Scalene Triangle Formula: A triangle is the smallest three-sided polygon. We can classify a triangle as equilateral, isosceles, or scalene based on its sides. A scalene triangle is one in which all three sides of the triangle are of different lengths, or none of the two sides is equal. Even though the angles are different, the sum of all of the interior angles of the scalene triangle is still \(180^\circ \).

This article will discuss scalene triangles, their perimeter, area, calculation of area using Heron’s formula, the formulas for circumcircle radius, and solved examples.

What is a Triangle?

A polygon with three sides is called a triangle. It is a figure bounded or enclosed by three-line segments. A triangle has three sides, three vertices and three angles, as you can see in the image below:

Learn All the Properties of Triangles

Classification of Triangles Based on Length of Sides

We can divide the triangles into three categories based on the length of their sides.

1. Isosceles triangles
2. Equilateral triangles
3. Scalene triangles

Isosceles Triangles

An isosceles triangle is one in which two of the three sides are equal.

Two equal sides are shown in the triangle above. The two angles opposite to the two equal sides are equal in the measure in an isosceles triangle.

Equilateral Triangles

When the lengths of all three sides of a triangle are equal, the triangle is known as an equilateral triangle.

In an equilateral triangle, the measure of all the angles is also equal to \(60^\circ \).

Scalene Triangles

A scalene triangle is one in which all three sides of the triangle are of different lengths, or none of the two sides is equal. A scalene triangle is a triangle shown below.

All three angles of this triangle also differ in measures.

What is Perimeter?

The term perimeter is derived from the Greek words ‘peri’ (meaning surrounding) and ‘metron’ (meaning measuring).

The perimeter of any form is the entire length of its edges. The perimeter of a polygon is equal to the sum of its side lengths. The perimeter of any closed figure, except a polygon, is the length of its boundary or outside line.

Perimeter of a Scalene Triangle

We can calculate the perimeter of a triangle by summing the lengths of its three sides.

If the three sides of a scalene triangle are \(a, b\) and \(c\), then the perimeter \((P)\) of the scalene triangle is given by

\(P=(a+b+c)\,\text {units}\)

Area of Scalene Triangle Formula

When the base and height of a scalene triangle is known, then the area of a scalene triangle is given by

\(\text {Area} =\frac{1}{2} \times \text {Base} \times \text {Height} =\frac{1}{2} b h\)

Area of Scalene Triangle Using Heron’s Formula

When the height of the scalene triangle is not known, then we use Heron’s formula to calculate the area of the scalene triangle.

If the length of all three sides is known or given, then according to Heron,s formula, the area of a scalene triangle is given by,
\( \text {Area} =\sqrt{s(s-a)(s-b)(s-c)}\),
where, \(a, b\) and \(c\) are the lengths of sides of the triangle and \(s\) is the semi-perimeter of the scalene triangle, given by \(s=\frac{a+b+c}{2}\).

Scalene Triangle Angles Formula

Let \(a, b, c\) be the sides of the scalene triangle and \(\alpha, \beta, \gamma\) are opposite angles to the sides.

When the other two sides and the angle between them are known, the Law of Cosines, also known as the cosine rule or cosine formula, is used to calculate the third side. When all three sides of a triangle are known, we can also find all of the angles.

By the law of cosines, we have
\( \cos \alpha=\frac{b^{2}+c^{2}-a^{2}}{2 b c}\)
\( \cos \beta=\frac{a^{2}+c^{2}-b^{2}}{2 a c}\)
\( \cos \gamma=\frac{a^{2}+b^{2}-c^{2}}{2 a b}\)

If two angles of a scalene triangle are given, we can calculate the third angle using the angle sum property in a triangle.

Circumcircle of a Triangle

The circumscribed circle is a circle that goes across all of a triangle’s vertices. It is also known as the circumcircle of a triangle. The centre of this circle is known as the circumcenter, and the radius is known as the circumradius.

Example:

Radius of Circumcircle of a Scalene Triangle

The radius of a circle circumscribed around a triangle can be calculated using the following formula.

\(R=\frac{a b c}{4 \sqrt{s(s-a)(s-b)(s-c)}}\)

Where \(a, b\) and \(c\) are the sides of the scalene triangle and \(s\) is the semi-perimeter of a scalene triangle. Let us see how we arrive at this.

