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The Pythagorean Theorem

Use a² + b² = c² to find a missing side of a right triangle, solve real-world problems, and find distances on a grid.

Practice this skill 35 min Pythagorean theorem Leer en español
Key idea

What you'll learn

The Pythagorean theorem connects the three sides of any right triangle. If you know two sides, you can always find the third. In this lesson you will:

  • recognize the legs and the hypotenuse of a right triangle,
  • find a missing hypotenuse (add the squares) or a missing leg (subtract the squares),
  • solve word problems with ladders, ramps, TV screens, roofs and walking shortcuts,
  • find the distance between two points on a coordinate grid, and
  • check whether three lengths make a right triangle.

Why it matters on the GED: The Pythagorean theorem is on the formula sheet, and it appears on almost every GED Math test — often as a word problem where you must draw the triangle yourself. It is also hidden inside other questions (the slant of a roof, the diagonal of a rectangle, the distance between points).

In real life: builders use it to make square corners and cut roof rafters; it tells you how tall a ladder can reach and what size a "55-inch" TV really is.

Words to know

  • Right triangle — a triangle with one 90^\circ (square) corner, marked with a small square.
  • Legs (a and b) — the two sides that form the right angle (the floor and the wall under a ladder).
  • Hypotenuse (c) — the side across from the right angle; always the longest side (the ladder itself).
  • Square of a number — the number times itself: 7^2 = 49.
  • Square root (\sqrt{\ }) — the number that, squared, gives you back the original: \sqrt{49} = 7.
  • Perfect square — a whole number whose square root is whole: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
  • Pythagorean triple — three whole numbers that fit a^2 + b^2 = c^2 (3-4-5, 5-12-13, 8-15-17, 7-24-25).
  • Diagonal — a segment joining opposite corners of a rectangle; it splits the rectangle into two right triangles.
  • Coordinate grid — the x-y plane where points are written (x, y).

All Math words in the glossary

Right triangle: the two sides that form the right angle are leg a and leg b; the longest side, across from the right angle, is the hypotenuse c leg b leg a hypotenuse c

The hypotenuse c is always across from the little square that marks the right angle. The two legs form the square corner.

The theorem and finding the hypotenuse

In every right triangle,

a^2 + b^2 = c^2

"The square of one leg plus the square of the other leg equals the square of the hypotenuse."

Finding the hypotenuse (both legs known): square, add, square-root.

  • Legs 6 and 8: c^2 = 6^2 + 8^2 = 36 + 64 = 100, so c = \sqrt{100} = 10.
A right triangle with legs 6 and 8 and hypotenuse 10, with a square drawn on each side. The green square on the 6 leg has area 36, the blue square on the 8 leg has area 64, and the red square on the hypotenuse has area 100: 36 + 64 = 100. 8² = 64 6² = 36 10² = 100 8 6 10 36 + 64 = 100, so c = √100 = 10
  • Legs 48 and 27 (a TV screen): c^2 = 2304 + 729 = 3033, so c = \sqrt{3033} \approx 55.1.

If the number under the square root is not a perfect square, use the calculator and round as the question asks.

Why not just add the legs? Because the shortcut across a corner is always shorter than going along both sides. 6 + 8 = 14, but the hypotenuse is only 10. The hypotenuse is longer than either leg but shorter than the two legs added together — a quick check for every answer.

Finding a missing leg

If you know the hypotenuse and one leg, the missing side is a leg. Put the numbers into the formula carefully — the hypotenuse always goes alone on the right side:

a^2 + b^2 = c^2 \quad\Rightarrow\quad a^2 = c^2 - b^2

Square, subtract, square-root.

  • Hypotenuse 13, leg 5: a^2 = 13^2 - 5^2 = 169 - 25 = 144, so a = 12.
  • Ladder 20 ft, base 6 ft from the wall: h^2 = 20^2 - 6^2 = 400 - 36 = 364, so h = \sqrt{364} \approx 19.1 ft.
A 20 ft ladder leaning against a wall with its base 6 ft from the wall. The wall and ground form the right angle; the ladder is the hypotenuse. The height h = √(400 − 36) = √364 ≈ 19.1 ft. h = ? 6 ft ladder 20 ft wall h² = 20² − 6² = 364, h = √364 ≈ 19.1 ft

How to tell which case you have: find the side across from the right angle. If that side is the unknown, add the squares. If it is known, subtract.

Pythagorean triples save time. Learn 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Multiples work too: 6-8-10, 9-12-15, 15-20-25, 30-40-50. If you see legs 9 and 12, you can predict the hypotenuse is 15.

Word problems and the coordinate grid

Word problems: draw the triangle. Look for a right angle in the story — a wall meeting the ground, north meeting east, the corner of a rectangle.

