Linear Algebra: Ch 2 – Determinants (47 of 48) Area of Parallelogram=? (Using Matrices)
Linear Algebra: Ch 2 – Determinants (47 of 48) Area of Parallelogram=? (Using Matrices)

How to find Area of a Parallelogram: Formula, Examples, Worksheet

Get the free Area of a Parallelogram worksheet and other resources for teaching & understanding how to find Area of a Parallelogram

Key Points about Area of a Parallelogram

  • A parallelogram is a two-dimensional shape with two pairs of parallel sides.
  • The formula for calculating the area of a parallelogram is base times height.
  • The height of a parallelogram is the perpendicular distance between the parallel sides.

A Quick Summary for finding Area of a Parallelogram

The area of a parallelogram is a basic concept in geometry that is used to calculate the amount of space enclosed by a parallelogram. A parallelogram is a two-dimensional shape that has two pairs of parallel sides. It is a type of quadrilateral, a four-sided polygon, and has various properties that make it unique. The formula for calculating the area of a parallelogram is simple and easy to use, making it an essential tool for solving problems in mathematics and other fields.

To understand the area of a parallelogram, it is important to first define what a parallelogram is. A parallelogram is a quadrilateral with opposite sides parallel to each other. This means that the opposite sides are equal in length and the opposite angles are also equal. The height of a parallelogram is the perpendicular distance between the parallel sides, and it is used to calculate the area of the parallelogram.

The formula for the area of a parallelogram is base times height, which is the same as the formula for the area of a rectangle. This is because a parallelogram can be divided into two congruent triangles, and the area of each triangle is half the area of the parallelogram. The height of the parallelogram is the length of the line segment perpendicular to the base, and it is used to calculate the area of the parallelogram.

A parallelogram is a 2-D four sided figure that has two different sets of parallel sides. The sides that are across from each other are always the same length in size. The angles that are across from each other always have the same degree measure. In order to find the Area of a Parallelogram, you have to multiply the length of the base by the length of the height. Finding the Area of a Parallelogram follows the same formula as finding the area of a rectangle. The difference is that a parallelogram may have slanted sides that are not at a 90 degree angle, so you cannot use the side length as the height. The height will always be at a 90 degree angle, or perpendicular, to the base.Typically, you will see a dotted line in a parallelogram that represents the height.

Common Core Standard: 6.G.1
Related Topics: Area of a Triangle, Area of a Trapezoid, Area of Composite Figures
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Definition of a Parallelogram

A parallelogram is a two-dimensional geometric shape that falls under the category of quadrilaterals. It is defined as a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are congruent and parallel to each other.

The properties of a parallelogram include the following:

  • Sides: A parallelogram has four sides that are opposite and parallel to each other. The opposite sides are congruent, which means they have the same length.

  • Base: The base of a parallelogram is any one of its sides. The length of the base is the distance between the parallel sides.

  • Height: The height of a parallelogram is the perpendicular distance between the base and the opposite side. It is also known as the altitude.

  • Angle: The opposite angles of a parallelogram are congruent. This means that if one angle measures x degrees, then the opposite angle also measures x degrees.

  • Opposite Sides: The opposite sides of a parallelogram are parallel and congruent. This means that if one side measures x units, then the opposite side also measures x units.

  • Diagonals: The diagonals of a parallelogram are the line segments that connect the opposite vertices. The diagonals of a parallelogram bisect each other.

  • Vertices: A parallelogram has four vertices, which are the points where the sides of the parallelogram intersect.

  • Interior Angles: The sum of the interior angles of a parallelogram is always equal to 360 degrees.

In summary, a parallelogram is a quadrilateral with two pairs of parallel sides. It has four sides, a base, a height, opposite angles, opposite sides, diagonals, vertices, and interior angles. Understanding the properties of a parallelogram is essential when calculating its area.

Area of a Parallelogram Formula

The formula for calculating the area of a parallelogram is simple and straightforward. It involves multiplying the base of the parallelogram by its height. The formula can be written as:

Area of a Parallelogram = Base x Height

The base of a parallelogram is the length of one of its sides, and the height is the perpendicular distance from the base to the opposite side. The height can be drawn from any vertex of the parallelogram to the opposite side, as long as it is perpendicular to the base.

In some cases, the height of the parallelogram may not be given directly. In such cases, the height can be calculated using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Another way to calculate the height is to use trigonometry. If the angle between the base and the height is known, the height can be calculated using the sine or cosine functions.

It is worth noting that the formula for the area of a parallelogram is the same as that for a rectangle. This is because a parallelogram can be divided into a rectangle and two congruent triangles. Therefore, the area of a parallelogram can also be calculated using the formula for a rectangle, which is:

Area of a Rectangle = Length x Width

Where the length and width are the two adjacent sides of the rectangle.

