Question:** An electrical engineer designs a triangular circuit board where the sides are \(7\) units, \(24\) units, and \(25\) units. Find the area of the triangle using Heron's formula.
["Title: Calculating the Area of a Triangle with Sides 7, 24, and 25 Using Heron’s Formula", "When designing innovative electronic components, precise geometric calculations are essential—especially for triangular circuit boards. One such compelling example involves a triangular circuit board with side lengths of 7 units, 24 units, and 25 units. Can this triangle be used effectively in circuit design? More importantly, how do engineers compute its area accurately? This article demonstrates the solution using Heron’s formula, a powerful method for finding the area of any triangle when all three side lengths are known.", "---", "### Step 1: Verify the Triangle Is Valid", "Before applying Heron’s formula, confirm the triangle is valid by checking the triangle inequality:", "- (7 + 24 = 31 > 25)\n- (7 + 25 = 32 > 24)\n- (24 + 25 = 49 > 7)", "Since all conditions hold, a triangle with these side lengths exists and is valid.", "---", "### Step 2: Calculate the Semi-Perimeter (s)", "Heron’s formula requires the semi-perimeter, denoted (s), defined as half the sum of the triangle’s three sides:", "[\ns = \frac{a + b + c}{2}\n]", "Here, (a = 7), (b = 24), (c = 25). Substitute:", "[\ns = \frac{7 + 24 + 25}{2} = \frac{56}{2} = 28\n]", "So, the semi-perimeter is (28) units.", "---", "### Step 3: Apply Heron’s Formula", "Heron’s formula for the area (A) of a triangle with sides (a), (b), and (c) is:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Plug in the known values:", "[\nA = \sqrt{28(28 - 7)(28 - 24)(28 - 25)}\n]", "Simplify each term inside the square root:", "- (s - a = 28 - 7 = 21)\n- (s - b = 28 - 24 = 4)\n- (s - c = 28 - 25 = 3)", "Now compute:", "[\nA = \sqrt{28 \ imes 21 \ imes 4 \ imes 3}\n]", "Group and simplify step-by-step:", "- (28 \ imes 4 = 112)\n- (21 \ imes 3 = 63)\n- (112 \ imes 63 = ?)", "Break down multiplication:", "[\n112 \ imes 63 = (112 \ imes 60) + (112 \ imes 3) = 6720 + 336 = 7056\n]", "Now take the square root:", "[\nA = \sqrt{7056} = 84\n]", "---", "### Final Result", "The area of the triangular circuit board with side lengths 7, 24, and 25 is exactly 84 square units.", "---", "### Why This Matters for Electrical Engineers", "Precise area calculations help engineers estimate material usage, heat dissipation across board surfaces, and placement of components requiring flat, stable mounting areas. Using Heron’s formula ensures accuracy even for scalene triangles—common in custom PCB layouts—without needing angle measurements.", "---", "### Conclusion", "Leveraging Heron’s formula allows electrical engineers to confidently compute triangle areas from side lengths alone. For a triangle with sides 7, 24, and 25, the area is 84 units²—simple to calculate and vital for designing efficient and reliable circuit boards.", "Keywords: triangular circuit board, electrical engineering, Heron’s formula, area calculation, SI units, PCB design, triangle area formula, 7-24-25 triangle, geometry in engineering, heron’s formula example, 84 square units.", "---", "### Additional Resources", "- Heron’s Formula Explanation\n- Applications of triangular geometry in electronics\n- Calculating board area for PCB layout tips", "---", "By mastering geometric tools like Heron’s formula, engineers strengthen their technical foundation—turning exact measurements into reliable innovation."]