From \(\triangle B D O\),
\( \sin \theta  = \frac{{\frac{a}{2}}}{R}\)
\( \sin \theta=\frac{a}{2 R}\)

We know that,
Area of \(\triangle A B C=\frac{1}{2} b c \sin \theta\)
Area of \(\triangle A B C=\frac{1}{2} b c\left(\frac{a}{2 R}\right)\)
\(\Rightarrow \sqrt{s(s-a)(s-b)(s-c)}=\frac{1}{4}\left(\frac{a b c}{R}\right)\)
\(\Rightarrow R=\frac{a b c}{4 \sqrt{s(s-a)(s-b)(s-c)}}\)

Altitude of Triangles

The altitude of a triangle from any vertex is the perpendicular drawn from that vertex to the opposite side.

A perpendicular \(A D\) is drawn from the vertex \(A\) to the side \(B C\) in the above figure. So, \(A D\) is known as the altitude of the \(\triangle A B C\).

You don’t have to draw a perpendicular from the triangle’s top vertex to the opposite side to get altitude. To get altitudes, we may draw perpendiculars from any vertex of the triangle to the opposite side, as shown in the diagram.

In the above figure, perpendiculars \(A D, B E\), and \(C F\) are the altitudes of \(\triangle A B C\) drawn from the vertices \(A, B\) and \(C\) on the opposite sides \(B C, C A\) and \(A B\), respectively.

Altitude of a Scalene Triangle

If we know the length of all the sides of the scalene triangle, we can easily find its height. Area of a scalene triangle
\(=\sqrt{s(s-a)(s-b)(s-c)}\)
Where \(a, b, c\) are the triangle sides, and \(s\) is the semi perimeter.

Area of scalene triangle \(=\frac{1}{2} \times \text {Base} \times \text {Altitude}\)
\(\Rightarrow \sqrt{s(s-a)(s-b)(s-c)}=\frac{1}{2} \times \text {Base} \times \text {Altitude}\)
\(\Rightarrow \text {Altitude} =\frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{\text { Base }}\)

Solved Examples – Scalene Triangle Formula

Q.1. Calculate the perimeter of a scalene triangle whose lengths of the sides are \(2 \mathrm{~cm}, 4 \mathrm{~cm}\) and \(8 \mathrm{~cm}\).
Ans: Let the lengths of the sides of the scalene triangle is \(a, b,\) and \(c\) respectively.
Then, \(a=2 \mathrm{~cm}, b=4 \mathrm{~cm}\) and \(c=8 \mathrm{~cm}\)
The perimeter of the scalene triangle is the sum of the lengths of the three sides of the triangle.
So, perimeter, \(P=a+b+c=2+4+8=14 \mathrm{~cm}\)
Hence, the perimeter of the scalene triangle is \(14 \mathrm{~cm}\).

Q.2. Find the area of the scalene triangle whose lengths of the sides are \(3 \mathrm{~cm}, 4 \mathrm{~cm}\) and \(5 \mathrm{~cm}\).
Ans: Let the lengths of the sides of the scalene triangle is \(a, b,\) and \(c\) respectively.
Here, \(a=3 \mathrm{~cm}, b=4 \mathrm{~cm}\) and \(c=5 \mathrm{~cm}\)
So, we shall use Heron’s formula to calculate the area of the scalene triangle.
According to this formula, the area of the scalene triangle is given by,
\(\text {Area} =\sqrt{s(s-a)(s-b)(s-c)}\)
where, \(a, b\) and \(c\) are the lengths of sides of the scalene triangle and \(s\) is the semi-perimeter of the scalene triangle, given by \(s=\frac{a+b+c}{2}\).
So, \(s=\frac{a+b+c}{2}=\frac{3+4+5}{2}=\frac{12}{2}=6 \mathrm{~cm}\)
Hence, the area of the scalene triangle
\(=\sqrt{s(s-a)(s-b)(s-c)}\)
\(=\sqrt{6(6-3)(6-4)(6-5)}\)
\(=\sqrt{6 \times 3 \times 2 \times 1}=\sqrt{36}=6 \mathrm{~cm}^{2}\)

