TutorialKart.comWhat Is a Scalene Triangle?
A scalene triangle is a triangle in which all three sides have different lengths. Since the side lengths are unequal, all three interior angles also have different measures.
For example, a triangle with side lengths 4 cm, 5 cm, and 6 cm is scalene because no two sides are equal.
How to Identify a Scalene Triangle
A triangle is scalene when its three side lengths satisfy both of these conditions:
- All three sides have different lengths.
- The side lengths satisfy the triangle inequality rule.
The triangle inequality rule states that the sum of any two sides of a triangle must be greater than the third side.
For sides a, b, and c, the following conditions must be true:
- a + b > c
- b + c > a
- c + a > b
Consider side lengths 3 cm, 4 cm, and 5 cm. All three lengths are different, and each pair of sides has a sum greater than the remaining side. Therefore, these sides form a scalene triangle.
Measurements That Determine a Scalene Triangle
A unique scalene triangle can be constructed when sufficient measurements are known. Common combinations include:
- All three side lengths, known as the SSS condition.
- Two side lengths and the included angle, known as the SAS condition.
- One side and the two angles at its endpoints, known as the ASA condition.
The given measurements must produce three unequal sides for the resulting triangle to be scalene.
Properties of a Scalene Triangle
All Three Sides Have Different Lengths
The defining property of a scalene triangle is that no two sides are congruent. If the sides are represented by a, b, and c, then:
a ≠ b, b ≠ c, and c ≠ a.
All Three Interior Angles Are Unequal
Each side of a triangle is opposite an interior angle. In a scalene triangle, the unequal side lengths produce three unequal opposite angles.
The Longest Side Is Opposite the Largest Angle
The largest interior angle lies opposite the longest side. Similarly, the smallest interior angle lies opposite the shortest side.
The Interior Angles Add Up to 180 Degrees
Like every triangle, the sum of the three interior angles of a scalene triangle is 180°.
If the angles are A, B, and C, then:
A + B + C = 180°
A Scalene Triangle Has No Line of Symmetry
A general scalene triangle has no line of reflection symmetry because its sides and angles do not match in equal pairs. It does, however, have rotational symmetry of order 1, meaning it matches itself after a full rotation of 360°.
Types of Scalene Triangles by Angle
A scalene triangle can also be classified according to its angles.
- Acute scalene triangle: All three angles are less than 90°.
- Right scalene triangle: One angle is exactly 90°.
- Obtuse scalene triangle: One angle is greater than 90°.
A triangle with side lengths 3 cm, 4 cm, and 5 cm is a right scalene triangle. A triangle with side lengths 4 cm, 5 cm, and 6 cm is an acute scalene triangle.
Perimeter of a Scalene Triangle
The perimeter of a scalene triangle is the sum of its three side lengths.
Perimeter = a + b + c
For a scalene triangle with sides 5 cm, 7 cm, and 9 cm:
Perimeter = 5 + 7 + 9 = 21 cm
Area of a Scalene Triangle
The area formula depends on the measurements that are available.
Area Using Base and Height
When the base and corresponding perpendicular height are known:
Area = 1/2 × base × height
For a base of 8 cm and a perpendicular height of 5 cm:
Area = 1/2 × 8 × 5 = 20 cm²
Area Using Heron’s Formula
When all three side lengths are known, the area can be calculated using Heron’s formula.
First calculate the semiperimeter:
s = (a + b + c) / 2
Then calculate the area:
Area = √[s(s − a)(s − b)(s − c)]
For side lengths 3 cm, 4 cm, and 5 cm:
- s = (3 + 4 + 5) / 2 = 6 cm
- Area = √[6(6 − 3)(6 − 4)(6 − 5)]
- Area = √(6 × 3 × 2 × 1) = √36 = 6 cm²
Area Using Two Sides and the Included Angle
When two sides and the angle between them are known:
Area = 1/2 × a × b × sin C
Here, a and b are the known sides, and C is their included angle.
Scalene Triangle Compared with Other Triangles
| Triangle type | Side-length relationship | Angle relationship |
|---|---|---|
| Scalene triangle | All three sides are unequal | All three angles are unequal |
| Isosceles triangle | At least two sides are equal | The angles opposite the equal sides are equal |
| Equilateral triangle | All three sides are equal | All three angles are 60° |
Worked Example: Checking Whether Three Sides Form a Scalene Triangle
Determine whether side lengths 5 cm, 7 cm, and 10 cm form a scalene triangle.
- The side lengths are all different: 5 ≠ 7, 7 ≠ 10, and 5 ≠ 10.
- Check the triangle inequality: 5 + 7 > 10, 7 + 10 > 5, and 10 + 5 > 7.
- All required conditions are satisfied.
Therefore, the three sides form a scalene triangle.
Common Mistakes When Working with Scalene Triangles
- Assuming that any three unequal lengths form a triangle without checking the triangle inequality.
- Using a slanted side as the height instead of the perpendicular distance to the selected base.
- Confusing a scalene triangle with an acute triangle. Scalene describes side lengths, while acute describes angle measures.
- Assuming that every right triangle is scalene. Some right triangles are isosceles.
- Forgetting to include square units when writing the area.
Scalene Triangle Questions and Answers
Can a scalene triangle have a right angle?
Yes. A scalene triangle can contain one right angle as long as all three side lengths are different. The 3-4-5 triangle is a common example.
Can a scalene triangle have two equal angles?
No. Equal angles in a triangle have equal opposite sides. A triangle with two equal angles would therefore be isosceles rather than scalene.
Can a scalene triangle be obtuse?
Yes. A scalene triangle is obtuse when one of its interior angles is greater than 90° and all three side lengths remain unequal.
Does a scalene triangle have a line of symmetry?
No. A scalene triangle has no line of reflection symmetry because it has no equal sides or equal angles.
How many unequal sides does a scalene triangle have?
A scalene triangle has three unequal sides. No pair of sides has the same length.