Geometry - Scalene Triangle
- What is a scalene triangle?
- Types of scalene triangle
- Properties of Scalene triangle
- Perimeter of Scalene triangle
- Area of Scalene triangle
- Comparison of Equilateral, Isosceles, and Scalene triangles
- Solved problems
- Practice problems
- Frequently asked questions
What is a scalene triangle?
A triangle is a three-sided polygon. There are three kinds of triangles: equilateral triangle, isosceles triangle, and scalene triangle.
A scalene triangle is a triangle that has all sides of different lengths. In other words, we can say that a scalene triangle has unequal sides or the sides of a scalene triangle are not congruent.
Following is the image showing all three kinds of triangles.
Equilateral triangle: All sides are equal as shown by the small line crossed on the sides.
Isosceles triangles: Two sides are equal as shown by two crossed lines.
Scalene triangle: All sides are unequal.
Also, all the angles in a scalene triangle are of different measures.
Types of scalene triangle
1. Acute scalene triangle
Acute scalene triangle means that each of the three angles has a measure of less than `90^\circ`. It is to be noted that none of the angles is a right angle (equal to `90^\circ`) or an obtuse angle (greater than `90^\circ`). The below figure shows an acute scalene triangle.
2. Right scalene triangle
A right scalene triangle is a triangle in which one angle has a measure of `90^\circ` whereas the other two acute angles have different measures. A right scalene triangle is shown below.
3. Obtuse scalene triangle
An obtuse scalene triangle is a triangle that has a measure of one of its angles greater than `90^\circ` whereas the other two acute angles have different measures.
Properties of scalene triangle
- Unequal side lengths
Scalene triangle has all three sides of different measure. They are incongruent.
- Unequal angles
All three angles of a scalene triangle have different measures i.e. they are not congruent.
- Absence of line of symmetry
A line of symmetry does not exist for scalene triangles. A line of symmetry divides the shape into two equal parts but the scalene triangle does not have equal sides and angles so it cannot be divided into two equal parts by any dividing line.
- Triangle inequality theorem
“It states that the sum of lengths of any two sides must be greater than the length of the remaining side”. For a scalene triangle, all sides have different lengths, so when the lengths of two sides are added it becomes greater than the length of the remaining side irrespective of the combination of sides.
- Perimeter and area formulae
The formulae for finding the perimeter and area of a scalene triangle are different from the formulae of an equilateral triangle that will be discussed in the upcoming sections.
- Versatility
The scalene triangle has versatility because its sides are of different lengths making them flexible and applicable to possible geometric problems involving triangles.
- Sum of angles property
This property corresponds to the property of any other triangle. For example, similar to an equilateral triangle, a scalene triangle also has the sum of all the interior angles as 180°. The image below shows the two angles of a scalene triangle. Its third angle can be calculated as
Sum of angles in a triangle `= ∠A + ∠B + ∠C =180°`
Here, `∠A = 39°`, and `∠B = 65°`
Therefore, `∠C = 180 -(39+65)=76°`.
Perimeter of Scalene triangle
We know that the sides of the scalene triangle are not equal. The sum of all the sides of a scalene triangle is defined as the perimeter of the scalene triangle.
Suppose `a, b,` and `c` are the measures of the sides of a scalene triangle as shown in the figure below. Then, the perimeter is given by,
Perimeter `= a+b+c`.
Area of Scalene triangle
The area of the scalene triangle can be calculated in two ways:
1. When the measure of base and height is given as shown in the image below.
The area of the scalene triangle is given in the same way as for an equilateral or right-angle triangle.
Area of the scalene triangle`=1/2 \times B\times H`
Where `B =` measure of the base of the scalene triangle
`H =` measure of the height of the scalene triangle.
Example: A triangle has a height of `7` cm and a base of `10` cm. Find the area of the triangle.
Solution: Here, `H=7` cm, `B=10` cm
Area of the triangle`=1/2 \times B\times H`
Area of the triangle`=1/2 \times 10\times 7=35text( cm)^2`.
2. When measure of all sides is given as shown in the image below.
To find out the area in this condition, Heron’s formula for the area of the triangle is used.
According to Heron’s formula
Area of triangle `=\sqrt(s(s-a)(s-b)(s-c))`
Where `s` is known as the semi perimeter and is given by `s=(a+b+c)/2`
Example: A triangle has its sides with a measure of `10` cm, `14` cm, and `8` cm. Find the area and the perimeter of the triangle.
