. To find the long side, square the legs, add, and take the square root. To find a leg, subtract instead of adding.

The rule also works backward. If the sides of a triangle pass the test, the triangle has a square corner. Builders use this to check corners.

Be careful. The rule works only for triangles that have a square corner.

See it

A square on each side of a right triangle leg a = 3 a² = 9 leg b = 4 b² = 16 hypotenuse c = 5 c² = 25 Add the two small areas square on leg a: 9 square on leg b: 16 square on hypotenuse c: 25 9 + 16 = 25 a² + b² = c² Legs a and b meet at the right angle. Right triangles only. Drawn to scale. The little square marks the right angle; the hypotenuse is across from it.
The squares on the legs have areas 32 = 9 and 42 = 16. The square on the hypotenuse has area 52 = 25, and 9 + 16 = 25.
Two ways to fill a square of side a + b Arrangement 1 a b b a c² 4 triangles + 1 tilted square area = 2ab + c² Arrangement 2 b a b a b² a² 4 triangles + 2 squares area = 2ab + a² + b² = same area Take away the same 4 triangles: c² = a² + b²
Same big square, same four triangles: 2ab + c2 = 2ab + a2 + b2, so c2 = a2 + b2. Drawn with a = 3 and b = 4, but the argument works for any right triangle.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Sports

Cutting across the field

A coach marks a rectangular practice field that is 30 yards long and 20 yards wide. Two players start at the same corner and race to the opposite corner. One runs along the two sides, which is 30 + 20 = 50 yards. The other runs straight across the field, which is √(302 + 202) = √1300 = 10√13 ≈ 36.06 yards.

How it connects: The two sides are the legs and the straight run is the hypotenuse, so the straight run is c = √(a2 + b2), not a + b. It is shorter because c2 = a2 + b2 is less than (a + b)2. The limit: the corner of a rectangle is a right angle. If the field is not a rectangle, or something blocks the way, this formula does not apply.

Food

Square pizzas

A pizza shop sells square pizzas cut into square slices. A 3-by-3 pizza has 9 slices, a 4-by-4 pizza has 16 slices, and a 5-by-5 pizza has 25 slices. Two friends order a 3-by-3 and a 4-by-4 pizza. Together they get 9 + 16 = 25 slices, exactly as many as one 5-by-5 pizza.

How it connects: The slice counts are the areas of squares with sides 3, 4 and 5, which are the squares on the sides of a right triangle: 32 + 42 = 52. The limit: this works only because 3, 4, 5 are the sides of a right triangle. Sides do not add: a 3-by-3 and a 4-by-4 pizza do not make a 7-by-7 pizza (49 slices). The areas add, and the square with area 25 has side 5. One example does not prove the rule for every triangle; the two-arrangement picture does that.

Proverb

Measure twice, cut once

This saying means: check your measurements carefully before doing something you cannot undo, such as cutting a board.

How it connects: A builder can check that a corner is square with the converse. Mark 3 feet along one edge and 4 feet along the other. If the straight-line distance between the two marks is exactly 5 feet, then 32 + 42 = 52, so the corner is a right angle. Measuring that third side is the second measurement. The limit: the check is only as good as the measuring. A larger multiple such as 6, 8, 10 also works, and a small slip in measuring matters less.

Idiom

As the crow flies

This idiom means the straight-line distance between two places, ignoring roads and obstacles. Suppose Town B is 7 miles east and 10 miles north of Town A, and the roads run only east-west and north-south, so driving is 7 + 10 = 17 miles. As the crow flies, the distance is √(72 + 102) = √149 ≈ 12.21 miles.

How it connects: The two road segments are the legs and the crow’s path is the hypotenuse: c = √(a2 + b2). The limit: the formula needs the two road directions to meet at a right angle on flat ground. Over very long distances the curve of the Earth matters, and a flat right triangle is no longer exact.

