8.G.6: Why the Pythagorean Theorem Works
Also written as 8.G.B.6 and CCSS.MATH.CONTENT.8.G.B.6 · Cluster: Understand and apply the Pythagorean Theorem.
The Official Wording
Explain a proof of the Pythagorean Theorem and its converse.
Common Core State Standards, Grade 8, Geometry · Read it at the source · Checked October 14, 2026
What 8.G.6 Means in Plain English
Eighth graders explain why the Pythagorean theorem is true, usually with a rearrangement proof. Four copies of a 3-4-5 triangle inside a 7 by 7 square leave a tilted square of area 49 − 24 = 25, which is 3² + 4²: the hypotenuse squared.
They also argue the converse: sides 5, 12 and 13 fit 25 + 144 = 169, so the angle between 5 and 12 must be right. That reasoning feeds 8.G.7 and 8.G.8, and high school proves the theorem again with similar triangles (HSG-SRT.4).
8.G.6 I Can Statements
- I can explain a proof that a² + b² = c² in right triangles.
- I can find one area two ways and set them equal.
- I can use the converse to decide whether a triangle is right.
8.G.6 Worked Example
The proof most classes meet first: one square, cut up two ways, with the same area counted twice.
Four copies of a right triangle with legs a and b and hypotenuse c are placed inside a square of side a + b, leaving a tilted square in the middle. Explain why a² + b² = c², and why the converse holds.
- Each side of the big square is one long leg plus one short leg, so its area can be written two ways. First, as one square.Area = (a + b)²
- Second, as its pieces: four triangles, each half of an a by b rectangle, plus the tilted square. Its side is c, and each corner is 90° because the two acute angles beside it on the straight edge already make 90°.Area = 4 × ½ab + c² = 2ab + c²
- Expand the first expression: an a by a square, a b by b square and two a by b rectangles.(a + b)² = a² + 2ab + b²
- Both expressions describe the same square, so set them equal and take 2ab from each side.a² + 2ab + b² = 2ab + c², so a² + b² = c²
- The converse runs backwards. If sides a, b and c fit a² + b² = c², build a right triangle with legs a and b. Its hypotenuse is c as well, so the two triangles match side for side, and the original angle is 90° too.a² + b² = c², so the angle across from c is 90°
Check it: Test it with legs 6 and 8: the outer square is 14 × 14 = 196, the four triangles are 4 × 24 = 96, and 196 − 96 = 100 = 10². And 6² + 8² = 36 + 64 = 100.
Common Mistakes with 8.G.6
Calling examples a proof
“3, 4, 5 works and 6, 8, 10 works, so it’s always true.”
Try this: Separate checking from proving: examples show the rule holds for those triangles, while the area argument with a and b covers every right triangle.
Mixing up the theorem and its converse
“The angle is 90°, so a² + b² = c²” offered as the reason a triangle is right
Try this: Write both as if-then sentences side by side. The theorem starts from a right angle, and the converse starts from the side lengths.
Testing the converse with the wrong side as c
Sides 5, 12, 13 tested as 12² + 13² = 5²
Try this: Put the longest side in the c spot every time. Only the longest side can sit across from a right angle.
Where 8.G.6 Fits
Practice 8.G.6
Quick Check
- Do sides of 8, 15 and 17 make a right triangle?
- Do sides of 6, 7 and 9 make a right triangle?
- In the four-triangle proof with legs 5 and 12, what are the outer square’s area and the tilted square’s area?