Calculating the Area of a Triangle with Equal Sides Using Heron's Formula

When given the lengths of the sides of a triangle, the area can be calculated using various methods. One of the most popular and accurate methods is Heron's Formula. This formula allows us to compute the area of a triangle when we only know the lengths of its sides. In this article, we will calculate the area of a triangle with sides 35 cm, 35 cm, and 35 cm, as well as the area of a triangle with sides 3 cm, 5 cm, and 6 cm, both using Heron's Formula.

Using Heron's Formula to Calculate the Area of an Equilateral Triangle

An equilateral triangle is a triangle with all sides of equal length. For an equilateral triangle with side length of 35 cm:

Step-by-Step Calculation:

Calculate the semi-perimeter (S): S (a b c) / 2 (35 35 35) / 2 52.5 / 2 26.25 cm Apply Heron's Formula: A sqrt(S(S-a)(S-b)(S-c))

Plugging in the values:

A sqrt(26.25(26.25-35)(26.25-35)(26.25-35)) sqrt(26.25 x 10.25 x 10.25 x 10.25) 54.106 square centimeters.

However, this result seems incorrect because we are dealing with an equilateral triangle. The area of an equilateral triangle can be calculated more easily with the formula:

Area (sqrt(3) × side2) / 4

Applying this to our triangle with sides 35 cm:

Area (sqrt(3) × 352) / 4 (1.732 × 1225) / 4 2142.05 / 4 535.5125 square centimeters

Therefore, the area of the equilateral triangle with sides 35 cm is approximately 535.51 square centimeters.

Using Heron's Formula to Calculate the Area of a Scalene Triangle

User then proceeds to calculate the area of a triangle with sides 3 cm, 5 cm, and 6 cm using Heron's formula. Here, Heron's Formula is defined as:

A sqrt(S(S-a)(S-b)(S-c))

Step-by-Step Calculation:

Calculate the semi-perimeter (S): S (a b c) / 2 (3 5 6) / 2 14 / 2 7 cm Apply Heron's Formula: A sqrt(S(S-a)(S-b)(S-c)) sqrt(7(7-3)(7-5)(7-6)) sqrt(7 × 4 × 2 × 1) sqrt(56) ≈ 7.48 square centimeters

The area of the triangle with sides 3 cm, 5 cm, and 6 cm is approximately 7.48 square centimeters.

Alternative Calculation Using Cosine Rule and Other Formulas

Additionally, the user calculates the area using the cosine rule and sine rule.

Cosine Rule:

cos x (32 52 - 62) / (2 × 3 × 5) (9 25 - 36) / 30 -1 / 15 x ≈ 93.82 degrees Area 1/2 × 3 × 5 × sin(93.82°) ≈ 7.48 square centimeters

To Validate with Another Formula:

The area of a triangle can also be calculated using the formula:

Area (sqrt(3)/4) × side2

For a triangle with sides 3.5 cm (equilateral triangle):

Area (sqrt(3)/4) × 3.52 (1.732 / 4) × 12.25 ≈ 5.3044 square centimeters

Hence, the area of an equilateral triangle with side length 3.5 cm is approximately 5.30 square centimeters.

Conclusion

Both the equilateral triangle with side length 35 cm and the scalene triangle with sides 3 cm, 5 cm, and 6 cm can be accurately computed using Heron's Formula. Heron's Formula is a versatile and powerful tool for determining the area of a triangle based solely on the lengths of its sides. The cosine rule and sine rule can also be applied to verify the computed area.

Keywords

Heron's Formula Area of Triangle Equilateral Triangle