Calculate the Area of a Rhombus with Diagonal Length 5 and Height 3

Video Solution

Solution Steps

00:00 Find the area of the rhombus

00:03 In the rhombus, the diagonals are perpendicular to each other

00:10 We'll use the Pythagorean theorem in triangle BEC

00:16 We'll substitute appropriate values and solve to find EC

00:25 This is the length of EC

00:31 We'll use the formula to calculate the area of a rhombus

00:35 (diagonal times diagonal) divided by 2

00:40 The diagonal equals the sum of its segments

00:50 In the rhombus, the diagonals intersect each other

01:07 Let's simplify what we can

01:15 We'll substitute appropriate values and solve to find the area

01:19 And this is the solution to the problem

Step-by-Step Solution

Remember there are two options to calculate the area of a rhombus:

1: The diagonal multiplied by the diagonal divided by 2.

2: The base multiplied by the height.

In the question, we are only given the data for one of the diagonals and one of the sides, which means we cannot use either of the above formulas.

We need to find more data. Let's begin by finding the second diagonal:

Remember that the diagonals of a rhombus are perpendicular to one another, which means that they form a 90-degree angle.

Therefore, all the triangles in a rhombus are right-angled.

Now we can focus on the triangle where the side and the height are given, and we will calculate the third side using the Pythagorean theorem:

$a²+b²=c²$ Insert the given data:

$3^2+x^2=5^2$ $9+x^2=25$ $x^2=25-9=16$ $x=\sqrt{16}=4$

Now that we have found the second half of the diagonal, we can calculate the area of the rhombus by multiplying the two diagonals together.

Since the diagonals in a rhombus are perpendicular and cross each other, they are equal. Hence, our diagonals are equal:

$3+3=6$ $4+4=8$ Therefore, the area of the rhombus is:

$\frac{6\times8}{2}=\frac{48}{2}=24$