Scalene triangle
A scalene triangle is a triangle in which none of the three sides or angles are equal. The figure below shows a scalene triangle example.
In the figure above, the differing number of tick marks and arcs indicate that each side and angle of the triangle has a different measure. This is commonly used notation in geometry; sides that are the same length (congruent) are indicated using the same number of tick marks and angles that are the same are indicated using the same number of arcs.
What is a scalene triangle
A scalene triangle is defined as a three-sided polygon whose side lengths and angles are all different. To identify if a triangle is scalene, check whether it has any congruent sides or angles; if it does, it is not a scalene triangle; if it doesn't, it is a scalene triangle.
What does a scalene triangle look like
Because a scalene triangle has no equal sides, it can take on a number of forms. It can have acute, obtuse, or right angles, which all have very different shapes. The sail on a sailboat is a real-world example of a scalene triangle:
The scalene triangle in this example is a right scalene triangle. It is just one of the types of scalene triangles (more are described below).
Properties of scalene triangles
There are some properties that all triangles share:
- The sum of all internal angles of a triangle is equal to 180°.
- The length of the side of a triangle corresponds to the size of the angle opposite the side; the longer the side, the larger the angle. It is worth noting that the ratios do not correspond: if side a is twice the length of b, it does not mean that angle A is twice the size of angle B.
- The sum of the length of any two sides of a triangle is greater than the length of the third side.
There are more properties that all triangles share, but these are a few commonly used properties in geometry.
Scalene triangles specifically have the following properties:
- No equal sides.
- No equal angles.
- No line of symmetry - a scalene triangle cannot be divided into two identical halves. In contrast, equilateral and isosceles triangles both have line(s) of symmetry.
- Scalene triangles can be right triangles, obtuse triangles, or acute triangles.
Types of scalene triangles
Scalene triangles can be acute, obtuse, or right.
Acute scalene triangle
An acute scalene triangle is a triangle that has no congruent sides or angles in which all the angles are acute. The figure below shows an acute scalene triangle example:
Obtuse scalene triangle
An obtuse scalene triangle is a triangle in which one of the angles is obtuse and none of the sides or angles are congruent; the remaining two angles are acute. The figure below shows an obtuse scalene triangle example:
Right scalene triangle
A right scalene triangle is a triangle that has one angle that measures 90° and has no congruent sides or angles; the remaining angles are acute. The figure below shows a scalene right triangle example.
Scalene triangle formulas
Below are formulas for finding the sides, angles, area, and perimeter of a scalene triangle.
How to find the sides of a scalene triangle
To find the sides and angles of a scalene triangle, we can use the law of sines and law of cosines.
Law of sines
The law of sines is a formula that relates the sides and angles of a triangle. It can be used to find the sides and angles of any triangle given 3 known values (angles or sides). The law of sines is as follows,
where a, b, and c are the sides of the triangle and A, B, and C are their corresponding angles.
Example
Find the missing sides and angles of the following triangle using the law of sines.
In the above triangle, we know the measure of one side and 2 angles. The angle 26.241° corresponds to the side with length 12, and the 57.936° angle corresponds to side b, which we can find using the law of sines:
Then, since we know that the sum of the angles of a triangle must equal 180°, we can find angle C by subtracting the known angles from 180°:
180° - 57.936° - 26.241° = 95.823°
Then, to find the final side, we can apply the law of sines again with either of the sides and angles to find that the final side c = 27.
Law of cosines
The law of cosines is a formula that can be used to find the angles and sides of a triangle in certain cases. The law of cosines is as follows,
where a, b, and c are sides of the triangle and C is the angle between the sides.
The law of cosines can also be rearranged specifically for angles as follows:
Example
Find the missing sides and angles of the scalene triangle below using the law of cosines.
Let a = 9, b = 13, and C = 98°. Plugging these into the law of cosines,
Thus, side c = 16.80971. To find angle A, we plug our known values into the law of cosine angle formula for A:
Using the same method, we can find B = 49.981°. We could also simply subtract angles A and C from 180° to find angle B without having to use the law of cosine formula since we know that the sum of the interior angles of all triangles must equal 180°.
Scalene triangle area
There are a number of different ways to find the area of a triangle. The general formula for the area (A) of a triangle is
where b is the base and h is the height of the triangle. This works for any triangle given that the information provided makes it possible to find the base and the height of the triangle.
For scalene triangles, since each side has a different length, in most cases the problem likely will provide the side lengths, or make it possible to find the side lengths.
Heron's formula
Heron's formula can be used to find the area of a triangle in cases where the length of each side of a triangle is known:
In Heron's formula, a, b, and c represent the lengths of the sides of the triangle, and s is the semiperimeter:
Perimeter of a scalene triangle
The perimeter of a scalene triangle is the sum of the lengths of its sides. The perimeter of a scalene triangle formula is,
where a, b, and c are the sides of the triangle.
Scalene, isosceles, and equilateral
The following table shows the differences between scalene, isosceles, and equilateral triangles.
| Scalene triangle | Isosceles triangle | Equilateral triangle |
|---|---|---|
| 0 congruent sides | 2 congruent sides | 3 congruent sides |
| 0 congruent angles | 2 congruent angles | 3 congruent angles (all 60°) |
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