High school · Mathematics · 20 minutes
Right Triangles and the Pythagorean Theorem
Apply a² + b² = c² to a right triangle and interpret the result with units.
High School Math Foundations · Lesson 3 · Starter collectionLearning objective
Apply a² + b² = c² to a right triangle and interpret the result with units.
Gather your materials
- Paper or notebook
- Pencil; a calculator for checking arithmetic when appropriate
The lesson time covers the core activity. Allow extra time for video viewing and extended discussion.
Common Core skill focus
8.G.B.7Standard level: Grade 8 · Prerequisite review
Use the Pythagorean relationship to calculate a missing side of a right triangle.
Read the official standard ↗Grade 8 prerequisite review in the high school collection; this is not labeled as a high school geometry standard. Code and level checked 2026-11-04; selected skill coverage.
Teach it step by step
- Ask what the learner already knows. Introduce the objective and read the explanation together.
- Work through the example aloud. Pause for the learner to explain why each step or piece of evidence matters.
- Choose a video if useful, then connect it to the example. Let the learner complete the independent practice before checking the explained answers.
- Discuss the question below and finish with the exit check. Revisit the example if the reasoning is unclear.
Watch, notice, discuss
Video choices for this lesson
Preview a video, then choose what fits your learners. “Show to class” opens a larger view; use the player’s play and full-screen controls when you’re ready.
Teacher preview needed. Check the language, level and length before showing. YouTube may show ads or related videos.
Math Antics - The Pythagorean Theorem
By mathantics
Use this explanation as an introduction, then pause to connect it to the worked example and discussion question.
Open on YouTube: Math Antics - The Pythagorean Theorem (opens a new tab) ↗Pythagorean theorem 1 | Right triangles and trigonometry | Geometry | Khan Academy
By Khan Academy
Compare this alternate explanation with the lesson. Ask learners to show their reasoning after watching.
Open on YouTube: Pythagorean theorem 1 | Right triangles and trigonometry | Geometry | Khan Academy (opens a new tab) ↗Which side is the hypotenuse, and why can this rule only be used for a right triangle?
An external player loads only after you choose Preview or Show to class. Playback connects to YouTube. Videos belong to their creators; availability and embedding can change.
Understand the idea
For a right triangle, the squares of the leg lengths add to the square of the hypotenuse: a² + b² = c². The hypotenuse is opposite the right angle and is the longest side. The relationship is not a formula for every triangle. A length is positive; take the positive square root when finding a physical distance.
Worked example
A right triangle has legs of 6 cm and 8 cm. Then c² = 6² + 8² = 36 + 64 = 100, so c = 10 cm. If the hypotenuse is 13 m and one leg is 5 m, the other leg satisfies b² = 13² − 5² = 144, giving b = 12 m. Draw and label the triangle before choosing addition or subtraction.
Try it yourself
- Find the hypotenuse when the legs are 9 and 12 units.
- Find a missing leg when the hypotenuse is 10 and the other leg is 6 units.
- Could lengths 2, 3 and 4 form a right triangle? Show the test.
Check your reasoning
- c = √(81 + 144) = √225 = 15 units.
- b = √(100 − 36) = 8 units.
- No. With 4 as the longest side, 2² + 3² = 13, while 4² = 16.
Make it work for your learner
For support, model the example one step at a time and ask the learner to explain each choice in their own words. Use a sketch, organizer or physical model where helpful. For a challenge, create a new example, test an alternative explanation or describe when the model would not apply.
Exit check
Apply a² + b² = c² to a right triangle and interpret the result with units. Explain your reasoning with a specific example before moving on.
Prepare · Discuss · Reflect
Use evidence in a conversation
Discussion question: Which side is the hypotenuse, and why can this rule only be used for a right triangle?
- Before discussing, write a tentative answer with two details from the lesson’s passage, example or data. For a longer discussion, the teacher can add an age-appropriate source.
- In pairs, a group or a tutor–learner conversation, agree on roles: facilitator, evidence finder and note taker. Combine roles when needed and allow equal time to respond.
- Share evidence, ask a clarifying question and respond to a different interpretation. Explain which details support or challenge a claim.
- Paraphrase another person’s point. Write a final conclusion explaining what changed, what remains uncertain and what evidence you would seek next.
Teacher look-for: Accurate use of the example, reasons linked to evidence, thoughtful questions and a response to another viewpoint. A well-supported alternative answer is welcome.