Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Heron's Area Formula to find the area of the triangle. $$a=12, \quad b=24, \quad c=18$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 330.68 square units.

Step 1: Compute the semi-perimeter

The semi-perimeter (s) of a triangle with sides of lengths a, b, and c is given by the formula \(s = \frac{a + b + c}{2}\). For this exercise, \(a = 12\), \(b = 24\), and \(c = 18\). Thus, the semi-perimeter is calculated as \(s = \frac{12 + 24 + 18}{2} = 27\).

Step 2: Compute the area

Once the semi-perimeter is known, Heron's formula can be used to compute the area (A) of the triangle. Heron's formula is \(A = \sqrt{s(s-a)(s-b)(s-c)}\). Substituting the known values \(s = 27\), \(a = 12\), \(b = 24\), and \(c = 18\), the area becomes \(A = \sqrt{27(27 - 12)(27 - 24)(27 - 18)} = \sqrt{27 * 15 * 3 * 9}\).

Step 3: Simplify the computation

Simplifying the computation gives \(A = \sqrt{109350} \approx 330.68\), depending on the number of decimal places you want round to.

Key Concepts

Semi-perimeter Calculation

When working with triangles, especially in the context of Heron's Area Formula, the concept of a semi-perimeter is crucial. The semi-perimeter is essentially half of the triangle's perimeter. It acts as an intermediate step when calculating the area of a triangle in a more complex manner than just using the base and height.
To compute the semi-perimeter, use the formula:

• \(s = \frac{a+b+c}{2}\),

where \(a\), \(b\), and \(c\) are the lengths of the triangle's sides.
This calculated semi-perimeter, \(s\), becomes a key player in Heron's formula and allows us to use all three side lengths for determining the triangle's area. By calculating the semi-perimeter, we are setting the stage for the next step in the process – finding the area of the triangle.

Triangle Area

Calculating the area of a triangle can be done in various ways. One common method is Heron's Formula, especially when the triangle's side lengths are known, and not its height. This formula is particularly useful for triangles that are not right-angled.
Heron's Formula states:

• \(A = \sqrt{s(s-a)(s-b)(s-c)}\),

where \(s\) is the semi-perimeter of the triangle and \(a\), \(b\), \(c\) are the lengths of its sides.
Substituting these values into the formula can sometimes seem intimidating, but it’s quite straightforward once the semi-perimeter is known. Each expression in the brackets \((s-a)\), \((s-b)\), and \((s-c)\) represents the difference between the semi-perimeter and each side of the triangle. This calculation ensures all parts of the triangle contribute to determining its area.

Geometric Formulas

Geometric formulas like Heron's provide tools to solve problems involving shapes and spaces whether in mathematics classrooms or practical scenarios. Specifically for triangles, Heron’s formula links algebra with geometry, elegantly calculating the area using only side lengths.
Heron's Formula is less dependent on orientation or additional information like height. It stands apart for its universal applicability to any type of triangle, whether scalene, isosceles, or equilateral.
Each formula in geometry has its unique utility, allowing us to calculate properties like area, perimeter, and volume without altering the shape, which is crucial in both theoretical and applied fields. Understanding these formulas enhances problem-solving skills as they apply to real-world contexts involving structural engineering, land surveying, and many other disciplines.