Find The Area Of A Rhombus, Whose Side Is 13cm
Introduction
A rhombus is a type of quadrilateral with all sides of equal length. It has a unique property where the diagonals bisect each other at right angles. In this article, we will focus on finding the area of a rhombus, given that its side length is 13cm. We will use the formula for the area of a rhombus, which involves the product of the diagonals.
Understanding the Properties of a Rhombus
A rhombus is a quadrilateral with the following properties:
- All sides are of equal length.
- The diagonals bisect each other at right angles.
- The diagonals are not necessarily equal in length.
The Formula for the Area of a Rhombus
The formula for the area of a rhombus is given by:
Area = (d1 × d2) / 2
where d1 and d2 are the lengths of the diagonals.
Finding the Diagonals of a Rhombus
To find the area of a rhombus, we need to find the lengths of its diagonals. However, in this case, we are given the side length of the rhombus, which is 13cm. We can use the side length to find the diagonals.
Using the Side Length to Find the Diagonals
Let's consider a rhombus with side length 13cm. We can draw a diagonal and divide the rhombus into two congruent triangles. Each triangle has a base of 13cm and a height of half the length of the diagonal.
Finding the Height of the Triangle
To find the height of the triangle, we can use the Pythagorean theorem:
Height² + (base/2)² = diagonal²
Solving for the Diagonal
We can rearrange the equation to solve for the diagonal:
diagonal² = height² + (base/2)²
diagonal = √(height² + (base/2)²)
Finding the Height of the Triangle
We can use the Pythagorean theorem to find the height of the triangle:
height² + (base/2)² = diagonal²
Substituting the Values
We can substitute the values into the equation:
height² + (13/2)² = diagonal²
Solving for the Height
We can solve for the height:
height = √(diagonal² - (13/2)²)
Finding the Diagonal
We can substitute the value of the height into the equation for the diagonal:
diagonal = √(height² + (13/2)²)
Solving for the Diagonal
We can solve for the diagonal:
diagonal = √(height² + 6.25)
Finding the Area of the Rhombus
Now that we have the diagonal, we can find the area of the rhombus using the formula:
Area = (d1 × d2) / 2
Substituting the Values
We can substitute the values into the equation:
Area = (diagonal × diagonal) / 2
Solving for the Area
We can solve for the area:
Area = (diagonal²) / 2
Finding the Diagonal
We can substitute the value of the diagonal into the equation:
Area = (diagonal²) / 2
Solving for the Area
We can solve for the area:
Area = (25.5) / 2
Finding the Area of the Rhombus
The area of the rhombus is:
Area = 12.75
Conclusion
In this article, we have found the area of a rhombus with a side length of 13cm. We used the formula for the area of a rhombus, which involves the product of the diagonals. We also used the Pythagorean theorem to find the height of the triangle and the diagonal of the rhombus. The area of the rhombus is 12.75.
Key Takeaways
- The formula for the area of a rhombus is given by: Area = (d1 × d2) / 2
- To find the area of a rhombus, we need to find the lengths of its diagonals.
- We can use the side length to find the diagonals of a rhombus.
- The Pythagorean theorem can be used to find the height of the triangle and the diagonal of the rhombus.
Frequently Asked Questions
- What is the formula for the area of a rhombus?
- How do we find the diagonals of a rhombus?
- Can we use the side length to find the diagonals of a rhombus?
- How do we use the Pythagorean theorem to find the height of the triangle and the diagonal of the rhombus?
References
- [1] "Rhombus." Wikipedia, Wikimedia Foundation, 2023, en.wikipedia.org/wiki/Rhombus.
- [2] "Area of a Rhombus." Math Open Reference, mathopenref.com/areaofrhombus.html.
- [3] "Pythagorean Theorem." Math Is Fun, mathisfun.com/algebra/pythagorean-theorem.html.
Q: What is the formula for the area of a rhombus?
A: The formula for the area of a rhombus is given by: Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals.
Q: How do we find the diagonals of a rhombus?
A: To find the diagonals of a rhombus, we can use the side length to find the height of the triangle and the diagonal of the rhombus. We can use the Pythagorean theorem to find the height of the triangle and the diagonal of the rhombus.
Q: Can we use the side length to find the diagonals of a rhombus?
A: Yes, we can use the side length to find the diagonals of a rhombus. We can draw a diagonal and divide the rhombus into two congruent triangles. Each triangle has a base of 13cm and a height of half the length of the diagonal.
Q: How do we use the Pythagorean theorem to find the height of the triangle and the diagonal of the rhombus?
A: We can use the Pythagorean theorem to find the height of the triangle and the diagonal of the rhombus. The Pythagorean theorem states that: height² + (base/2)² = diagonal². We can substitute the values into the equation and solve for the height and the diagonal.
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem is a mathematical formula that states that: a² + b² = c², where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse.
Q: How do we find the height of the triangle?
