Study formulas and calculation examples related to rhombuses. These formulas are useful for calculating different properties of a rhombus and solving math problems involving rhombuses, whether it's finding their area, perimeter, diagonal lengths, angles, or the radii of circles related to them.
Definition: A rhombus is a special type of quadrilateral with four equal sides and opposite angles that are congruent.
Rhombus Formulas
Area of a Rhombus (A):
- Formula 1: A = (p * q) / 2 (based on diagonals)
- Formula 2: A = a * h (where 'a' is the length of one side, and 'h' is the height perpendicular to that side)
- Formula 3: A = (e * f) / 2 (where 'e' and 'f' are the lengths of two adjacent sides and the included angle)
- Formula 4: A = s^2 * sin(angle) (where 's' is the length of a side, and 'angle' is one of the acute angles)
Perimeter of a Rhombus (P):
Formula: P = 4 * a (side length)
Description: The perimeter of a rhombus is the sum of all four equal sides.
Diagonal Lengths (d1 and d2):
- Formula for d1: d1 = (2 * A) / q (where 'A' is the area, and 'q' is the length of the horizontal diagonal)
- Formula for d2: d2 = (2 * A) / p (where 'A' is the area, and 'p' is the length of the vertical diagonal) Description: These formulas allow you to calculate the lengths of the diagonals of a rhombus using the area and the length of the other diagonal. You can use either formula depending on the diagonal for which you have information.
Inradius (r):
Formula: r = (A) / (2 * a) (where 'A' is the area, and 'a' is the side length)
Description: The inradius of a rhombus is the radius of the largest circle that can fit inside the rhombus. It can be calculated by dividing the area by twice the side length.
Circumradius (R):
Formula: R = 0.5 * √(p² + q²) (where 'p' and 'q' are the lengths of the diagonals)
Description: The circumradius is the radius of the circle that can be circumscribed around a rhombus. This formula calculates it using the lengths of the diagonals ('p' and 'q').
Side of Rhombus (a):
Formula: a = √((p/2)² + (q/2)²)
Description: This formula calculates the length of one side (a) of a rhombus based on the lengths of its diagonals (p and q).
Rhombus Calculation Examples
Study the next calculation examples for rhombus:
Example 1: Finding the Area of a Rhombus
Let's say you have a rhombus with diagonals of length:
- d1 = 8 inches
- d2 = 6 inches
Using Formula 1: A = (d1 * d2) / 2 A = (8 inches * 6 inches) / 2 A = (48 square inches) / 2 A = 24 square inches
So, the area of the rhombus is 24 square inches.
Example 2: Finding the Perimeter of a Rhombus
Let's say the side length of the rhombus is:
- s = 5 feet
Using the perimeter formula: P = 4 * s P = 4 * 5 feet P = 20 feet
The perimeter of the rhombus is 20 feet.
Example 3: Finding the Length of Diagonals
Suppose you know the area of the rhombus is:
- A = 36 square centimeters
Using Formula for d1: d1 = (2 * A) / d2 d1 = (2 * 36 square centimeters) / 6 centimeters d1 = (72 square centimeters) / 6 centimeters d1 = 12 centimeters
So, the length of d1 is 12 centimeters.
Now, you can use Formula for d2 to find the length of d2: d2 = (2 * A) / d1 d2 = (2 * 36 square centimeters) / 12 centimeters d2 = (72 square centimeters) / 12 centimeters d2 = 6 centimeters
The length of d2 is 6 centimeters.
Example 4: Finding the Inradius
Assuming you have the area and the side length of the rhombus:
- A = 48 square inches
- s = 4 inches
Using the inradius formula: r = A / s r = 48 square inches / 4 inches r = 12 square inches / inch r = 12 inches
The inradius of the rhombus is 12 inches.
Example 5: Finding the Circumradius
Let's assume you know the lengths of the diagonals:
- d1 = 10 feet
- d2 = 8 feet
Using the circumradius formula: R = 1/2 × √(d1² + d2²) R = 1/2 × √((10 feet)² + (8 feet)²) R = 1/2 × √(100 square feet + 64 square feet) R = 1/2 × √(164 square feet) R = 1/2 × 12.81 feet (approximately)
So, the circumradius of the rhombus is approximately 6.41 feet.
Example 6: Finding the Side Length of a Rhombus
Let's assume you know the lengths of the diagonals:
- p = 7 inches
- q = 9 inches
Using the side length formula: a = [√(p^2 + q^2)] / 2 a = [√((7 inches)^2 + (9 inches)^2)] / 2 a = [√(49 square inches + 81 square inches)] / 2 a = [√(130 square inches)] / 2 a = (√130 square inches) / 2 a ≈ 6.40 inches
The length of one side of the rhombus is approximately 6.40 inches.
Rhombuses Real-life calculations
Real-Life Calculation 1: Finding the Area of a Baseball Diamond
A baseball diamond is typically a square with two 90-degree rhombuses added to each side. The length of each side of the square is 90 feet, and the length of each rhombus side is 60 feet.
Using Formula 2 for the area of a rhombus (A = a * h), where 'a' is the length of one side and 'h' is the height perpendicular to that side:
For the square part:
- a = 90 feet
- h = 90 feet (same as the side length of the square)
Area of the square = a * h = 90 feet * 90 feet = 8100 square feet
For each rhombus:
- a = 60 feet
- h = 90 feet (the same height as the square)
Area of one rhombus = a * h = 60 feet * 90 feet = 5400 square feet
Since there are two identical rhombuses, the total area contributed by the rhombuses is 2 * 5400 square feet = 10800 square feet.
Now, add the area of the square and the area of the rhombuses to find the total area of the baseball diamond:
Total Area = Area of Square + Area of Rhombuses = 8100 square feet + 10800 square feet = 18900 square feet
So, the total area of a baseball diamond is 18,900 square feet.
Real-Life Calculation 2: Finding the Inradius of a Traffic Sign
Imagine a traffic sign in the shape of a rhombus. The sign has an area of 12 square feet, and the length of one of its sides is 4 feet.
Using the inradius formula: r = A / s
- A = 12 square feet (area of the rhombus)
- s = 4 feet (length of one side)
r = 12 square feet / 4 feet = 3 feet
The inradius of the traffic sign is 3 feet. This means that the largest circle that can fit inside the rhombus-shaped traffic sign has a radius of 3 feet.