Solution: Using Heron's formula, $s = \frac{10 + 13 + 14}{2} = 18.5$. Area $= \sqrt{18.5(18.5-10)(18.5-13)(18.5-14)} = \sqrt{18.5 \times 8.5 \times 5.5 \times 4.5}$. Simplify: $18.5 \times 4.5 = 83.25$, $8.5 \times 5.5 = 46.75$, so area $= \sqrt{83.25 \times 46.75} \approx \sqrt{3890.9375} \approx 62.38$. The shortest altitude corresponds to the longest side (14 units): $h = \frac{2 \times 62.38}{14} \approx 8.91$. Exact calculation yields $h = \frac{2 \times \sqrt{18.5 \times 8.5 \times 5.5 \ti

["How to Calculate the Area and Shortest Altitude of a Triangle Using Heron’s Formula", "Triangles are fundamental shapes in geometry, but calculating their area and important altitudes—especially the shortest one—requires precise tools like Heron’s formula. This SEO-optimized guide explains step-by-step how to use Heron’s formula to find the area and determine the shortest altitude, using an example with sides 10, 13, and 14 units.", "---", "### What is Heron’s Formula?", "Heron’s formula lets you compute the area of any triangle when you know the lengths of all three sides. The formula relies on the semi-perimeter, defined as:", "$$\ns = \frac{a + b + c}{2}\n$$", "---", "### Step 1: Compute the Semi-Perimeter", "Given triangle sides:\n$a = 10$, $b = 13$, $c = 14$", "Calculate the semi-perimeter:\n$$\ns = \frac{10 + 13 + 14}{2} = \frac{37}{2} = 18.5\n$$", "---", "### Step 2: Apply Heron’s Formula for the Area", "The area $A$ is:", "$$\nA = \sqrt{s(s - a)(s - b)(s - c)}\n$$", "Substitute values:\n$$\nA = \sqrt{18.5 \ imes (18.5 - 10) \ imes (18.5 - 13) \ imes (18.5 - 14)}\n= \sqrt{18.5 \ imes 8.5 \ imes 5.5 \ imes 4.5}\n$$", "Compute products step-by-step:\n- $18.5 \ imes 4.5 = 83.25$\n- $8.5 \ imes 5.5 = 46.75$", "Then:\n$$\nA = \sqrt{83.25 \ imes 46.75} = \sqrt{3890.9375} \approx 62.38 \ ext{ square units}\n$$", "---", "### Step 3: Calculate the Shortest Altitude", "The shortest altitude corresponds to the longest side, which is 14 units. The formula for altitude $h$ opposite side $c$ is:", "$$\nh = \frac{2A}{c}\n$$", "Using $A \approx 62.38$ and $c = 14$:\n$$\nh = \frac{2 \ imes 62.38}{14} \approx \frac{124.76}{14} \approx 8.91\n$$", "For a more precise answer, keep the exact square root:\n$$\nh = \frac{2 \ imes \sqrt{3890.9375}}{14} = \frac{\sqrt{3890.9375}}{7}\n$$", "Simplify inside:\n$$\n3890.9375 = \frac{62271}{16}, \quad \sqrt{62271} \approx 249.54\n\Rightarrow h \approx \frac{249.54}{7} \approx 35.65 \quad \ ext{(Wait — this seems inconsistent)}\n$$", "Correction: Since $A = \sqrt{3890.9375} \approx 62.38$, the exact expression is:\n$$\nh = \frac{2 \ imes 62.38}{14} = \frac{124.76}{14} \approx 8.91\n$$", "---", "### Alternative Exact Approach Using Symmetric Formula", "Heron’s formula also offers a simplified exact expression:", "$$\nA = \frac{1}{4} \sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)}\n$$", "Plug in $a=10$, $b=13$, $c=14$:\n$$\nA = \frac{1}{4} \sqrt{37 \ imes 17 \ imes 11 \ imes 9} = \frac{1}{4} \sqrt{62271} \approx \frac{1}{4} \ imes 249.54 = 62.39\n$$", "Then altitude via base 14:\n$$\nh = \frac{2A}{14} = \frac{2 \ imes \sqrt{62271}/4}{14} = \frac{\sqrt{62271}}{28}\n$$", "But for practical clarity, use decimal approximation.", "---", "### Final Altitude Result", "From posterior calculation:", "$$\n\ ext{Shortest altitude } h \approx 8.91 \ ext{ units}\n$$", "Although algebraic simplification does not yield a clean whole number, using numerical approximation gives confidence:", "> Shortest altitude ≈ 8.91 units (exact: $ h = \frac{2 \ imes \sqrt{3890.9375}}{14} $)", "For an exact answer, write:", "$$\n\boxed{h = \frac{2\sqrt{3890.9375}}{14} = \frac{\sqrt{3890.9375}}{7}}\n$$", "Or, recognize the complexity — sometimes changing side lengths improves simplicity. But for 10, 13, 14:", "---", "### ✅ Summary", "- Semi-perimeter: $ s = 18.5 $\n- Area: $ A = \sqrt{3890.9375} \approx 62.38 $\n- Shortest altitude (opposite side 14): $ h = \frac{2 \ imes A}{14} \approx 8.91 $\n- Exact expression: $ h = \frac{2\sqrt{3890.9375}}{14} $", "---", "Why This Matters SEO:\nThis article explains Heron’s formula clearly, addresses a common problem (shortest altitude), uses exact and approximate values, and highlights real geometry applications. It incorporates keywords like “Heron’s formula,” “shortest altitude calculation,” “triangle area from sides,” and “geometric formulas” to improve search visibility for students and geometry learners.", "---", "Note: While the altitude is approximately 8.91, exact symbolic form is easiest maintained as $ \boxed{\dfrac{\sqrt{62271}}{7}} $ or the decimal approximation. Use context-dependent clarity."]