Difference Between Scalene, Isosceles & Equilateral Triangles
The concept of scalene triangle plays a key role in mathematics and is widely applicable to both exam scenarios and real-life shapes where all three sides or angles are different. Understanding scalene triangles helps in mastering geometry problems and is frequently tested in school exams and competitive Olympiads.
What Is a Scalene Triangle?
A scalene triangle is defined as a triangle where all three sides are of different lengths and all three angles are of different measures. You’ll find this concept applied in topics such as area calculation, comparing types of triangles, and triangle identification. Unlike equilateral or isosceles triangles, a scalene triangle has no equal sides and no equal angles.
Properties of a Scalene Triangle
- All sides have different lengths.
- All angles are different.
- The sum of the three interior angles is always 180°.
- There are no lines of symmetry in a scalene triangle.
- A scalene triangle can be acute-angled, obtuse-angled, or right-angled.
Comparison with Other Triangles
| Triangle Type | Sides | Angles | Symmetry Lines |
|---|---|---|---|
| Equilateral | All equal | All 60° | 3 |
| Isosceles | Two equal | Two equal | 1 |
| Scalene | All different | All different | 0 |
Key Formulas for Scalene Triangle
Here are the standard formulas for a scalene triangle:
- Perimeter: \( a + b + c \) (sum of all sides)
- Area (Heron’s Formula): \( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)
where \( s = \frac{a+b+c}{2} \) is the semi-perimeter, and a, b, c are the sides. - Area (Base and Height): \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
Solved Examples: Step-by-Step Illustration
Example 1: Identifying a Scalene Triangle
If a triangle has sides of 7 cm, 10 cm, and 12 cm, is it scalene?
1. Check if all sides are different:7 cm, 10 cm, 12 cm — all are different.
Final Answer: Yes, it is a scalene triangle.
Example 2: Perimeter Calculation
Find the perimeter of a scalene triangle with sides 5 cm, 6 cm, and 7 cm.
1. Use the perimeter formula:Perimeter = 5 + 6 + 7 = 18 cm
Final Answer: 18 cm
Example 3: Area Using Heron's Formula
Find the area if the sides are 8 cm, 15 cm, and 17 cm.
1. Find semi-perimeter: \( s = \frac{8+15+17}{2} = 20 \) cm2. Use Heron’s formula:
\( \text{Area} = \sqrt{20 \times (20-8) \times (20-15) \times (20-17)} \)
3. Work out the values:
\( = \sqrt{20 \times 12 \times 5 \times 3} \)
\( = \sqrt{20 \times 12 \times 15} \)
\( = \sqrt{3600} \)
\( = 60 \) cm2
Final Answer: 60 cm2
Speed Trick: Visualizing & Remembering Scalene Triangles
You can quickly spot a scalene triangle in diagrams by checking if no two sides look even close to equal. Imagine drawing any odd-angled triangle with a ruler and no sides matching—chances are it’s scalene! For MCQs, just remember: “All Different = Scalene.”
Try These Yourself
- Draw a scalene triangle using a scale with sides 5 cm, 7 cm, and 9 cm.
- Which of these sets can make a scalene triangle: 5, 5, 8 or 7, 9, 12?
- Calculate the perimeter and area of a triangle with sides 6 cm, 8 cm, 10 cm.
- List one household object shaped like a scalene triangle.
Frequent Errors and Misunderstandings
- Mistaking isosceles triangles (two equal sides) for scalene.
- Forgetting no lines of symmetry in a scalene triangle.
- Using the wrong formula for area when base and height are not given.
Relation to Other Concepts
Understanding scalene triangles helps you master concepts like symmetry, and angles. It connects closely with isosceles triangles, equilateral triangles, and area calculations using Heron's formula in geometry.
Classroom Tip
A quick way to remember scalene triangles: Think of the word “scalene” as “scattered lengths”—all sides and all angles are scattered, or different! Vedantu’s teachers often use visual puzzles and colour-code the sides in class to make this easy to spot.
We explored scalene triangles—from definition, formulas, examples, mistakes to practice ideas. Keep practicing with Vedantu to gain confidence in solving geometry problems and understanding all types of triangles!
Explore Related Topics
FAQs on What is a Scalene Triangle? Meaning, Properties & Formulas Explained
1. What is a scalene triangle?
A scalene triangle is a polygon with three unequal sides and three unequal angles. Unlike isosceles or equilateral triangles, it possesses no congruent sides or angles. The sum of its interior angles always equals 180°.
2. What are the properties of a scalene triangle?
Key properties of a scalene triangle include:
• All three sides have different lengths.
• All three angles have different measures.
• It has no lines of symmetry.
• It can be an acute, obtuse, or right-angled triangle.
3. How do you calculate the area of a scalene triangle?
The area of a scalene triangle can be calculated using Heron's formula if you know the lengths of all three sides (a, b, c):
1. Find the semi-perimeter (s): s = (a + b + c) / 2
2. Apply Heron's formula: Area = √[s(s - a)(s - b)(s - c)]
Alternatively, if you know the base (b) and height (h), use: Area = (1/2) * b * h
4. How is a scalene triangle different from an isosceles triangle?
In a scalene triangle, all three sides and angles are unequal. An isosceles triangle has at least two equal sides and two equal angles.
5. Does a scalene triangle have any lines of symmetry?
No, a scalene triangle has no lines of symmetry because all its sides and angles are different.
6. Can a scalene triangle be a right triangle?
Yes, a scalene triangle can also be a right-angled triangle. This occurs when one of its angles measures 90° and the other two angles are unequal and acute.
7. What real-life objects are shaped like scalene triangles?
Many everyday objects approximate a scalene triangle shape. Examples include certain traffic signs, pieces of broken glass, and some sections of architectural structures.
8. How does Heron's formula work for obtuse scalene triangles?
Heron's formula works for all triangles, including obtuse scalene triangles. The formula itself doesn't change; only the side lengths are inputted to calculate the area.
9. Why don’t scalene triangles have congruent angles?
Scalene triangles lack congruent angles because their sides are unequal in length. The angles opposite longer sides are always larger, ensuring no two angles are the same.
10. Are scalene triangles always acute or can they be obtuse or right-angled?
Scalene triangles can be acute, obtuse, or right-angled. The classification depends on the measures of its interior angles.
11. What is the perimeter of a scalene triangle?
The perimeter of any triangle, including a scalene triangle, is simply the sum of the lengths of its three sides: Perimeter = a + b + c, where a, b, and c are the lengths of the three sides.
12. How can I identify a scalene triangle in a diagram?
Look for a triangle where all three sides have different lengths. You can often visually estimate this, or if side lengths are labeled, check for inequality. Remember that unequal sides imply unequal angles.
