To find the area of a rhombus when given the height and one diagonal, you can use two approaches depending on the available information:

1. Using Height and Side (if side is known): The area of a rhombus can be calculated using the formula: $\text{Area} = \text{Base} \times \text{Height}$ But since you only know the height, this method would require the length of a side.

2. Using Diagonals: Another formula for the area of a rhombus is: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the lengths of the diagonals. Here, you know one diagonal ($d_1 = 30 \, \text{cm}$), but you still need the other diagonal ($d_2$).

Approach to Find the Missing Diagonal:

Since the height is given as 24 cm, and the height of the rhombus is the perpendicular distance between two parallel sides, we can use Pythagoras' theorem to find the other diagonal.

Step 1: Relationship Between Diagonal and Height

The height forms a right-angled triangle with half of each diagonal. So, the length of half the unknown diagonal $d_2/2$ can be found using the following relationship: $\left( \frac{d_1}{2} \right)^2 + \text{Height}^2 = \left( \frac{d_2}{2} \right)^2$ Substitute the values: $\left( \frac{30}{2} \right)^2 + 24^2 = \left( \frac{d_2}{2} \right)^2$ $15^2 + 24^2 = \left( \frac{d_2}{2} \right)^2$ $225 + 576 = \left( \frac{d_2}{2} \right)^2$ $801 = \left( \frac{d_2}{2} \right)^2$ $\frac{d_2}{2} = \sqrt{801} \approx 28.3 \, \text{cm}$ Thus, the full length of the second diagonal is: $d_2 \approx 2 \times 28.3 = 56.6 \, \text{cm}$

Step 2: Calculate the Area

Now, using the formula for the area with diagonals: $\text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 30 \times 56.6 = 849 \, \text{cm}^2$

So, the area of the rhombus is approximately 849 cm².

Would you like to explore any of the steps in more detail, or do you have any questions?

Related Questions:

1. How do you find the area of a rhombus if both diagonals are given?
2. What is the relationship between the side length and the diagonals of a rhombus?
3. How would the area change if the height of the rhombus were increased by 10 cm?
4. How does the height relate to the angles of a rhombus?
5. Can you use trigonometry to find the area of a rhombus?

Tip:

In any quadrilateral, if you know both diagonals, you can easily calculate the area by splitting it into two triangles.