SCALENE TRIANGLE: CHARACTERISTICS, FORMULA AND AREAS, CALCULATION - MATHS - 2025

A scalene triangle is a polygon with three sides, all of which have different measures or lengths; for that reason it is given the name of scalene, which in Latin means climbing.

Triangles are polygons considered the simplest in geometry, because they are made up of three sides, three angles, and three vertices. In the case of the scalene triangle, by having all the sides different, it implies that its three angles will be too.

Characteristics of scalene triangles

Scalene triangles are simple polygons because none of their sides or angles have the same measure, unlike isosceles and equilateral triangles.

Because all of their sides and angles have different measures, these triangles are considered irregular convex polygons.

Based on the amplitude of the internal angles, scalene triangles are classified as:

• Scalene right triangle: all sides are different. One of its angles is right (90 ^or) and the others are sharp and with different measures.
• Obtuse scalene triangle: all sides are different and one of its angles is obtuse (> 90 ^or).
• Scalene acute triangle: all sides are different. All angles are acute (<90 ^or) with different measures.

Another characteristic of scalene triangles is that due to the incongruity of their sides and angles, they do not have an axis of symmetry.

Components

The median: it is a line that starts from the midpoint of one side and reaches the opposite vertex. The three medians meet at a point called the barycenter or centroid.

The bisector: it is a ray that divides each angle into two angles of equal measure. The bisectors of a triangle meet at a point called the incenter.

The bisector: it is a segment perpendicular to the side of the triangle, which has its origin in the middle of it. There are three bisectors in a triangle and they meet at a point called the circumcenter.

The height: it is the line that goes from the vertex to the side that is opposite and also this line is perpendicular to that side. All triangles have three heights that coincide at a point called the orthocenter.

Properties

Scalene triangles are defined or identified because they have several properties that represent them, originating from the theorems proposed by great mathematicians. They are:

Internal angles

The sum of the interior angles is always equal to 180 ^°.

Sum of the sides

The sum of the measures of two sides must always be greater than the measure of the third side, a + b> c.

Incongruous sides

All sides of scalene triangles have different measures or lengths; that is, they are incongruous.

Incongruous angles

Since all the sides of the scalene triangle are different, its angles will be too. However, the sum of the internal angles will always be equal to 180º, and in some cases, one of its angles can be obtuse or right, while in others all its angles will be acute.

Height, median, bisector, and bisector are not coincident

Like any triangle, scalene has various line segments that compose it, such as: height, median, bisector and bisector.

Due to the particularity of its sides, in this type of triangle none of these lines will coincide in one.

Orthocenter, barycenter, incenter and circumcenter are not coincident

As the height, median, bisector and bisector are represented by different line segments, in a scalene triangle the meeting points - the orthocenter, the incenter and the circumcenter - will be found at different points (they do not coincide).

Depending on whether the triangle is acute, right, or scalene, the orthocenter has different locations:

to. If the triangle is acute, the orthocenter will be inside the triangle.

b. If the triangle is right, the orthocenter will coincide with the vertex of the right side.

c. If the triangle is obtuse, the orthocenter will be on the outside of the triangle.

Relative heights

Heights are relative to the sides.

In the case of the scalene triangle, these heights will have different measurements. Every triangle has three relative heights and Heron's formula is used to calculate them.

How to calculate the perimeter?

The perimeter of a polygon is calculated by adding the sides.

Since in this case the scalene triangle has all its sides with different measures, its perimeter will be:

P = side a + side b + side c.

How to calculate the area?

The area of ​​the triangles is always calculated with the same formula, multiplying the base times height and dividing by two:

Area = (base _* h) ÷ 2

In some cases the height of the scalene triangle is not known, but there is a formula that was proposed by the mathematician Herón, to calculate the area knowing the measure of the three sides of a triangle.

Where:

• a, b and c, represent the sides of the triangle.
• sp, corresponds to the semiperimeter of the triangle, that is, half of the perimeter:

sp = (a + b + c) ÷ 2

In the case that we only have the measure of two of the sides of the triangle and the angle formed between them, the area can be calculated by applying the trigonometric ratios. So you have to:

Area = (side _* h) ÷ 2

Where the height (h) is the product of one side and the sine of the opposite angle. For example, for each side, the area will be:

• Area = (b _* c _* sin A) ÷ 2
• Area = (a _* c _* sin B) ÷ 2.
• Area = (a _* b _* sin C) ÷ 2

How to calculate the height?

Since all sides of the scalene triangle are different, it is not possible to calculate the height with the Pythagorean theorem.

From Heron's formula, which is based on the measurements of the three sides of a triangle, the area can be calculated.

Height can be cleared from the general formula of the area:

The side is replaced by the measure of side a, b, or c.

Another way to calculate the height when the value of one of the angles is known, is by applying the trigonometric ratios, where the height will represent a leg of the triangle.

For example, when the angle opposite the height is known, it will be determined by the sine:

How to calculate the sides?

