feature

Right Triangle Checker

Enter three side lengths to check if they form a right triangle. Also classifies the triangle as acute or obtuse if it is not a right triangle.

Back to Pythagorean Theorem Calculator

Pythagorean Theorem Tips

Click to show tips

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Ladder Against Wall

A 10-foot ladder leans against a wall with the base 6 feet away. How high does it reach?

Key values: Leg a = 6 ft · Hypotenuse c = 10 ft · Solve for leg b

TV Screen Diagonal

Find the diagonal of a TV with a 16:9 aspect ratio (width 40", height 22.5").

Key values: Width = 40 in · Height = 22.5 in · Find diagonal

Classic 3-4-5 Triangle

Verify that sides 3, 4, and 5 form a right triangle (the most famous Pythagorean triple).

Key values: Sides: 3, 4, 5 · Right triangle · Primitive triple

Documentation

The Converse of the Pythagorean Theorem

The Pythagorean theorem says: if a triangle is right, then $a^2 + b^2 = c^2$ . The converse goes the other direction: if three sides satisfy $a^2 + b^2 = c^2$ (where $c$ is the largest side), then the triangle must be a right triangle.

But the converse tells us even more. Given three sides with $c$ being the largest:

Condition	Triangle Type	Largest Angle
$a^2 + b^2 = c^2$	Right	Exactly 90°
$a^2 + b^2 > c^2$	Acute	Less than 90°
$a^2 + b^2 < c^2$	Obtuse	Greater than 90°

Key insight: The converse works because the Pythagorean theorem characterizes right triangles uniquely. No other triangle type satisfies the exact equality. This makes $a^2 + b^2 = c^2$ a perfect test—both necessary and sufficient for right-angle verification.

Worked Examples

Example 1: Classic Right Triangle

Sides: 3, 4, 5. The largest side is 5:

3^2 + 4^2 = 9 + 16 = 25 = 5^2 \quad \Rightarrow \text{Right triangle } \checkmark

Example 2: Acute Triangle

Sides: 5, 6, 7. The largest side is 7:

5^2 + 6^2 = 25 + 36 = 61 > 49 = 7^2 \quad \Rightarrow \text{Acute triangle}

Example 3: Obtuse Triangle

Sides: 3, 4, 6. The largest side is 6:

3^2 + 4^2 = 9 + 16 = 25 < 36 = 6^2 \quad \Rightarrow \text{Obtuse triangle}

Practical Applications

The 3-4-5 Rule in Construction

Builders use the 3-4-5 triple (or multiples like 6-8-10 or 9-12-15) to verify right angles. To check if a corner is square:

Measure 3 feet along one wall from the corner
Measure 4 feet along the other wall from the corner
The diagonal between these points should be exactly 5 feet

If the diagonal is shorter than 5 feet, the angle is acute (walls angle inward). If longer, the angle is obtuse (walls angle outward).

Surveying and Engineering

Surveyors use the converse to verify right angles in property boundaries and building foundations. The advantage over protractors is that distance measurements are much more precise than angle measurements in the field.

Digital Verification

In computer-aided design (CAD) and computational geometry, checking whether a triangle is right is a fundamental operation. The converse provides an exact algebraic test that avoids floating-point issues with trigonometric functions. For integer coordinates, the test is exact with no rounding errors.

Frequently Asked Questions

How do you check if a triangle is a right triangle?

Square all three sides and check whether the sum of the two smaller squares equals the largest square. If $a^2 + b^2 = c^2$ (where $c$ is the longest side), the triangle is right. This is the converse of the Pythagorean theorem.

What if $a^2 + b^2$ does not equal $c^2$ ?

If $a^2 + b^2 > c^2$ , the triangle is acute (all angles less than $90^\circ$ ). If $a^2 + b^2 < c^2$ , the triangle is obtuse (the largest angle exceeds $90^\circ$ ). The relationship between $a^2 + b^2$ and $c^2$ fully classifies the triangle.

What is the 3-4-5 rule used in construction?

Builders verify right angles by measuring 3 feet along one wall, 4 feet along the other, and checking that the diagonal is exactly 5 feet. Multiples like 6-8-10 or 9-12-15 also work. If the diagonal is off, the corner is not square.

Why is the converse of the Pythagorean theorem valid?

The Pythagorean theorem uniquely characterizes right triangles. No other triangle type satisfies the exact equality $a^2 + b^2 = c^2$ . This makes the equation both necessary and sufficient for verifying a $90^\circ$ angle.

Can the right triangle check be done with decimal or irrational side lengths?

Yes. The check $a^2 + b^2 = c^2$ works for any positive real numbers. For non-integer measurements, allow a small tolerance for rounding. With integer coordinates, the test is exact with no rounding errors.

Related feature Variants

Explore more feature options

Triple Check

Pythagorean Triple Calculator & Checker

Use this calculator

More Math Calculators

Explore the category

Absolute Value Calculator

Area Calculator

Circle Calculator

Percentage Calculator

Prime Factorization Calculator

Unit Circle Explorer