Classifying Triangles Calculator
The Classifying Triangles Calculator is a free, user-friendly, and intelligent online tool designed to help you determine the exact classification of a triangle based on the given side lengths, angle measures, or both.
Whether you’re a student, teacher, or professional, this calculator provides instant, clear, and accurate results—eliminating guesswork and enhancing your understanding of triangle geometry.
How to Use the Classifying Triangles Calculator
The Classifying Triangles Calculator allows you to determine a triangle’s type based on its sides and/or angles. It accepts any three known values, as long as at least one of them is a side. The calculator then classifies the triangle by sides (Scalene, Isosceles, Equilateral) and by angles (Acute, Right, Obtuse).
1. Enter any three values in the six available input fields:
- Side lengths: a, b, and c (e.g., in meters or other)
- Angles: ∠α, ∠β, and ∠γ (in degrees or radian)
- Note: You must enter at least one side for a valid calculation.
2. Select decimal precision: Select your preferred units from the “Units” dropdown (e.g., meters for sides, degrees for angles).
3. Adjust decimal places (optional) to control the precision of your result (default is 3 decimal places).
4. Click the “Calculate” button. The tool will:
- Validate your inputs
- Calculate missing values if needed
- Classify the triangle by side lengths and angles
- Show a step-by-step explanation for transparency
5. Use “Hide Steps” to simplify the view or “Reset” to clear the fields and start over.
Note: The Classifying Triangles Calculator not only identifies the triangle type based on given sides and/or angles but can also compute various important geometric properties such as the perimeter, area, semiperimeter, heights (altitudes), medians, inradius, circumradius, and properties of the circumcircle to deepen your understanding of the triangle’s characteristics.
Classifying Triangles Calculator
Classification of Triangles?
Triangles can be classified based on two primary criteria:
- By Side Lengths
- By Angle Measures
Each classification reveals different geometric properties and helps in solving triangle-related problems in geometry, construction, trigonometry, and real-world applications.
Classification by Side Lengths
This method looks at how many sides are equal in length.
a. Scalene Triangle
- Definition: A triangle where all three sides have different lengths.
- Properties:
- All three angles are different.
- No line of symmetry.
- The triangle has no equal sides or angles.
- Example: Side a = 5 cm, Side b = 6 cm, Side c = 7 cm
b. Isosceles Triangle
- Definition: A triangle with two sides of equal length.
- Properties:
- The angles opposite the equal sides are also equal.
- Has one line of symmetry.
- Example: Side a = 5 cm, Side b = 5 cm, Side c = 7 cm
c. Equilateral Triangle
- Definition: A triangle with all three sides equal.
- Properties:
- All angles are 60°.
- Highly symmetrical – has three lines of symmetry.
- It is also a regular polygon.
- Example: Side a = b = c = 6 cm
Classification by Angle Measures
This method considers the largest angle in the triangle.
a. Acute Triangle
- Definition: A triangle in which all three angles are less than 90°.
- Properties:
- Can be scalene, isosceles, or equilateral.
- The triangle looks “sharp” or narrow.
- Example: Angles = 60°, 65°, 55°
b. Right Triangle
- Definition: A triangle that has one angle exactly 90°.
- Properties:
- The side opposite the 90° angle is called the hypotenuse.
- Follows the Pythagorean Theorem: a2 + b2 = c2
- Common in construction and trigonometry.
- Example: Angles = 90°, 60°, 30° or Sides = 3 cm, 4 cm, 5 cm
c. Obtuse Triangle
- Definition: A triangle that has one angle greater than 90°.
- Properties:
- Only one angle can be obtuse.
- The other two angles must be acute.
- Example: Angles = 120°, 40°, 20°
| Classification Type | Subtype | Key Feature | Example |
|---|---|---|---|
| By Sides | Scalene | All sides different | 5 cm, 6 cm, 7 cm |
| Isosceles | Two sides equal | 5 cm, 5 cm, 7 cm | |
| Equilateral | All sides equal | 6 cm, 6 cm, 6 cm | |
| By Angles | Acute | All angles < 90° | 60°, 60°, 60° |
| Right | One angle = 90° | 90°, 60°, 30° | |
| Obtuse | One angle > 90° | 120°, 30°, 30° |
How to Classify Triangles Manually: Step-by-Step Procedure
Step 1: Gather Triangle Information
Collect the measurements you have:
- Side lengths: a,b,c
- Angle measures: α,β,γ (if available)
At minimum, you need three values, including at least one side length.