Q.3. The sides of a scalene triangular board are \(6 \mathrm{~m}, 8 \mathrm{~m}\), and \(10 \mathrm{~m}\). Find the cost of painting it at the rate of \(9\) paise per \(\text {m}^{2}\).
Ans: Given, \(a=6 \,\text {m}, b=8 \,\text {m}\) and \(c=10 \mathrm{~m}\)
According to this formula, the area of the scalene triangle is given by,
\(\text {Area} =\sqrt{s(s-a)(s-b)(s-c)}\),
where, \(a, b\) and \(c\) are the lengths of sides of the scalene triangle and \(s\) is the semi-perimeter of the scalene triangle, given by \(s=\frac{a+b+c}{2}\).
\(s=\frac{6+8+10}{2}=\frac{24}{2}=12\)
\(\Rightarrow \text {Area} =\sqrt{12(12-6)(12-8)(12-10)}\)
\(=\sqrt{12 \times 6 \times 4 \times 2}\)
\(=24\)
Hence, the area of the scalene triangular board is \(24 \mathrm{~m}^{2}\).
Therefore, the cost of painting at the rate of \(9\) paise per \(\text {m}^{2}=₹\,(24 \times 0.09)=₹\, 2.16\)

Q.4. Calculate the radius of the circumcircle of a scalene triangle, whose sides are given as sides are \(3 \mathrm{~cm}, 4 \mathrm{~cm}\) and \(5 \mathrm{~cm}\) respectively.
Ans: We know that the radius of the circumcircle of a scalene triangle is given by
\(R=\frac{a b c}{4 \sqrt{s(s-a)(s-b)(s-c)}}\)
\(s=\frac{a+b+c}{2}=\frac{3+4+5}{2}=\frac{12}{2}=6 \mathrm{~cm}\)
\(\Rightarrow R=\frac{3 \times 4 \times 5}{4 \sqrt{6(6-3)(6-4)(6-5)}}\)
\(=\frac{3 \times 4 \times 5}{4 \sqrt{6 \times 3 \times 2 \times 1}}\)
\(=\frac{3 \times 4 \times 5}{4 \times 6}=\frac{5}{2}\)
Hence, the radius of the circumcircle \(=2.5 \mathrm{~cm}\)

Q.5. Calculate the third angle of a scalene triangle given two angles are \(100^{\circ}\) and \(30^{\circ}\), respectively.
Ans: Let the third angle be \(x\).
We know that the sum of interior angles of a triangle is \(180^{\circ}\).
So, \(100^{\circ}+30^{\circ}+x=180^{\circ}\)
\(\Rightarrow 130^{\circ}+x=180^{\circ}\)
\(\Rightarrow x=180^{\circ}-130^{\circ}\)
\(\Rightarrow x=50^{\circ}\)
Hence, the third angle of the scalene triangle is \(50^{\circ}\).

Summary

The fundamental concept of a triangle, its perimeter, and circumcircle has been covered in this article. Then we learned how to classify triangles depending on their side lengths. We also went through the scalene triangle, its perimeter, area, area of a scalene triangle using Heron’s formula, the formula for circumcircle radius of a scalene triangle, and angle of a scalene triangle, as well as solved examples.

Learn All the Concepts on Area of a Triangle

FAQs

Q.1. What is a scalene triangle?
Ans: If all the three sides of the triangle are different in length or if none of the triangle sides is equal, then the triangle is called a scalene triangle.

Q.2. What is the area of the scalene triangle formula without height?
Ans: If we know the length of all the triangle sides, we can easily find its height. Area of a scalene triangle \(=\sqrt{s(s-a)(s-b)(s-c)}\)
Where \(a, b, c\) are the triangle sides, and \(s\) is the semi perimeter.

Q.3. What is the formula for finding the perimeter of a scalene triangle?
Ans: If a triangle has three sides \(a, b\) and \(c\), then the perimeter of the triangle
\(P=(a+b+c) \,\text {units}\)

Q.4. What is the unique property of the scalene triangle?
Ans: All three sides of the scalene triangle are unequal. There is no symmetry line in it. Acute, obtuse, or right-angle interior angles exist in a scalene triangle.

Q.5. How many altitudes does a scalene triangle have?
Ans: A scalene triangle has three altitudes.

We hope this detailed article on the scalene triangle formulas helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!