Table of Contents

** Scalene Triangle Formula**

**Introduction: **

When categorizing a triangle based on its sides, we examine the lengths of the sides to determine whether it falls into the categories of equilateral, isosceles, or scalene. An equilateral triangle has all three sides of equal length, while in a scalene triangle, all sides have different lengths. In this article, we will focus on the formula and examples for calculating scalene triangles.

**What is a Scalene Triangle?**

scalene triangle is a type of triangle in which all three sides have different lengths. Unlike an equilateral or isosceles triangle, where at least two sides are equal in length, a scalene triangle does not have any sides of equal length. The angles of a scalene triangle are also unequal, making it a versatile and asymmetrical geometric shape.

**Scalene Triangle Formulas:**

To work with scalene triangles, there are various formulas available to find different properties of the triangle.

**Area Formula:**The area of a scalene triangle can be calculated using the Heron’s formula. Let’s say the lengths of the three sides of the scalene triangle are a, b, and c. The semi-perimeter (s) of the triangle is calculated as (a + b + c) / 2. Then, the area (A) of the triangle can be found using the formula:

A = √(s x (s – a) x (s – b) x (s – c))

**Perimeter Formula:**The perimeter of a scalene triangle is the sum of the lengths of all three sides. It can be calculated by adding the lengths of the three sides together:

Perimeter = a + b + c

These formulas provide a way to calculate various properties of a scalene triangle, such as its area, and perimeter. By using these formulas, one can work with and solve problems involving scalene triangles in geometry and mathematics.

**Solved Examples on Scalene Triangle Formula:**

**Example 1:** Consider a scalene triangle with side lengths of 5 cm, 7 cm, and 9 cm. Find its area using the Scalene triangle area formula.

Solution:

Using the formula, we first calculate the semi-perimeter:

s = (5 + 7 + 9) / 2 = 21 / 2 = 10.5 cm

Now, we can calculate the area using the Heron’s formula:

A = √(10.5 x (10.5 – 5) x (10.5 – 7) x (10.5 – 9))

A = √(10.5 x 5.5 x 3.5 x 1.5)

A ≈ √423.75 ≈ 20.59 cm²

Therefore, the area of the scalene triangle is approximately 20.59 cm².

**Example 2: **Consider a scalene triangle with side lengths of 6 cm, 8 cm, and 10 cm. Find its perimeter using the scalene triangle perimeter formula.

Solution:

The perimeter of a scalene triangle is simply the sum of its three side lengths. In this case, we have:

Perimeter = 6 cm + 8 cm + 10 cm

Perimeter = 24 cm

Therefore, the perimeter of the scalene triangle is 24 cm.

**Frequently Asked Questions on Scalene Triangle Formula:**

1: How do you find the area of a scalene triangle?

Answer: To find the area of a scalene triangle, you can use the formula: Area = (1/2) x base x height. Multiply half of the base by the corresponding height to calculate the area. Ensure that the height is perpendicular to the base for accurate results.

2: How do you find the area of a scalene triangle without height?

Answer: To find the area of a scalene triangle without the height, you can use Heron’s formula. Heron’s formula states that the area of a triangle can be calculated using its side lengths. Let’s assume the side lengths of the scalene triangle are a, b, and c. The semi-perimeter (s) can be calculated as (a + b + c) / 2. Then, the area (A) can be found using the formula A = √(s x (s – a) x (s – b) x (s – c)). Heron’s formula allows you to find the area of a scalene triangle even without knowing its height.

3: What is the height of a scalene triangle?

Answer: The height of a scalene triangle is the perpendicular distance from one of the vertices to the opposite side (base) of the triangle. Unlike an equilateral or isosceles triangle, which have special properties related to their heights, a scalene triangle does not have a specific formula or relationship to determine its height. The height of a scalene triangle can be found by various methods, such as using trigonometry, the Pythagorean theorem, or by dividing the triangle into smaller triangles. The height can also be given as a known value in a given problem or scenario.

4: Can we use Heron’s formula for scalene triangle?

Answer: Yes, Heron’s formula can be used to find the area of a scalene triangle. It does not require the height of the triangle and instead uses the side lengths directly. By substituting the side lengths into the formula, you can calculate the area of the scalene triangle.

5: Is scalene triangle irregular?

Answer: Yes, a scalene triangle is considered an irregular triangle. It is characterized by having three sides of different lengths and three angles of different measures. In contrast, an equilateral triangle has all three sides and angles equal, while an isosceles triangle has two sides and two angles equal. The irregularity of a scalene triangle distinguishes it from the other two types of triangles.

6: What are the 3 properties of a scalene triangle?

Answer: The three properties of a scalene triangle are:

Unequal sides: In a scalene triangle, all three sides have different lengths. This is in contrast to an equilateral triangle where all three sides are equal in length.

Unequal angles: The three angles of a scalene triangle are also unequal. None of the angles are congruent, meaning they have different measures.

No symmetry: A scalene triangle does not possess any lines of symmetry. This means that there is no line that can be drawn through the triangle to divide it into two congruent halves.

7: How do you find the area of a scalene triangle using Heron’s formula?

Answer: To find the area of a scalene triangle using Heron’s formula, follow these steps

Measure the lengths of all three sides of the triangle and denote them as a, b, and c

Calculate the semi-perimeter of the triangle, denoted as s, by adding the lengths of all three sides and dividing the sum by 2: s = (a + b + c) / 2.

Apply Heron’s formula: Area = √(s x (s – a) x (s – b) x (s – c)).

Substitute the values of the side lengths and calculate the area using the formula.

The resulting value will be the area of the scalene triangle.

8: Can you use Pythagorean theorem on a non-right triangle?

Answer: No, the Pythagorean theorem can only be used on right triangles, which are triangles that have one angle measuring 90 degrees. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

For non-right triangles, the Pythagorean theorem cannot be directly applied.