  • Ladder: wall and ground are the legs; the ladder is the hypotenuse.
  • Walking/driving north then east: the two directions are legs; the straight-line distance back is the hypotenuse. If there are several moves, first combine all east–west moves and all north–south moves.
  • Rectangles (screens, gates, fields): the length and width are legs; the diagonal is the hypotenuse. A "55-inch TV" is 55 in diagonally.
  • Isosceles roof or triangle: the height splits the base in half. Use half the base as a leg.

Distance between two points on a grid. Make a right triangle:

  1. Horizontal leg: the difference of the x-values.
  2. Vertical leg: the difference of the y-values.
  3. Use a^2 + b^2 = c^2.
  • From (-3, -2) to (3, 6): horizontal 3 - (-3) = 6; vertical 6 - (-2) = 8; distance = \sqrt{36 + 64} = 10.
Coordinate grid with points (−3, −2) and (3, 6). A right triangle is drawn: the horizontal leg is 3 − (−3) = 6, the vertical leg is 6 − (−2) = 8, and the distance between the points is the hypotenuse, √(36 + 64) = 10. x y -4 -2 2 4 2 4 (−3, −2) (3, 6) 6 8 10 distance = √(6² + 8²) = √100 = 10

Watch the signs: subtracting a negative means adding. Count grid squares to double-check.

Is it a right triangle? Test the lengths: if (shorter)² + (middle)² = (longest)², it is a right triangle. 5, 12, 13: 25 + 144 = 169 ✓. 4, 5, 6: 16 + 25 = 41 \ne 36 ✗.

Formula

Formula (on the GED formula sheet)

a^2 + b^2 = c^2 c = \sqrt{a^2 + b^2} \qquad a = \sqrt{c^2 - b^2}
  • a, b = the legs (the two sides that form the right angle).
  • c = the hypotenuse (across from the right angle; the longest side).

On the GED formula sheet: yes, a^2 + b^2 = c^2 is printed on the sheet. The sheet does not tell you which side is c — you must find the side across from the right angle — and it does not give a separate distance formula; on the GED you build a right triangle on the grid instead.

Worked example

Example 1 (easy): missing hypotenuse

A right triangle has legs of 5 cm and 12 cm. How long is the hypotenuse?

A right triangle with legs 5 cm and 12 cm drawn to scale. The hypotenuse c, across from the right angle, is unknown. 12 cm 5 cm c = ?
  1. c^2 = 5^2 + 12^2 = 25 + 144 = 169.
  2. c = \sqrt{169} = 13.

Answer: 13 cm

Check: 13 is longer than 12 but shorter than 5 + 12 = 17 ✓ (and 5-12-13 is a known triple).

Worked example

Example 2 (easy): missing leg

A right triangle has a hypotenuse of 10 in and one leg of 6 in. Find the other leg.

A right triangle with hypotenuse 10 in and one leg 6 in. The other leg b is unknown. b = ? 6 in 10 in (hypotenuse)
  1. The hypotenuse is known, so subtract: b^2 = 10^2 - 6^2 = 100 - 36 = 64.
  2. b = \sqrt{64} = 8.

Answer: 8 in

Check: 6^2 + 8^2 = 36 + 64 = 100 = 10^2 ✓.

Worked example

Example 3 (word problem): how high does the ladder reach?

A 16 ft ladder leans against a house with its base 5 ft from the wall. How high up the wall does it reach, to the nearest tenth of a foot?

A 16 ft ladder leaning against a house with its base 5 ft from the wall. The ladder is the hypotenuse and the ground distance is a leg; the height h up the wall is unknown. h = ? 5 ft ladder 16 ft wall
  1. The ladder is the hypotenuse (c = 16); the ground distance is a leg (b = 5); the height h is the other leg.
  2. h^2 = 16^2 - 5^2 = 256 - 25 = 231.
  3. h = \sqrt{231} \approx 15.199 \approx 15.2.

Answer: about 15.2 ft

Check: 15.2^2 + 5^2 = 231.04 + 25 = 256.04 \approx 256 = 16^2 ✓, and 15.2 < 16 as a leg must be ✓.

Worked example

Example 4 (grid): distance between two points

Find the distance between P(-2, 1) and Q(3, -3), to the nearest tenth.

Coordinate grid with P(−2, 1) and Q(3, −3). A right triangle has a horizontal leg of 5 and a vertical leg of 4; the distance d from P to Q is the hypotenuse. x y -4 -2 2 4 -4 -2 2 P(−2, 1) Q(3, −3) 5 4 d = ?
  1. Horizontal leg: 3 - (-2) = 5.
  2. Vertical leg: 1 - (-3) = 4.
  3. d^2 = 5^2 + 4^2 = 25 + 16 = 41.
  4. d = \sqrt{41} \approx 6.4.

Answer: about 6.4 units

Check: the distance must be more than the longer leg (5) and less than 5 + 4 = 9 ✓.

Worked example

Example 5 (GED-hard, work backwards): is the corner square?

A carpenter wants to check that the corner of a deck is a perfect right angle. She measures 6 ft along one edge and 8 ft along the other, then measures the diagonal between those two marks: 10 ft 2 in. Is the corner square? If not, what should the diagonal be?

