Table of Contents
Introduction to Area of parallelogram
The area of a parallelogram is a fundamental geometric concept that measures the space enclosed by the sides of a parallelogram. Just as we use the length of a line to understand its extent, the area provides insight into the size of a flat, quadrilateral shape. It’s found by multiplying the length of one side (base) by the perpendicular distance (altitude) between the base and the opposite side. The formula, A = base × altitude, applies universally to all parallelograms. Understanding the area of a parallelogram is essential in various fields, from geometry and engineering to physics and architecture, where accurate spatial measurements are vital.
What is the area of parallelogram
The formula for calculating the area of a parallelogram is:
base height = area
The “base” of the parallelogram is represented by one of its parallel sides, while the “height” is the perpendicular distance between the two parallel sides. The height is measured perpendicular to the base.
If you have a parallelogram with a base (b) and a height (h), the area (A) is provided by:
A = b × h
Understanding and estimating the area of a parallelogram is essential in geometry, engineering, and other professions that need spatial measurements.
Area of the parallelogram formula without height
If you want to find the area of a parallelogram without knowing the height directly, you can use the length of both the base and one of the sides along with the sine of the angle between them. The formula for finding the area of a parallelogram using these parameters is:
Area = base × side × sin(θ)
Here’s how you can do it: Identify the length of the base of the parallelogram (b) and the length of one of the sides adjacent to the base (s).
Measure the angle (θ) between the base and the chosen side. Make sure to use the angle in radians if you’re using a calculator that requires trigonometric functions in radians.
Use the formula: Area = b × s × sin(θ)
This method is based on the fact that the area of a parallelogram is equal to the product of the base, the adjacent side, and the sine of the angle between them. It’s particularly useful when you don’t have the height of the parallelogram readily available.
Remember to ensure that the angle used in the formula is the angle between the base and the chosen side, and that the lengths of the base and side are measured consistently with each other.
How to find the area of parallelogram
To find the area of a parallelogram, you can use one of the following methods, depending on the information you have:
Method 1: Using Base and Height
Identify the length of the base (b) of the parallelogram.
Measure the perpendicular distance (h) between the base and the opposite side. This distance is called the height.
Use the formula: Area = base × height
Area = b × h
Method 2: Using Base, Side, and Angle
Identify the length of the base (b) of the parallelogram.
Measure the length of one of the sides adjacent to the base (s).
Measure the angle (θ) between the base and the chosen side. Ensure the angle is in radians if using trigonometric functions.
Use the formula: Area = base × side × sin(θ)
Area = b × s × sin(θ)
These methods work for any parallelogram, whether you have the height or angle available. Just make sure to use the appropriate formula based on the information you have.
Area of the parallelogram with vectors
The area of a parallelogram can also be calculated using vectors. Given two vectors a and b that form adjacent sides of the parallelogram, the area (A) of the parallelogram they define is given by the magnitude of their cross product:
A = |a × b|
Here’s how you can calculate it step by step:
Compute the cross product of the two vectors a and b.
a × b = |a| |b| sin(θ) n̂
Where |a| and |b| are the magnitudes of the vectors, θ is the angle between them, and n̂ is the unit vector perpendicular to the plane defined by a and b.
Find the magnitude of the cross product:
A = |a × b|
Calculating the area of a parallelogram using vectors is particularly useful when you have vector representations of the sides or diagonals of the parallelogram. It provides an alternative approach to finding the area without needing the height or angles.
To find the area of a parallelogram using its diagonals in vector form, you can use the following formula:
Area = 0.5 * |d₁ × d₂|
Here, d₁ and d₂ are the vectors representing the diagonals of the parallelogram, and d₁ × d₂ is their cross product. The magnitude of the cross product gives you the area of the parallelogram.
Also Check For:
Solved Examples on Area of Parallelogram
Example 1:
Given a parallelogram with a base length of 6 units and a height of 4 units, calculate its area.
Solution: Area = base × height
Area = 6 × 4 = 24 square units
Example 2:
Consider a parallelogram with a base length of 8 units, an adjacent side length of 5 units, and an angle of 60 degrees between them. Find its area.
Solution:
Area = base × side × sin(θ)
Area = 8 × 5 × sin(60°)
Area = 40 × √3 / 2
Area ≈ 34.64 square units
Example 3:
Suppose you have a parallelogram with diagonals of lengths 10 units and 8 units, forming an angle of 45 degrees between them. Determine its area.
