The area \(A\) of a triangle is given by \(A = \frac{1}{2} \times \text{base} \times \text{height}\). Solving for height, \(45 = \frac{1}{2} \times 9 \times \text{height}\), gives \(\text{height} = \frac{45 \times 2}{9} = 10\) units.

["How to Solve for Height in a Triangle Area Formula: A Step-by-Step Guide", "Understanding how to calculate the area of a triangle is essential in geometry, especially when solving for unknown dimensions like height. The formula for the area (A) of a triangle is:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "This straightforward yet powerful equation allows you to determine the area when two sides are known—and it becomes even more useful when you need to find an unknown side, such as height.", "---", "### The Problem in Practice", "Suppose we know the area of a triangle and one side’s length, but need to find the height. For example:\nIf the area (A = 45) square units and the base is (9) units, how do we find the height?", "---", "### Step 1: Start with the Area Formula", "We begin with the basic formula:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Substituting the known values:", "[\n45 = \frac{1}{2} \ imes 9 \ imes \ ext{height}\n]", "---", "### Step 2: Simplify the Equation", "Multiply both sides of the equation by 2 to eliminate the fraction:", "[\n2 \ imes 45 = 9 \ imes \ ext{height}\n]", "[\n90 = 9 \ imes \ ext{height}\n]", "---", "### Step 3: Solve for Height", "Now divide both sides by 9 to isolate the height:", "[\n\ ext{height} = \frac{90}{9} = 10\n]", "---", "### Final Answer", "The height of the triangle is 10 units.", "---", "### Why This Formula Matters", "knowing how to solve for height is valuable in real-world applications such as architecture, land measurement, and design. This approach uses algebraic manipulation to extract critical geometric information—proving that a simple formula becomes powerful when combined with clear problem-solving steps.", "---", "Summary:\nGiven ( A = \dfrac{1}{2} \ imes \ ext{base} \ imes \ ext{height} ), solving for height in a triangle involves multiplying area and base, dividing by half the base, yielding the height. In our example, ( 45 = \dfrac{1}{2} \ imes 9 \ imes \ ext{height} ) leads to:", "[\n\ ext{height} = \frac{45 \ imes 2}{9} = 10 \ ext{ units}.\n]", "This method is fast, reliable, and fundamental to mastering triangle geometry."]