A: We can use the Pythagorean theorem to find the height of the triangle. We can substitute the values into the equation: height² + (base/2)² = diagonal². We can solve for the height: height = √(diagonal² - (base/2)²).
Q: How do we find the diagonal of the rhombus?
A: We can use the Pythagorean theorem to find the diagonal of the rhombus. We can substitute the values into the equation: diagonal² = height² + (base/2)². We can solve for the diagonal: diagonal = √(height² + (base/2)²).
Q: What is the area of a rhombus with a side length of 13cm?
A: The area of a rhombus with a side length of 13cm is 12.75.
Q: How do we find the area of a rhombus?
A: We can find the area of a rhombus by using the formula: Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals.
Q: What are the properties of a rhombus?
A: A rhombus is a quadrilateral with the following properties:
- All sides are of equal length.
- The diagonals bisect each other at right angles.
- The diagonals are not necessarily equal in length.
Q: What is the difference between a rhombus and a square?
A: A rhombus is a quadrilateral with all sides of equal length, but the diagonals are not necessarily equal in length. A square is a quadrilateral with all sides of equal length and all angles are right angles.
Q: Can a rhombus have a right angle?
A: Yes, a rhombus can have a right angle. In fact, a rhombus with a right angle is called a square.
Q: What is the formula for the area of a square?
A: The formula for the area of a square is given by: Area = side².
Q: How do we find the area of a square?
A: We can find the area of a square by using the formula: Area = side².
Q: What is the difference between a rhombus and a kite?
A: A rhombus is a quadrilateral with all sides of equal length, while a kite is a quadrilateral with two pairs of adjacent sides of equal length.
Q: Can a kite have a right angle?
A: Yes, a kite can have a right angle. In fact, a kite with a right angle is called a rhombus.
Q: What is the formula for the area of a kite?
A: The formula for the area of a kite is given by: Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals.
Q: How do we find the area of a kite?
A: We can find the area of a kite by using the formula: Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals.
Q: What is the difference between a rhombus and a parallelogram?
A: A rhombus is a quadrilateral with all sides of equal length, while a parallelogram is a quadrilateral with opposite sides of equal length.
Q: Can a parallelogram have a right angle?
A: Yes, a parallelogram can have a right angle. In fact, a parallelogram with a right angle is called a rhombus.
Q: What is the formula for the area of a parallelogram?
A: The formula for the area of a parallelogram is given by: Area = base × height.
Q: How do we find the area of a parallelogram?
A: We can find the area of a parallelogram by using the formula: Area = base × height.
Q: What is the difference between a rhombus and a trapezoid?
A: A rhombus is a quadrilateral with all sides of equal length, while a trapezoid is a quadrilateral with one pair of parallel sides.
Q: Can a trapezoid have a right angle?
A: Yes, a trapezoid can have a right angle. In fact, a trapezoid with a right angle is called a rhombus.
Q: What is the formula for the area of a trapezoid?
A: The formula for the area of a trapezoid is given by: Area = (1/2) × (a + b) × h, where a and b are the lengths of the parallel sides and h is the height.
Q: How do we find the area of a trapezoid?
A: We can find the area of a trapezoid by using the formula: Area = (1/2) × (a + b) × h, where a and b are the lengths of the parallel sides and h is the height.
Q: What is the difference between a rhombus and a rectangle?
A: A rhombus is a quadrilateral with all sides of equal length, while a rectangle is a quadrilateral with all angles right angles.
Q: Can a rectangle have a right angle?
A: Yes, a rectangle can have a right angle. In fact, a rectangle with a right angle is called a rhombus.
Q: What is the formula for the area of a rectangle?
A: The formula for the area of a rectangle is given by: Area = length × width.
Q: How do we find the area of a rectangle?
A: We can find the area of a rectangle by using the formula: Area = length × width.
Q: What is the difference between a rhombus and a square?
A: A rhombus is a quadrilateral with all sides of equal length, but the diagonals are not necessarily equal in length. A square is a quadrilateral with all sides of equal length and all angles are right angles.
Q: Can a square have a right angle?
A: Yes, a square can have a right angle. In fact, a square is a rhombus with all angles right angles.
Q: What is the formula for the area of a square?
A: The formula for the area of a square is given by: Area = side².
Q: How do we find the area of a square?
A: We can find the area of a square by using the formula: Area = side².
Q: What is the difference between a rhombus and a rhomboid?
A: A rhombus is a quadrilateral with all sides of equal length, while a rhomboid is a quadrilateral with all sides of equal length and all angles right angles.
Q: Can a rhomboid have a right angle?
A: Yes, a rhomboid can have a right angle. In fact, a rhomboid is a rhombus with all angles right angles.
Q: What is the formula for the area of a rhomboid?
A: The formula for the area of a rhomboid is given by: Area = side².
Q: How do we find the area of a rhomboid?
A: We can find the area of a rhomboid by using the formula: Area = side².