Understanding and Calculating Properties of a Rhombus: Perimeter, Area, and Formula Examples

Rhombus

A rhombus is a quadrilateral with four equal sides

A rhombus is a quadrilateral with four equal sides. It is a special type of parallelogram where all sides are congruent. Additionally, opposite angles of a rhombus are equal, making it a symmetric shape.

Properties of a Rhombus:
1. All sides are congruent.
2. Opposite angles are congruent.
3. Diagonals bisect each other at a 90-degree angle.
4. The diagonals are perpendicular to each other.
5. The diagonals are not congruent unless the rhombus is a square.

Formulas for a Rhombus:
1. Perimeter: The perimeter of a rhombus is calculated by adding up the lengths of all four sides. Thus, P = 4a, where “P” is the perimeter and “a” is the length of one side.
2. Area: The area of a rhombus can be calculated using two different formulas. One way is by multiplying the lengths of the diagonals and dividing by 2. Therefore, A = (d1 * d2) / 2, where “A” is the area and “d1” and “d2” are the lengths of the diagonals. Another way to find the area is by multiplying the base length by the height. Thus, A = base * height.

Example:
Let’s consider a rhombus with a side length of 6 cm and diagonals measuring 8 cm and 10 cm. We can find its perimeter and area using the formulas mentioned above.

1. Perimeter: P = 4a = 4 * 6 cm = 24 cm
Therefore, the perimeter of the rhombus is 24 cm.

2. Area: First, using the formula A = (d1 * d2) / 2:
A = (8 cm * 10 cm) / 2 = 80 cm² / 2 = 40 cm²

Alternatively, using the base and height formula:
To find the height, we can split the rhombus into two congruent right triangles by drawing a perpendicular line from one corner to the opposite side. The base of each triangle would be half of the long diagonal (10 cm / 2 = 5 cm), and the height would be one side (6 cm). Thus, using the formula A = base * height:
A = 5 cm * 6 cm = 30 cm²

Therefore, the area of the rhombus is either 40 cm² or 30 cm², depending on the method used.

Remember, when solving problems involving a rhombus, it is essential to use the correct formulas and understand the properties to find accurate results.

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