Solution: Here, `a=10` cm, `b=14` cm, and `c=8` cm
`s=(10+14+8)/2=(32)/2=16\text( cm)`
Area of triangle `=\sqrt(16(16-10)(16-14)(16-8))=39.19\text( cm)^2`.
Comparison of Equilateral, Isosceles, and Scalene triangles
| Characteristic |  Equilateral Triangle |  Isosceles Triangle |  Scalene Triangle |
| Side Lengths | All sides are equal. | At least two sides are equal. | All sides are unequal. |
| Angle Measurements | All angles are `60°`. | Two angles are congruent. | All angles are incongruent. |
| Number of Lines of Symmetry | Three lines of symmetry | Typically, one line of symmetry (if base angles are congruent) | No lines of symmetry. |
| Perimeter Formula | `3a` | `2a+b` | `a+b+c` |
| Area Formula | `\sqrt(3)/4 a^2` | `1/2 \times b\times h` | `\sqrt(s(s-a)(s-b)(s-c))` Where, `s=(a+b+c)/2` Also, `1/2 \times b\times H` |
| Examples | Mercedes-Benz logo, regular stop signs | Flagpole with a triangular flag, many roof designs | Everyday triangles in various context |
Solved problems
- A land surveyor found the dimensions of a scalene triangular field as `30` ft, `30` ft, and `40` ft. Calculate the perimeter and area of the triangular field.
Solution:
Here, `a=30` ft, `b=30` ft and `c=40` ft
Perimeter `=30+30+40=100` ft.
Area of the field
`s=(30+30+40)/2=(100)/2=50\ \text(ft)`
Area of the field `=sqrt(50(50-30)(50-30)(50-40))= 447.21\ \text(ft)^2`
- A student draws a volcano of base `24` cm and a height of `32` cm. He is not sure about the measurement of the sides of the volcano, but still wants to find the area of the volcano. How can he find the area of the volcanic figure?
Solution:
Here `B=24` cm and `H=32` cm
Area of the volcano`= 1/2\times B\times H=1/2 \times 24\times 32=384\ \text(cm)^2`.
- An architecture is designing a scalene triangle shaped roof of an official space. He measures two angles as `45°` and `57°`. He wants to find the third angle. What will be the measure of the third angle?
Solution:
Consider the angles as `∠A=45°` and `∠B=57°`.
Apply the angle sum property of the triangle.
`∠A+∠B+∠C=180°`
`∠C=180°-(∠A+∠B)`
`∠C=180°-(45°+57°)`
`∠C=78°`
Practice problems
1. Identify the correct statement for a scalene triangle.
- Scalene triangles have at least one right angle.
- Only one obtuse angle is present in a scalene triangle.
- All the angles are congruent for a scalene triangle.
- Two of the sides have the same length in a scalene triangle.
2. The side lengths of the scalene triangle are `6` inches, `7` inches, and `11` inches. What is the area of the triangle?
- `19.67`
- `15.78`
- `18.97`
- `20`
3. A scalene triangle has a measure of `43°`, and `57°`. Find the other angle.
- `70°`
- `80°`
- `90°`
- `100°`
4. The side lengths of a scalene triangle are consecutive integers. If the perimeter of the triangle is `45` cm. Find all its sides.
- `13` cm, `14` cm, `15` cm
- `14` cm, `15` cm, `16` cm
- `15` cm, `16` cm, `17` cm
- `16` cm, `17` cm, `18` cm
Frequently asked questions
Q`1`. Can a scalene triangle have one of its angles as an obtuse angle, but still be categorized as an acute scalene triangle?
No, it will not be called as an acute angle triangle because the condition of an acute angle triangle is that each angle of an acute angle triangle must have a measure of less than `90°`, otherwise it will be categorized as an obtuse angle scalene triangle.
Q`2`. Measures of how many angles are required to define a scalene triangle?
A triangle has three angles, so if you have the measure of two angles then you can use the angle sum property of the triangle to find the third angle.
Q`3`. What are the maximum lines of symmetry that can pass through the scalene triangle?
A scalene triangle like the equilateral and isosceles triangles cannot be divided equally into two parts because by nature the scalene triangle has different dimensions of its sides and that restricts it from having lines of symmetry.