When you have the measure of two sides and the angle opposite them, it is possible to determine the third side by applying the cosine theorem.

For example, in a triangle AB, the height relative to segment AC is plotted. In this way the triangle is divided into two right triangles.

To calculate side c (segment AB), apply the Pythagorean theorem for each triangle:

• For the blue triangle we have:

c ² = h ² + m ²

Since m = b - n, we substitute:

c ² = h ² + b ² (b - n) ²

c ² = h ² + b ² - 2bn + n ².

• For the pink triangle you have to:

h ² = a ² - n ²

It is substituted in the previous equation:

c ² = a ² - n ² + b ² - 2bn + n ²

c ² = a ² + b ² - 2bn.

Knowing that n = a _* cos C, it is substituted in the previous equation and the value of side c is obtained:

c ² = a ² + b ² - 2b _* a _* cos C.

By the Law of Cosines, the sides can be calculated as:

• a ² = b ² + c ² - 2b _* c _* cos A.
• b ² = a ² + c ² - 2a _* c _* cos B.
• c ² = a ² + b ² - 2b _* a _* cos C.

There are cases where the measures of the sides of the triangle are not known, but rather their height and the angles formed at the vertices. To determine the area in these cases it is necessary to apply the trigonometric ratios.

Knowing the angle of one of its vertices, the legs are identified and the corresponding trigonometric ratio is used:

For example, the leg AB will be opposite for angle C, but adjacent to angle A. Depending on the side or leg corresponding to the height, the other side is cleared to obtain the value of this.

Exercises

First exercise

Calculate the area and a height of the scalene triangle ABC, knowing that its sides are:

a = 8 cm.

b = 12 cm.

c = 16 cm.

Solution

As data, the measurements of the three sides of the scalene triangle are given.

Since the height value is not available, the area can be determined by applying Heron's formula.

First the semiperimeter is calculated:

sp = (a + b + c) ÷ 2

sp = (8 cm + 12 cm + 16 cm) ÷ 2

sp = 36 cm ÷ 2

sp = 18 cm.

Now the values ​​are substituted in Heron's formula:

Knowing the area, the height relative to side b can be calculated. From the general formula, clearing it, we have:

Area = (side _* h) ÷ 2

46, 47 cm ² = (12 cm _* h) ÷ 2

h = (2 _* 46.47 cm ²) ÷ 12 cm

h = 92.94 cm ² ÷ 12 cm

h = 7.75 cm.

Second exercise

Given the scalene triangle ABC, whose measures are:

• Segment AB = 25 m.
• Segment BC = 15 m.

At vertex B an angle of 50º is formed. Calculate the height relative to side c, perimeter and area of ​​that triangle.

Solution

In this case we have the measurements of two sides. To determine the height it is necessary to calculate the measurement of the third side.

Since the angle opposite to the given sides is given, it is possible to apply the law of cosines to determine the measure of side AC (b):

b ² = a ² + c ² - 2a _* c _* cos B

Where:

a = BC = 15 m.

c = AB = 25 m.

b = AC.

B = 50 ^o.

The data is replaced:

b ² = (15) ² + (25) ² - 2 _* (15) _* (25) _* cos 50

b ² = (225) + (625) - (750) _* 0.6427

b ² = (225) + (625) - (482,025)

b ² = 367.985

b = √367,985

b = 19.18 m.

Since we already have the value of the three sides, the perimeter of that triangle is calculated:

P = side a + side b + side c

P = 15 m + 25 m + 19, 18 m

P = 59.18 m

Now it is possible to determine the area by applying Heron's formula, but first the semiperimeter must be calculated:

sp = P ÷ 2

sp = 59.18 m ÷ 2

sp = 29.59 m.

The measurements of the sides and the semiperimeter are substituted in Heron's formula:

Finally knowing the area, the height relative to side c can be calculated. From the general formula, clearing it you have to:

Area = (side _* h) ÷ 2

143.63 m ² = (25 m _* h) ÷ 2

h = (2 _* 143.63 m ²) ÷ 25 m

h = 287.3 m ² ÷ 25 m

h = 11.5 m.

Third exercise

In the scalene triangle ABC side b is 40 cm, the side c is 22 cm, and the apex A, an angle 90 is formed ^or. Calculate the area of ​​that triangle.

Solution

In this case, the measures of two sides of the scalene triangle ABC are given, as well as the angle that is formed at the vertex A.

To determine the area, it is not necessary to calculate the measure of side a, since through the trigonometric ratios the angle is used to find it.

Since the angle opposite the height is known, it will be determined by the product of one side and the sine of the angle.

Substituting in the area formula we have:

• Area = (side _* h) ÷ 2
• h = c _* sin A

Area = (b _* c _* sin A) ÷ 2

Area = (40 cm _* 22 cm _* sin 90) ÷ 2

Area = (40 cm _* 22 cm _* 1) ÷ 2

Area = 880 cm ² ÷ 2

Area = 440 cm ².

References

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