  1. If the corner were a right angle, the diagonal would be the hypotenuse: c^2 = 6^2 + 8^2 = 36 + 64 = 100, so c = 10 ft exactly.
  2. Her measurement is 10 ft 2 in, which is longer than 10 ft.
  3. A diagonal that is too long means the angle is a bit wider than 90^\circ.
A deck corner with marks 6 ft along one edge and 8 ft along the other. For a square (90 degree) corner the blue diagonal between the marks must be exactly 10 ft, because 6² + 8² = 100. The measured diagonal of 10 ft 2 in is too long, so the corner is a little wider than 90 degrees. 8 ft 6 ft diagonal must be 10 ft measured: 10 ft 2 in (too long) → corner is wider than 90° 6² + 8² = 100, so a square corner has diagonal 10 ft

Answer: No — for a square corner the diagonal must be exactly 10 ft.

Check: 6-8-10 is the 3-4-5 triple doubled (3 \times 2, 4 \times 2, 5 \times 2) ✓. Builders call this the "3-4-5 rule."

Calculator tip

Calculator tip: TI-30XS MultiView

  • Squares: use the x² key: 16 x² → 256.
  • Square root: press 2nd then x² to get the √ template.
  • Hypotenuse in one line: 2nd x² then 48 x² + 27 x² then enter. Type the whole sum under the root. The screen may show an exact root (3\sqrt{337}); press toggle → 55.07267925.
  • Missing leg in one line: 2nd x² then 20 x² − 6 x² enter, then toggle if needed → 19.07878403.
  • Negative coordinates: use the (−) key for a negative number, e.g. 3 − (−) 2, or just remember that subtracting a negative is adding.

When not to use it: if the numbers are a Pythagorean triple (3-4-5, 5-12-13, 6-8-10…), you can answer in seconds without a calculator. Some easy versions appear in the no-calculator section.

Common mistake

Common mistakes

  • Adding the legs instead of their squares. Legs 6 and 8: the hypotenuse is \sqrt{36 + 64} = 10, not 6 + 8 = 14.
  • Forgetting the square root. c^2 = 100 means c = 10. An answer of 100 is c^2, not c.
  • Adding when you should subtract. Ladder 20 ft (hypotenuse), base 6 ft: the height is \sqrt{400 - 36} \approx 19.1, not \sqrt{400 + 36} \approx 20.9. A leg can never be longer than the hypotenuse.
  • Subtracting the lengths instead of the squares. 20 - 6 = 14 is not the height; you must square first.
  • Using the full base of an isosceles triangle. The height cuts the base in half: roof 24 ft wide, 5 ft tall → rafter \sqrt{12^2 + 5^2} = 13, not \sqrt{24^2 + 5^2}.
An isosceles roof 24 ft wide and 5 ft tall. The dashed height splits the base into two halves of 12 ft, making right triangles with legs 12 and 5, so each rafter is √(144 + 25) = 13 ft. 12 ft 12 ft 24 ft wide 5 ft rafter 13 ft rafter = √(12² + 5²) = √169 = 13 ft
  • Sign errors on the grid. From x = -3 to x = 3 is 3 - (-3) = 6 units, not 0.
Test strategy

GED test strategy

  • Draw and label. Sketch the right triangle, mark the right angle, and write c on the side across from it.
  • Ask "add or subtract?" Hypotenuse unknown → add squares. Leg unknown → subtract.
  • Reasonableness check: the hypotenuse is the longest side but shorter than the two legs added. Eliminate choices that break this.
  • Look for triples (3-4-5, 5-12-13, 8-15-17 and their multiples) before reaching for the calculator.
  • Plug answers back in: for a choice of 9 with hypotenuse 15 and leg 12, check 81 + 144 = 225 = 15^2 ✓.
  • Read the rounding instruction ("nearest tenth") and the units, and answer exactly what is asked (the difference, the new height, the total distance).

Check yourself

Pythagorean theoremNo calculatorHot spot

Click the hypotenuse of the right triangle.

Click on the figure to mark your answer (or focus it and use the arrow keys).

Right triangle with the right angle at the bottom left corner
Hint

Find the small square that marks the right angle. The hypotenuse does not touch that corner.

Check yourself

Pythagorean theoremNo calculator

Find the length of the hypotenuse c.

Right triangle with legs 6 m (vertical) and 8 m (horizontal); the hypotenuse c is unknown 8 m 6 m c
Answer choices
Hint

Square each leg, add, and then undo the squaring.

Summary

  • In a right triangle, a^2 + b^2 = c^2; c is the hypotenuse, across from the right angle (on the formula sheet).
  • Missing hypotenuse: square, add, square-root. Missing leg: square, subtract, square-root.
  • In word problems, find the right angle: wall + ground, north + east, length + width of a rectangle; halve the base of an isosceles triangle.
  • Grid distance: legs are the differences in x and in y; then use the theorem.
  • Know the triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples; check that c is the longest side.
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