Pythagorean Theorem: Formulas, School and AMC Examples

The Pythagorean Theorem is one of the most important formulas in middle school math, used to solve right triangles by relating the lengths of their sides. It’s a core skill in geometry and shows up frequently in advanced math competitions like AMC 8 and AMC 10. In this guide, you’ll learn how to apply the Pythagorean Theorem formula through clear explanations and step-by-step examples. Plus, we’ve included free worksheets and a diagnostic assessment to help your child master this concept with confidence.

What Is the Pythagorean Theorem?

The Pythagorean Theorem is a key concept in middle school geometry, taught primarily in Grade 8 and sometimes earlier in advanced Grade 7 math classes. It applies specifically to right-angled triangles—triangles with one 90-degree angle—and shows how the three side lengths are mathematically related.

More than just a formula, the Pythagorean Theorem helps students connect algebra with spatial reasoning. It’s one of the first big ideas that bridges shapes and numbers, and it plays a major role in both school math and real-world problem-solving.

Pythagorean Theorem Formula

The Pythagorean Theorem equation is:

a² + b² = c²

Where:

• a and b are the lengths of the two legs (the shorter sides)
• c is the hypotenuse, the side opposite the right angle and the longest side in the triangle

How to Use Pythagorean Theorem

School-Level Examples

Let’s look at how this formula is used in school math problems. These examples are typical in Grade 7 or 8 textbooks and worksheets.

Pythagorean Theorem Example 1: Find the Hypotenuse

Problem

A right triangle has legs of length 6 cm and 8 cm. What is the length of the hypotenuse?

Solution

a² + b² = c²

6² + 8² = c²

36 + 64 = c²

c² = 100

c = √100 = 10

Answer

The hypotenuse is 10 cm.

Pythagorean Theorem Example 2: Find a Missing Leg

Problem

A triangle has one leg of 9 m and a hypotenuse of 15 m. What is the length of the other leg?

Solution

a² + b² = c²

9² + b² = 15²

81 + b² = 225

b² = 144

b = √144 = 12

Answer

The missing leg is 12 meters.

These types of problems are often included in Pythagorean Theorem worksheets to build fluency with identifying triangle parts and using square roots.

AMC 8 & 10-Style Application of Pythagorean Theorem

In math contests like AMC 8 and 10, problems involving the Pythagorean Theorem are more advanced. They often integrate the formula into layered scenarios or require a mix of skills, such as spatial reasoning or coordinate geometry.

Pythagorean Theorem Example 3

Problem (2020 AMC 8 Problem 18)

Rectangle ABCD is inscribed in a semicircle with diameter FE. Let DA = 16, and let FD = AE = 9. What is the area of rectangle ABCD?

(A) 240 (B) 248 (C) 256 (D) 264 (E) 272

Solution

Let O be the center of the semicircle.

• Diameter FE = FD + DA + AE = 9 + 16 + 9 = 34
• Radius OC = 17
• Since O is the midpoint of DA, we have OD = OA = 16/2 = 8

Now apply the Pythagorean Theorem in triangle ODC:

OC² = OD² + CD²

289 = 64 + CD²

CD² = 225 → CD = 15

Area of rectangle ABCD = CD x DA = 16 × 15 = 240

Answer

(A) 240

Pythagorean Theorem Example 4: Diagonal in a Rectangle

Problem (2023 AMC 10A Problem 11)

A square of area 2 is inscribed in a square of area 3, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?

(A) 1/5 (B) 1/4 (C) 2 – √3 (D) √3 – √2 (E) √2 – 1

Solution

This is a symmetric chord geometry problem.

• Side length of the large square = √3
• Side length of the small square = √2

The four triangles formed are congruent. Assume the shorter leg of the shaded right triangle is x.

Then, the longer leg is √3 – x, and the hypotenuse is √2.

Using the Pythagorean Theorem:

x² + (√3 – x)² = (√2)²

Expand and simplify:

x² + (√3 – x)² = 2

x² + (3 – 2√3·x + x²) = 2

2x² – 2√3·x + 3 = 2

2x² – 2√3·x + 1 = 0

Solve using the quadratic formula:

x = [√3 ± √( (√3)² – 4·1·0.5 )] / 2

x = [√3 ± √(3 – 2)] / 2

x = (√3 ± 1) / 2

Since x is the shorter leg and must be less than √3 / 2, we take the smaller root:

x = (√3 – 1) / 2

Then the longer leg = √3 – x = √3 – (√3 – 1)/2 = (√3 + 1)/2

Now compute the ratio:

(shorter leg) / (longer leg)

= [(√3 – 1)/2] ÷ [(√3 + 1)/2]

= (√3 – 1) / (√3 + 1)

Rationalize:

Multiply by the conjugate

[(√3 – 1) / (√3 + 1)] × [(√3 – 1) / (√3 – 1)]

= (√3 – 1)² / ( (√3 + 1)(√3 – 1) )

= (3 – 2√3 + 1) / (3 – 1)

= (4 – 2√3) / 2

= 2 – √3

Answer

(C) 2 – √3

Difference Between School Math and AMC Application

In traditional school math, Pythagorean Theorem problems tend to be:

• Direct and formula-based
• Easy to identify from diagrams
• Focus on procedural fluency and accurate calculation

In contrast, enrichment programs like AMC use this concept in more challenging ways. Problems often:

• The triangle may be hidden within a complex figure or context
• Involves irrational numbers, estimation, or multiple steps
• Requires reasoning, pattern recognition, and flexible problem-solving

In competitions, reading carefully is just as important as knowing the formula. Students need to not only apply the Pythagorean Theorem but also think critically about how and when to use it.

Ready to Practice?

🎯 Download your Pythagorean Theorem practice worksheets. Go to the [G6 and above] Tab, and find the Geometry worksheets.

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