How to Calculate the Third Side of a Triangle Given Its Area and Other Side Lengths

When dealing with triangles, particularly when the area and one or more side lengths are known, we can use various methods to determine the third side. This article will explore the process of finding the third side of a triangle given the longest side, another side, and the area.

Introduction

A triangle with a longest side of 20 cm and another side of 10 cm has an area of 80 cm2. This tutorial will guide you through the mathematical steps to find the exact length of the third side.

Using Trigonometric Formulas

The area of a triangle can be expressed as (T frac{abc}{4R}), where (a), (b), and (c) are the sides of the triangle and (R) is the circumradius. Given that the area (T 80) cm2, the longest side (a 20) cm, and another side (b 10) cm, we can use the following steps to find the remaining side (c). Express the area formula: (80 frac{20 cdot 10 cdot c}{4R}). Solve for c: (c frac{40R}{5} 8R). Determine the sine of the angle opposite to side c: (sin C frac{4}{5}). Use the Pythagorean identity to find (cos^2 C): (cos^2 C 1 - sin^2 C 1 - frac{16}{25} frac{9}{25}). Apply the cosine rule: (cos C frac{500 - c^2}{400}). Equate (cos^2 C) with the expression from the cosine rule: (frac{500 - c^2}{400} frac{9}{25}). Solve the resulting quadratic equation for c: (500 - c^2 14400/25 576). Find the possible values for c: (c 2sqrt{65}) or (2sqrt{185}). Discard negative values and round for practical purposes: (c approx 26.83) cm.

Using Heron's Formula

Alternatively, we can use Heron's formula to find the third side, which states that the area of a triangle is (A sqrt{s(s - a)(s - b)(s - c)}), where (s) is the semi-perimeter of the triangle.

Express the semi-perimeter s: (s frac{a b c}{2} frac{20 10 c}{2}). Set up the area equation: (80 sqrt{frac{20 10 c}{2} left( frac{20 10 c}{2} - 10 right) left( frac{20 10 c}{2} - c right) left( frac{20 10 c}{2} - 20 right)}). Substitute s into the area equation and simplify: (80 sqrt{left( frac{30 c}{2} right) left( frac{10 c}{2} right) left( frac{30 - c}{2} right) left( frac{c - 10}{2} right)}). Square both sides and simplify: (6400 left( frac{30 c}{2} right) left( frac{10 c}{2} right) left( frac{30 - c}{2} right) left( frac{c - 10}{2} right)). Multiply through to eliminate fractions and simplify further: (102400 (30 c)(10 c)(30 - c)(c - 10)). Solve the resulting polynomial equation: (c^4 - 300c^2 - 91200 0). Use the quadratic formula to solve for c2: (c^2 frac{300 pm sqrt{90000 364800}}{2} frac{300 pm sqrt{454800}}{2}). Find the valid root and take the square root to find c: (c sqrt{187.32} approx 13.66) cm.

Conclusion

Both methods lead us to find the exact length of the third side, which is approximately 26.83 cm or 13.66 cm depending on the approach.

Keywords: triangle area, side length calculation, longest side