If some values are missing, you can use the Law of Cosines and Law of Sines to calculate unknown sides or angles:
Law of Cosines:
$a^2 = b^2 + c^2 – 2bc \cdot \cos(\alpha)$
$b^2 = a^2 + c^2 – 2ac \cdot \cos(\beta)$
$c^2 = a^2 + b^2 – 2ab \cdot \cos(\gamma)$
(Use to find a side when you know two sides and the included angle, or find an angle when all sides are known.)
Law of Sines:
$\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$
(Use to find unknown angles or sides when you know one side and its opposite angle, along with another angle or side.)
Step 2: Verify Triangle Validity
Before classification, confirm the values can form a valid triangle using the Triangle Inequality Theorem:
- a + b > c
- a + c > b
- b + c > a
If all three inequalities hold true, proceed. If not, the given sides cannot form a triangle.
Step 3: Classify by Side Lengths
Compare the side lengths:
Equilateral Triangle: All sides equal a = b = c
Isosceles Triangle: Exactly two sides equal a = b ≠ c or a ≠ b = c
Scalene Triangle: All sides different a ≠ b ≠ c
Step 4: Classify by Angles
If you have angles or can calculate them (e.g., using Law of Cosines), classify the triangle by angles:
Right Triangle: One angle equals 90°
Acute Triangle: All angles less than 90°
Obtuse Triangle: One angle greater than 90°
Note(Very Important): If you only have sides, use the Pythagorean relation to check for right triangles:
$a^2 + b^2 = c^2$ (where c is the longest side)
If equality holds → Right Triangle
$a^2 + b^2 < c^2$ → Obtuse Triangle
$a^2 + b^2 > c^2$ → Acute Triangle
Step 5: Summarize Your Classification
Combine the results from Steps 3 and 4 for full classification, e.g.:
- Scalene Right Triangle
- Isosceles Acute Triangle
- Equilateral (always acute)
Example 1: Classify Triangle with Sides $a = 5$, $b = 5$, $c = 8$
Step 1: Given
Side lengths: $a = 5$, $b = 5$, $c = 8$ units
No angles given.
Step 2: Check Triangle Validity (Triangle Inequality Theorem)
$5 + 5 = 10 > 8$ ✔
$5 + 8 = 13 > 5$ ✔
$8 + 5 = 13 > 5$ ✔
All inequalities are satisfied ⇒ Triangle is valid.
Step 3: Classify by Side Lengths
$a = 5$, $b = 5$, $c = 8$
Two sides are equal ⇒ Isosceles Triangle
Step 4: Classify by Angles (Using Pythagorean Relation)
Longest side $c = 8$
$a^2 + b^2 = 5^2 + 5^2 = 25 + 25 = 50$
$c^2 = 8^2 = 64$
$50 < 64 \Rightarrow a^2 + b^2 < c^2$ ⇒ Obtuse Triangle
Step 5: Final Classification
Isosceles Obtuse Triangle
Example 2: Classify Triangle with Sides $a = 6$, $b = 7$, $c = 8$
Step 1: Given
$a = 6$, $b = 7$, $c = 8$
Step 2: Triangle Inequality
$6 + 7 = 13 > 8$ ✔
$6 + 8 = 14 > 7$ ✔
$7 + 8 = 15 > 6$ ✔
Step 3: Side Classification
All sides are different ⇒ Scalene Triangle
Step 4: Angle Classification
Longest side $c = 8$
$a^2 + b^2 = 6^2 + 7^2 = 36 + 49 = 85$
$c^2 = 8^2 = 64$
$85 > 64 \Rightarrow a^2 + b^2 > c^2$ ⇒ Acute Triangle
Step 5: Final Classification
Scalene Acute Triangle
Example 3: Classify Triangle with Sides $a = 6$, $b = 8$, $c = 10$
Step 1: Given
$a = 6$, $b = 8$, $c = 10$
Step 2: Triangle Inequality
$6 + 8 = 14 > 10$ ✔
$6 + 10 = 16 > 8$ ✔
$8 + 10 = 18 > 6$ ✔
Step 3: Side Classification
All sides are different ⇒ Scalene Triangle
Step 4: Angle Classification
Longest side $c = 10$
$a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100$
$c^2 = 10^2 = 100$
$100 = 100 \Rightarrow a^2 + b^2 = c^2$ ⇒ Right Triangle
Step 5: Final Classification
Scalene Right Triangle
Example 4: Classify Triangle with Sides $a = 4$, $b = 5$, $c = 7$
Step 1: Given
$a = 4$, $b = 5$, $c = 7$
Step 2: Triangle Inequality
$4 + 5 = 9 > 7$ ✔
$4 + 7 = 11 > 5$ ✔
$5 + 7 = 12 > 4$ ✔
Step 3: Side Classification
All sides are different ⇒ Scalene Triangle