Solution:
Area = 0.5 × d₁ × d₂ × sin(θ) = 0.5 × 10 × 8 × sin(45°)
= 0.5 × 10 × 8 × √2 / 2 = 20√2
Area = 28.28 square units
Example 4:
Given two diagonal vectors d₁ = (3, 4) and d₂ = (-2, 5), calculate the area of the parallelogram they form.
Solution:
Calculate the cross product:
d₁ × d₂ = 3 * 5 – 4 * (-2) = 15 + 8 = 23
Find the magnitude of the cross product:
Area = 0.5 * |d₁ × d₂| = 0.5 * 23 = 11.5 square units
Frequently Asked Questions on Area of parallelogram
What is parallelogram?
A parallelogram is a four-sided polygon with opposite sides that are parallel and equal in length. Its opposite angles are also equal. This geometric shape has various properties, including diagonals that bisect each other, and is commonly used in mathematics, geometry, and real-world applications like architecture and engineering.
What is the area of parallelogram?
The area of a parallelogram is the measure of the space enclosed by its sides. It's calculated using formulas involving the base (a side) and height (perpendicular distance between sides) or using the lengths of diagonals and angles. The area quantifies the two-dimensional extent of the shape in a plane.
What is the perimeter of the parallelogram ?
The perimeter of a parallelogram is the sum of the lengths of all its sides. It involves adding the lengths of the parallel sides and multiplying by 2. The perimeter represents the total distance around the outer boundary of the parallelogram in a two-dimensional space.
What is the area of the parallelogram whose height is 5 cm, and base is 4 cm?
The area (A) of a parallelogram can be found using the formula: Area = base × height Given a height of 5 cm and a base of 4 cm: Area = 4 cm × 5 cm = 20 square cm The area of the parallelogram is 20 square centimetres.
How do you find the area of parallelogram using vectors?
To find the area of a parallelogram using vectors, follow these steps: Identify Diagonal Vectors: Select two non-adjacent sides of the parallelogram as diagonal vectors. These will form the diagonals of the parallelogram. Calculate the Cross Product: Compute the cross product of the two diagonal vectors. The resulting vector will be perpendicular to the plane of the parallelogram. Find the Magnitude: Determine the magnitude (length) of the cross product vector. This magnitude represents the area of the parallelogram. Calculate Area: The area of the parallelogram is half the magnitude of the cross product: Area = 0.5 * |d₁ × d₂| Where d₁ and d₂ are the diagonal vectors. This approach leverages vector properties and avoids the need for base-height measurements, providing a concise method to calculate the parallelogram's area.
What is the formula for the area of parallelogram of vectors
The formula for calculating the area of a parallelogram using vectors involves the cross product of two vectors. If you have two vectors a and b that form adjacent sides of the parallelogram, the area (A) of the parallelogram they define is given by the magnitude of their cross product: A = |a × b| Here, a × b represents the cross product of the vectors a and b, and |a × b| is the magnitude of their cross product. This formula allows you to find the area of a parallelogram directly using vector operations, without needing traditional measurements like base and height.
How to find the area of the parallelogram when four vertices are give?
To find the area of a parallelogram when the vertices are given, follow these steps: Identify Vertices: Write down the coordinates of the given vertices of the parallelogram. Let's call these vertices A, B, C, and D. Find Vectors: Calculate the vectors representing two sides of the parallelogram. For example, if you have vertices A and B, the vector AB can be found as: AB = B - A. Calculate Cross Product: Compute the cross product of the two vectors you obtained. The magnitude of the cross product is the area of the parallelogram: Area = |AB × AC| (or any other pair of non-parallel sides). Determine Area: Find the magnitude of the cross product calculated in the previous step. This value represents the area of the parallelogram. Using vector methods, this approach allows you to calculate the area directly from the coordinates of the vertices, without needing the traditional base-height measurements.
What will be the formula for the area of the parallelogram?
The formula for finding the area of a parallelogram using vectors involves the cross product of two vectors that represent its sides. If you have two vectors a and b representing adjacent sides of the parallelogram, the area (A) of the parallelogram is given by the magnitude of their cross product: A = |a × b| Here, a × b represents the cross product of vectors a and b, and |a × b| is the magnitude of that cross product. This vector-based formula allows you to calculate the area of a parallelogram without needing traditional measurements like base and height.