In summary, the formula for calculating the area of a parallelogram is simple and involves multiplying the base by the height. The height can be calculated using the Pythagorean theorem or trigonometry if it is not given directly. The formula for a rectangle can also be used to calculate the area of a parallelogram.

Area of a Parallelogram Without Height

When calculating the area of a parallelogram, the height is usually required. However, there are situations where the height is not provided, and it can be challenging to calculate the area. Fortunately, there are alternative methods to find the area of a parallelogram without using height.

One method to calculate the area of a parallelogram without height is to use trigonometry. If the length of the two adjacent sides and the included angle are known, the area can be calculated using the formula:

Area = a * b * sin(θ)

Where a and b are the lengths of the adjacent sides, and θ is the included angle.

Another method is to use the Pythagorean theorem to find the height. If the length of the two adjacent sides and one angle are known, the height can be calculated using the formula:

h = b * sin(θ)

Where b is the length of the adjacent side, and θ is the included angle. Once the height is found, the area can be calculated using the formula:

Area = b * h

Where b is the length of the adjacent side, and h is the height.

It is important to note that these methods only work for parallelograms with acute angles. For obtuse angles, the height falls outside the parallelogram, and the formula cannot be used.

In summary, there are alternative methods to calculate the area of a parallelogram without using height, such as using trigonometry or the Pythagorean theorem. However, these methods only work for parallelograms with acute angles.

4 Quick Steps for Solving Area of a Parallelogram Examples

To understand how to calculate the area of a parallelogram, let’s take a look at some examples.

  1. Determine the length of the base of the parallelogram examples.
  2. Determine the height of the parallelogram.
  3. Multiple the length of the base by the height of the parallelogram.
  4. Make sure you use the correct units once you know the parallelogram area.

Example 1:

Find the area of a parallelogram with base 8 cm and height 5 cm.

Solution:

Using the formula for the area of a parallelogram, A = base × height, we can plug in the given values to get:

A = 8 cm × 5 cm = 40 cm²

Therefore, the area of this parallelogram is 40 square centimeters.

Example 2:

Suppose a parallelogram has a base of 12 inches and a height of 7 inches. Find its area.

Solution:

Using the formula for the area of a parallelogram, A = base × height, we can plug in the given values to get:

A = 12 in × 7 in = 84 in²

Therefore, the area of this parallelogram is 84 square inches.

Example 3:

Consider a parallelogram that has a base of 10 meters and a height of 6 meters. Find its area.

Solution:

Using the formula for the area of a parallelogram, A = base × height, we can plug in the given values to get:

A = 10 m × 6 m = 60 m²

Therefore, the area of this parallelogram is 60 square meters.

Example 4:

Suppose a parallelogram has a base of 5 units and a height of 3 units. Find its area in terms of square units.

Solution:

Using the formula for the area of a parallelogram, A = base × height, we can plug in the given values to get:

A = 5 units × 3 units = 15 square units

Therefore, the area of this parallelogram is 15 square units.

In conclusion, calculating the area of a parallelogram is a simple process that involves multiplying the base by the height. By understanding this concept, one can easily find the area of any parallelogram, regardless of its size or orientation.

Types of Parallelograms

Parallelograms are a type of quadrilateral that has two pairs of parallel sides. There are three special types of parallelograms: rectangle, rhombus, and square.

Rectangle

A rectangle is a parallelogram with four right angles. It has two pairs of parallel sides of equal length. The opposite sides of a rectangle are equal in length, and the diagonals bisect each other. A rectangle is also an equiangular parallelogram, meaning all four angles are equal.

Rhombus

A rhombus is a parallelogram with four sides of equal length. It has opposite sides parallel and equal in length, and the opposite angles are equal. A rhombus is also an equilateral parallelogram, meaning all four angles are equal. The diagonals of a rhombus bisect each other at right angles.

Square

A square is a parallelogram with four sides of equal length and four right angles. It is both an equilateral and equiangular parallelogram. The opposite sides of a square are parallel and equal in length, and the diagonals bisect each other at right angles.

Other types of parallelograms include the trapezoid and the simple quadrilateral. Trapezoids have one pair of parallel sides and one pair of non-parallel sides. A simple quadrilateral is a four-sided polygon that does not intersect itself and has no holes.

Understanding the different types of parallelograms is important when calculating their areas. The formulas for finding the area of each type of parallelogram are different and knowing which formula to use can save time and effort.

FAQ about how to Find Area of a Parallelogram

How do you find the area of a parallelogram with base and height?

To find the area of a parallelogram with base and height, you need to multiply the base by the height. The formula for finding the area of a parallelogram is:

Area = base x height

How do you find the area of a parallelogram with vectors?

