8.G.7: Use the Pythagorean Theorem
Also written as 8.G.B.7 and CCSS.MATH.CONTENT.8.G.B.7 · Cluster: Understand and apply the Pythagorean Theorem.
The Official Wording
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Common Core State Standards, Grade 8, Geometry · Read it at the source · Checked October 14, 2026
What 8.G.7 Means in Plain English
Eighth graders use a² + b² = c² to find a missing side in a real setting. A 13-foot ladder with its foot 5 feet from a wall reaches 12 feet up, because 13² − 5² = 144 = 12². Some problems want a leg, some the hypotenuse.
In three dimensions they use the theorem twice: across the floor of a box, then up to the far top corner. The same right-triangle habit leads to distance on a grid in 8.G.8 and, in high school, to solving right triangles with trigonometry (HSG-SRT.8).
8.G.7 I Can Statements
- I can decide whether a problem asks for a leg or the hypotenuse.
- I can use the Pythagorean theorem to find a missing side length.
- I can find the diagonal of a box by using the theorem twice.
8.G.7 Worked Example
A box problem that needs the theorem twice, once across the floor and once up to the far corner.
A shipping box is 12 in long, 9 in wide and 8 in tall. Will a 16-inch rod fit inside if it lies from one bottom corner to the opposite top corner?
- The rod runs corner to corner through the middle of the box. First find the diagonal across the floor, the hypotenuse of a right triangle with legs 12 and 9.d² = 12² + 9² = 144 + 81 = 225
- Take the square root to get the floor diagonal.d = √225 = 15 in
- Now stand a second right triangle on that diagonal. One leg is the floor diagonal, the other is the height, and the rod is the hypotenuse.r² = 15² + 8² = 225 + 64 = 289
- Square root again, then compare with the rod.r = √289 = 17 in, and 16 < 17
Check it: Do it in one line: 12² + 9² + 8² = 144 + 81 + 64 = 289 = 17². And 17 is longer than every edge, as a diagonal through a box must be.
Common Mistakes with 8.G.7
Adding when the missing side is a leg
Ladder 13 ft, foot 5 ft from the wall: 13² + 5² = 194, so about 13.9 ft up
Try this: Find the hypotenuse first: the side across from the right angle. If it is given, subtract the squares instead of adding them.
Forgetting the square root
12² + 9² = 225, so the floor diagonal is 225 inches
Try this: Ask whether the answer is a sensible length. The diagonal of a 12 by 9 floor cannot be 225, so take √225 = 15.
Pairing the height with an edge in 3D
Rod length found as √(12² + 8²), about 14.4 in, from the length and height alone
Try this: Sketch the triangle the rod actually makes. Its legs are the height and the floor diagonal, not the height and one edge.
Where 8.G.7 Fits
Practice 8.G.7
Quick Check
- A TV screen is 24 inches wide and 18 inches tall. How long is its diagonal?
- A ramp is 10 m long and rises 6 m. How far along the ground does it reach?
- A box is 3 ft by 4 ft by 12 ft tall. How long is the diagonal from one bottom corner to the opposite top corner?