A triangle has sides of lengths 13 cm, 14 cm, and 15 cm. Calculate its area using Heron's formula.
["Calculating the Area of a Triangle with Sides 13 cm, 14 cm, and 15 cm Using Heron’s Formula", "When solving geometric problems involving unknown areas, Heron’s formula offers a powerful and elegant solution—especially when the triangle’s side lengths are known but its height isn’t readily available. One classic example is a triangle with sides measuring 13 cm, 14 cm, and 15 cm. In this article, we’ll walk through how to calculate the area using Heron’s formula, explaining each step clearly to help you master this essential geometric technique.", "---", "### Understanding Heron’s Formula", "Heron’s formula allows you to compute the area of a triangle purely from the lengths of its three sides. Introduced by the ancient Greek mathematician Heron of Alexandria, the formula is independent of the triangle’s orientation and avoids the need for measuring angles or heights directly.", "The formula states:", "[\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)}\n]", "where:\n- (a), (b), and (c) are the lengths of the triangle’s sides\n- (s) is the semi-perimeter of the triangle, calculated as (s = \frac{a + b + c}{2})", "---", "### Step-by-Step Calculation", "Let’s apply Heron’s formula to a triangle with sides:\n- (a = 13 , \ ext{cm})\n- (b = 14 , \ ext{cm})\n- (c = 15 , \ ext{cm})", "#### 1. Compute the semi-perimeter (s)", "[\ns = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 , \ ext{cm}\n]", "#### 2. Apply Heron’s formula", "Now substitute into the area equation:", "[\n\ ext{Area} = \sqrt{21 \ imes (21 - 13) \ imes (21 - 14) \ imes (21 - 15)}\n]\n[\n\ ext{Area} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Multiply the values inside the square root:", "[\n21 \ imes 8 = 168\n]\n[\n7 \ imes 6 = 42\n]\n[\n168 \ imes 42 = 7056\n]", "Now compute the square root:", "[\n\ ext{Area} = \sqrt{7056} = 84 , \ ext{cm}^2\n]", "---", "### Results and Conclusion", "Using Heron’s formula, we’ve determined that the area of a triangle with sides 13 cm, 14 cm, and 15 cm is exactly:", "[\n\boxed{84 , \ ext{cm}^2}\n]", "This result is both accurate and derived without requiring trigonometric functions or angle measurements—making Heron’s formula a renowned tool in geometry, especially for olympiad problems, surveying, and architectural design.", "If you’re studying triangles or preparing for standardized exams, remember: Heron’s formula is a reliable and efficient method to find the area when side lengths are known.", "---", "Keywords: Heron’s formula, triangle area calculation, sides 13 cm, 14 cm, 15 cm, geometry, semi-perimeter, Heronian formula, area of triangle, math tutorial"]