Step 4: Angle Classification
Longest side $c = 7$
$a^2 + b^2 = 4^2 + 5^2 = 16 + 25 = 41$
$c^2 = 7^2 = 49$
$41 < 49 \Rightarrow a^2 + b^2 < c^2$ ⇒ Obtuse Triangle
Step 5: Final Classification
Scalene Obtuse Triangle
Example 5: Classify Triangle with Sides $a = 6$, $b = 6$, $c = 6$
Step 1: Given
All sides are $6$
Step 2: Triangle Inequality
$6 + 6 = 12 > 6$ ✔ (all combinations hold)
Step 3: Side Classification
All sides equal ⇒ Equilateral Triangle
Step 4: Angle Classification
All internal angles in equilateral triangle = $60^\circ$ ⇒ Acute Triangle
Step 5: Final Classification
Equilateral Acute Triangle
Example 6: Classify Triangle with Sides $a = 7$, $b = 7$, $c = \sqrt{98} \approx 9.899$
Step 1: Given
$a = 7$, $b = 7$, $c = \sqrt{98} \approx 9.899$
Step 2: Triangle Inequality
$7 + 7 = 14 > 9.899$ ✔
$7 + 9.899 > 7$ ✔
$7 + 9.899 > 7$ ✔
Step 3: Side Classification
Two equal sides ⇒ Isosceles Triangle
Step 4: Angle Classification
$a^2 + b^2 = 49 + 49 = 98$
$c^2 = (\sqrt{98})^2 = 98$
$98 = 98 \Rightarrow a^2 + b^2 = c^2$ ⇒ Right Triangle
Step 5: Final Classification
Isosceles Right Triangle
Example 7: Classify Triangle with Sides $a = 7$, $b = 7$, $c = 12$
Step 1: Given
$a = 7$, $b = 7$, $c = 12$
Step 2: Triangle Inequality
$7 + 7 = 14 > 12$ ✔
$7 + 12 = 19 > 7$ ✔
$7 + 12 = 19 > 7$ ✔
Step 3: Side Classification
Two equal sides ⇒ Isosceles Triangle
Step 4: Angle Classification
$a^2 + b^2 = 49 + 49 = 98$
$c^2 = 144$
$98 < 144 \Rightarrow a^2 + b^2 < c^2$ ⇒ Obtuse Triangle
Step 5: Final Classification
Isosceles Obtuse Triangle
Frequently Asked Questions (FAQs) on Classifying Triangles Calculator
1. What does the Classifying Triangles Calculator do?
The calculator classifies a triangle based on the lengths of its sides and/or the measures of its angles. It determines the type of triangle by sides (Equilateral, Isosceles, Scalene) and by angles (Acute, Right, Obtuse).
2. What input values do I need to provide?
You need to enter the lengths of the three sides of the triangle: a, b, and c. Optionally, you can also input the angles if you have them, but side lengths alone are sufficient for classification.
3. How does the calculator check if the triangle is valid?
It applies the Triangle Inequality Theorem, which states that for any triangle, the sum of any two sides must be greater than the third side. If this condition is not met, the triangle is invalid.
4. How are the types of triangles classified by sides?
- Equilateral: All three sides are equal.
- Isosceles: Exactly two sides are equal.
- Scalene: All three sides are different lengths.
5. How are the types of triangles classified by angles?
- Acute Triangle: All angles are less than 90°.
- Right Triangle: One angle is exactly 90°.
- Obtuse Triangle: One angle is greater than 90°.
6. What if I don’t know the angles? Can the calculator still classify the triangle by angles?
Yes! If you only provide side lengths, the calculator uses the Pythagorean relation between the squares of the sides to determine the angle classification:
- If a² + b² = c² , it’s a Right Triangle.
- If a² + b² > c² , it’s an Acute Triangle.
- If a² + b² < c², it’s an Obtuse Triangle.
Here, c is the longest side.
Final Thoughts: The Classifying Triangles Calculator is a simple powerful tool that helps students, teachers, and professionals quickly determine the type of a triangle based on side lengths and angles. Whether you’re learning geometry or verifying triangle properties in real-world problems, this calculator ensures accurate classification with instant results—making it an essential aid for anyone working with triangles.
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