Gaming

The tower rope

In a made-up building game, a player ties a rope from the top of a tower that is 6 blocks tall to a stake in flat ground 13 blocks from the base of the tower. The rope is pulled tight, so how long must it be? The rope length is √(62 + 132) = √205 ≈ 14.32 blocks.

How it connects: The tower and the ground meet at a right angle, so the rope is the hypotenuse: 62 + 132 = c2. A reasonableness check: 14.32 is longer than 13 and shorter than 6 + 13 = 19. The limit: a loose rope would sag and need more length. The formula gives the length of a tight, straight rope.

Everyday joke

Always right

Why do right triangles win every argument? Because they are always right.

How it connects: The pun uses two meanings of “right”. A right angle is exactly 90°, and the Pythagorean theorem is true for every right triangle. The limit: a triangle without a right angle is not “wrong”; the theorem just does not apply to it. And a right angle does not make anyone’s argument correct.

Memory help

Acronym

Spot, Square, Combine, Root

For a missing side, say Spot, Square, Combine, Root. Spot the hypotenuse (across from the right angle). Square the two sides you know. Combine: add the squares if the hypotenuse is the unknown, or subtract the smaller square from the larger if a leg is the unknown. Root: take the positive square root. Say it as “Silly Seals Climb Rocks.”

Rhyme

Add for long, subtract for short

Add for the long side, subtract for the short, and the root at the end is the last support. Here “long side” means the hypotenuse and “short side” means a leg, since a leg is always shorter than the hypotenuse.

Picture it

Look across the corner

Stand in the right-angle corner and look straight across. The side you are facing is the hypotenuse. The two sides you could touch with your hands are the legs.

Story

The triple family

3, 4, 5 is the parent: doubled it becomes 6, 8, 10. For 5, 12, 13 think “five fingers, a dozen eggs, a baker’s dozen (13).” In 3, 4, 5 and 5, 12, 13 the hypotenuse is 1 more than the longer leg, and in 8, 15, 17 it is 2 more. That pattern is just a memory help for these three; it does not hold for every triple.

Trap alert

Not c, but c squared

For legs 3 and 4, 32 + 42 = 25. The 25 is c2, not c. Stop there and you are only halfway: c = √25 = 5. If an answer is longer than the sum of the legs, check that the root was taken.

Trap alert

The longest side goes alone

In the test a2 + b2 = c2, the c is the longest side. Squares of the two shorter sides go together on one side of the equals sign, and the square of the longest side goes by itself on the other.

Sketch it

Draw the three squares

Sketch a right triangle and put a square on each side. Label the areas a2, b2 and c2. The biggest square is always on the side across from the right angle, and the two smaller squares together fill the same area.

Study notes

Key ideas

  • a2 + b2 = c2 for a right triangle with legs a and b and hypotenuse c. The hypotenuse is across from the right angle and is the longest side.
  • Hypotenuse unknown: add the squares of the legs, then take the square root. A leg unknown: subtract (big square minus small square), then take the square root.
  • Give exact answers as simplified radicals and decimals with ≈. Keep the exact value, or extra decimals, until the last step, and round once at the end.
  • The converse: if three side lengths satisfy a2 + b2 = c2 with c the longest side, the triangle is a right triangle. If they do not, it is not a right triangle.
  • Pythagorean triples such as 3-4-5, 5-12-13 and 8-15-17, and their multiples, give whole-number sides.
  • The theorem is for right triangles only.

Steps

  1. Draw the triangle, mark the right angle, and find the hypotenuse: the side across from it.
  2. Write a2 + b2 = c2 with the numbers you know and a letter for the unknown side. Keep the hypotenuse alone on its own side.
  3. Solve for the square of the unknown side: add the squares if the hypotenuse is unknown, subtract if a leg is unknown.
  4. Take the positive square root. Simplify the radical, and give a decimal with ≈ if one is asked for.
  5. Check that the hypotenuse is longer than each leg and shorter than the sum of the legs. Give the units.

