Calculating the Area of a Rhombus Given the Diagonal Ratio and Side Length

The importance of accurate calculations in geometry: Understanding how to calculate the area of complex shapes like a rhombus is a fundamental skill in geometry. This article will teach you how to determine the area of a rhombus when you know the ratio of its diagonals and the length of its sides. This knowledge is applicable not only in academic settings but also in practical situations where such geometrical calculations are essential.

The area of a rhombus can be calculated using the formula Area (d_1 * d_2) / 2, where d_1 and d_2 are the lengths of the diagonals. In this article, we will solve a specific problem to demonstrate the process step-by-step.

Problem Statement

Given a rhombus where the ratio of the diagonals is 3:4 and the side length is 40 units, find the area of the rhombus.

Step-by-Step Solution

Step 1: Establishing the Relationship Between the Diagonals

Let's denote the lengths of the diagonals as d_1 and d_2. According to the given ratio, we have:

d_1 3k d_2 4k

where k is a constant.

Step 2: Using the Side Length to Find k

Given that the diagonals bisect each other at right angles, we can use the Pythagorean theorem to relate the side length to the diagonals. The side length s 40 units.

The Pythagorean theorem states that:

s sqrt{left(frac{d_1}{2}right)^2 left(frac{d_2}{2}right)^2}

Substituting the expressions for d_1 and d_2, we get:

40 sqrt{left(frac{3k}{2}right)^2 left(frac{4k}{2}right)^2}

40 sqrt{left(frac{3k}{2}right)^2 left(2kright)^2}

40 sqrt{frac{9k^2}{4} 4k^2}

40 sqrt{frac{9k^2 16k^2}{4}}

40 sqrt{frac{25k^2}{4}}

40 frac{5k}{2}

Multiplying both sides by 2:

80 5k

Dividing by 5:

k 16

Step 3: Calculating the Lengths of the Diagonals

Now we can find the lengths of the diagonals:

d_1 3k 3 * 16 48 units d_2 4k 4 * 16 64 units

Step 4: Calculating the Area

Finally, we can calculate the area using the formula for the area of a rhombus:

Area (d_1 * d_2) / 2

Area (48 * 64) / 2 3072 / 2 1536 square units

Conclusion

The area of the rhombus is 1536 square units.