Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Once the theorem is understood, it becomes a measuring tool for lengths that are hard to reach directly. A ladder leaning against a wall, the diagonal of a TV screen, the shortcut across a rectangular park and the height of a ramp are all hypotenuses or legs of right triangles hiding in plain sight. Students sketch the triangle, label what is known, and choose whether to add squares (to find the hypotenuse) or subtract them (to find a leg).
The three-dimensional part of the standard is where many students are stretched. To find the longest straight rod that fits in a box measuring 3 by 4 by 12, first find the diagonal across the base, √(3² + 4²) = 5, then use that diagonal as a leg of a second right triangle standing up inside the box: √(5² + 12²) = 13. Problems like this require spotting right angles in solids and using the theorem twice. Answers that are not whole numbers are rounded sensibly, often to the nearest tenth.
If the hypotenuse is known, students must subtract: leg² = c² - other leg². Adding gives a 'leg' longer than the hypotenuse, which is impossible.
Students stop at c² = 169 and report 169. The final step is c = √169 = 13.
The diagonal across one face of a box is shorter than the diagonal through the box. The space diagonal needs the height as well.
In context problems the right angle is often implied, such as a wall meeting the ground. Drawing the triangle first makes it visible.
A 13-foot ladder leans against a vertical wall. The foot of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?
Answer: The ladder reaches 12 feet up the wall.
Use the classroom itself: measure the length and width of the floor and ask students to predict the diagonal before checking with a tape measure, then extend to the diagonal from a floor corner to the opposite ceiling corner. Physical boxes and straws make the 3D case concrete.
Assessment items often hide the right triangle in a diagram or story, include a 3D prism, and ask for an answer to the nearest tenth. Teach a routine: sketch, label legs and hypotenuse, decide add or subtract, solve, and check that the hypotenuse is the longest side.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 10 (also accepted: 10 cm)
c² = 36 + 64 = 100, so c = 10 cm.
Answer: 50 (also accepted: 50 m)
The path is the hypotenuse: √(1600 + 900) = √2500 = 50 m.
Answer: 15
leg² = 17² - 8² = 289 - 64 = 225, so the leg is 15.
Answer: B) 13 in
The base diagonal is √(9 + 16) = 5. The space diagonal is √(5² + 12²) = √169 = 13 in.
Answer: 37 (also accepted: 37 in)
√(32² + 18²) = √(1024 + 324) = √1348 ≈ 36.7, which rounds to 37 inches.
Answer: D) 10.4 ft
√(3² + 10²) = √109 ≈ 10.4 feet.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
A full lesson with slides, activities and an exit ticket on applying the pythagorean theorem, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.G.B.7 with an answer key, ready in about a minute.
Make a worksheet →Turn applying the pythagorean theorem into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Students solve flat problems, such as a ladder or rectangle diagonal, and solid problems, such as the diagonal through a box, which uses the theorem twice.
In context, answers are usually rounded to a sensible accuracy, often the nearest tenth. Leaving √109 is fine when an exact value is asked for.