To find the area of a parallelogram with vectors, you need to take the cross product of the two vectors that form the parallelogram. The magnitude of the resulting vector is equal to the area of the parallelogram.

What is the formula for finding the area of a parallelogram?

The formula for finding the area of a parallelogram is:

Area = base x height

How do you find the area of a parallelogram with diagonals?

To find the area of a parallelogram with diagonals, you need to multiply the length of the diagonals and divide the result by 2. The formula is:

Area = (d1 x d2) / 2

where d1 and d2 are the lengths of the diagonals.

What is the difference between a parallelogram and a rhombus?

A parallelogram is a quadrilateral with opposite sides that are parallel. A rhombus is a parallelogram with four congruent sides. Therefore, all rhombuses are parallelograms, but not all parallelograms are rhombuses.

How do you find the perimeter of a parallelogram?

To find the perimeter of a parallelogram, you need to add the lengths of all four sides. The formula is:

Perimeter = 2a + 2b

where a and b are the lengths of the two parallel sides.

Is a trapezoid a parallelogram?

No, a trapezoid is not a parallelogram. A trapezoid is a quadrilateral with one pair of parallel sides, while a parallelogram has two pairs of parallel sides.

How to find the height of a parallelogram?

To find the height of a parallelogram, you need to divide the area of the parallelogram by the length of the base. The formula is:

Height = Area / Base

Watch the video explanation for our Area of a Parallelogram Worksheet

Watch our free video on how to find Area of a Parallelogram. This video shows how to solve problems that are on our free Area of Parallelogram worksheet that you can get by submitting your email above.

Watch the free Area of a Parallelogram video on YouTube here: Area of a Parallelogram Video

Video Transcript:

This video is about how to find the area of a parallelogram. You can get the parallelogram problems worksheets used in this video for free by clicking on the link in the description below. Finding the area of a parallelogram is the same as finding the area of a rectangle. The formula for finding area of a rectangle is area equals base times the height. The formula for finding area of a parallelogram is also area equals base times the height. A parallelogram is just a figure where the opposite sides are parallel to each other. In the case of the example shown here you can see that this side is parallel with this side and then the top and bottom sides this side and this side are also parallel with each other. You can think of a parallelogram as kind of like a rectangle that has been pushed over or slanted at its angle. Whereas, rectangles have 90-degree angles where the sides meet the base parallelograms will not be. In order to find the area of a parallelogram you will use the same formula that you use to find the area of a rectangle. That means we’re going to use the area is equal to the base times the height.

In the case of this example here, our base which is the distance of the long side of the parallelogram is 12 feet. We know the base is 12 feet and we also know the height is 6 feet and I know this is the height because it’s the distance between the two sides of the parallelogram. In order to find the area, we’re just going to multiply the base which is 12 feet. We’re going to do 12 times the height which is 6 feet. Then when you multiply 12 times 6 you will get 72 feet squared and that’s going to be the area of this parallelogram. Let’s do a couple practice problems on our area of a parallelogram worksheet.

The first problem on our area of a parallelogram worksheets gives us a parallelogram where our base is 15 inches and our height is 8 inches. Remember the area of a parallelogram is base times the height. In the case of this problem I know that 15 is the base because it’s the long side on the outside of the parallelogram and I know that eight is the height because it’s the distance between the two parallel sides. In our formula we’re going to use 15 for the base times 8 for the height. When you multiply 15 times 8 you get 120 and our units are inches so it will be 120 inches squared and that’s the area of this parallelogram.

The second problem for showing you how to find the area of a parallelogram that we’re going to use is number two. Remember our formula is area equals base times the height. In the case of number two I know that the base is 20 because it’s the long side on the outer edge of our parallelogram and I know the height is 10 because it’s the distance between the two parallel sides. We’re going to multiply 20 inches for the base times 10 inches for the height. The area will be 20 times 10 which is 200 and our units are inches. We’re going to use inches and that is squared because we’re talking about area. The area of this parallelogram is 200 inches squared.

Number three on our area of a parallelogram worksheet gives us a parallelogram that has a base of five inches and a height of two inches. Now this one’s a little different because the orientation of this parallelogram is vertical. I know that this 5 is the base because this 5 refers to the outer edge of the parallelogram. I also know that this 2 represents the height because even though it’s running horizontally it’s still showing you the distance between this side over to this side. It still represents the distance between the two parallel sides and that means that it’s going to represent the height. When inputting this into the formula for finding area of a parallelogram we’re going to use 5 for the base multiplied by 2 for the height. 5 times 2 is 10 and then our units are feet so it’s going to be 10 feet squared. hopefully you found this video helpful for answering the question how to find the area of a parallelogram.

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