Formulas

  • Pythagorean theorem: a2 + b2 = c2
  • Hypotenuse: c = √(a2 + b2)
  • Missing leg: b = √(c2 − a2)
  • Converse test: a2 + b2 = c2, with c the longest side, means a right triangle
  • Multiples of a triple: (ka)2 + (kb)2 = (kc)2

Watch out

  • Only the side across from the right angle is the hypotenuse. Do not call a leg the hypotenuse, and do not assume the unknown side is always c.
  • Square first, then add: for legs 3 and 4, 32 + 42 = 25, which is not (3 + 4)2 = 49 and not 3 + 4 = 7.
  • a2 + b2 gives c2, not c. Take the square root.
  • A missing leg means subtract. Add only when the hypotenuse is the unknown.
  • Do not round until the end, and write ≈ for rounded values.
  • Do not use the theorem on a triangle with no right angle. In the converse test, put the longest side in the c spot.

Why it works

  • The squares on the sides of a right triangle have areas a2, b2 and c2. Filling a big square in two ways shows that the two small areas always add up to the big one.
  • It turns lengths that are hard to measure (a diagonal, the height of a ladder, a straight-line shortcut) into lengths you can measure along the right angle.
  • Other lessons build on it: special right triangles and trigonometry come next, and the distance formula on a coordinate grid is the same relationship.

Common mistakes

✗ MistakeThe unknown side is always c. With hypotenuse 20 and one leg 12, the other leg is √(122 + 202) = √544.
✓ InsteadOnly the side across from the right angle is c. Here 20 is the hypotenuse, so a2 + 122 = 202 and a2 = 400 − 144 = 256, which gives a = 16.
✗ MistakeLegs 8 and 14 give a hypotenuse of 8 + 14 = 22 (adding instead of squaring).
✓ InsteadSquare first: 82 + 142 = 64 + 196 = 260, so c = √260 = 2√65 ≈ 16.12. The hypotenuse is always shorter than the sum of the legs.
✗ MistakeLegs 11 and 3 give 112 + 32 = 130, so the hypotenuse is 130.
✓ Instead130 is c2. The hypotenuse is c = √130 ≈ 11.40, only a little longer than the leg 11.
✗ MistakeFor legs 10 and 40, √1700 = 10√17 and √17 ≈ 4.1, so the hypotenuse is 10 × 4.1 = 41.0.
✓ InsteadRounding √17 to 4.1 before multiplying loses accuracy. Keep 10√17, or use √17 ≈ 4.1231, until the end: 10√17 ≈ 41.23.
✗ MistakeA triangle has sides 8 and 9 and no right angle, so its third side is √(82 + 92) = √145.
✓ InsteadThe theorem needs a right angle. Without one, sides 8 and 9 can have many different third sides, depending on the angle between them. So √145 means nothing here.
✗ MistakeTo test sides 24, 32 and 40, I compared 242 + 402 = 2176 with 322 = 1024, so it is not a right triangle.
✓ InsteadThe longest side must be c. Compare 242 + 322 = 576 + 1024 = 1600 with 402 = 1600. They match, so it is a right triangle.
✗ MistakeThe equation is always a2 + b2 = c2 with those exact letters, even if the hypotenuse of my triangle is named a.
✓ InsteadThe letter c is only a name for the hypotenuse. If the hypotenuse is called a, the equation is b2 + c2 = a2. Always keep the hypotenuse alone on its own side.

Teach it back

Explain to a friend (or a pet!) how to find the missing side of a right triangle, and how to test whether three side lengths make a right triangle. Make up your own numbers, and say why the longest side goes by itself in the test.

Explaining it out loud shows what you really know. Did you…

  • I said that the hypotenuse is across from the right angle and is the longest side.
  • I showed adding the squares to find a hypotenuse and subtracting to find a leg, and I took the square root at the end.
  • I gave an exact answer (a radical) and a decimal with ≈, and I rounded only at the end.
  • For the test, I used the longest side as c and compared a2 + b2 with c2.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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Order of OperationsSolving Two-Step EquationsSpecial Products of BinomialsAngle Relationships