Topics
7. Polygons
topic badge
United States of AmericaVirginia
Grade 6

7.04 Area of parallelograms

Lesson
Practice

Area of parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides.

Three parallelograms with different side lengths and orientations

A rectangle is a special type of parallelogram, with all angles measuring 90 \degree.

# rectangle images with 90 degree angle and parallel side markings. The first is a square, the 2nd and 3rd are rectangles with one pair of sides longer than the other.
A rectangle with its length and width labeled. Opposite sides are marked as equal, and all corners are right angles.

\,\\\,\\\,Recall, the area of a rectangle is given by\text{Area} = \text{length} \cdot \text{width}\\ \\ \text{ or }\\ \\ A = l \cdot w

Exploration

Use the slider to rearrange the parallelogram with a base b and a perpendicular height h into a rectangle.

Loading interactive...

  1. What relationships do you notice?

  2. Write a formula that could be used to find the area of the parallelogram.

A rectangle with 7cm length and 4cm height, a triangle is cut off from one side. The second image is the same rectangle but the triangle cut off is moved to the right side, forming a parallelogram with height of 4cm and length of 7cm

We know the area of a rectangle is length times width.

We can take the rectangle, cut off one side, and move it to the other side to create a parallelogram. This does not change the size of the figure so both the rectangle and the parallelogram have the same area.

In this case, both the rectangle and the parallelogram have an area of: 7 \text{ cm} \cdot 4 \text{ cm}=28 \text{ cm}^2.

We can use the base and perpendicular height of the parallelogram to find its area, just like we do for a rectangle.

Image of a parallelogram showing a height and parallel bases. The base is also perpendicular to the height.

The area of a parallelogram is found by

\displaystyle A=b \cdot h
\bm{b}
is the base
\bm{h}
is the height
Two parallelograms showing two base height pairs.

The height is always measured perpendicular to the base (at a right angle).

Every parallelogram has two base height pairs.

Since we are finding the product of two lengths, area is always measured in square units.

Examples

Example 1

Find the area of this parallelogram.

A parallelogram with a base of 13 metres and a height of 8 metres.
Worked Solution
Create a strategy

Use the formula for the area of a parallelogram: A=bh.

Apply the idea

We know b=13 and h=8.

\displaystyle A\displaystyle =\displaystyle b\cdot hFormula for area of a parallelogram
\displaystyle =\displaystyle 13\cdot 8Substitute b=13 and h=8
\displaystyle =\displaystyle 104\text{ m}^2Evaluate

Example 2

Find the area of this parallelogram.

A parallelogram with a base of 3 millimetres and height of 11 millimetres.
Worked Solution
Create a strategy

Use the formula for the area of a parallelogram: A=bh.

Apply the idea

We have b=3 and h=11.

\displaystyle A\displaystyle =\displaystyle b\cdot hFormula for area of a parallelogram
\displaystyle =\displaystyle 3\cdot11Substitute b=3 and h=11
\displaystyle =\displaystyle 33\text{ mm}^2Evaluate

Example 3

A school is adding a new athletic field that is parallelogram-shaped. The longest side of the field measures 100 yards and the shortest distance (height) from this side to its opposite side is 60 yards. Calculate the area of the athletic field.

Worked Solution
Create a strategy

Start by drawing a diagram and then use the formula for the area of a parallelogram: A=bh.

Apply the idea
Parallelogram with length of 100 yards and height of 60 yards.

We know b=100 and h=60.

\displaystyle A\displaystyle =\displaystyle b\cdot hFormula for area of a parallelogram
\displaystyle =\displaystyle 100\cdot 60Substitute b=100 and h=60
\displaystyle =\displaystyle 6\,000\text{ yards}^2Evaluate
Idea summary

The area of a parallelogram is found by:

\displaystyle A=b\cdot h
\bm{A}
is the area of a parallelogram
\bm{b}
is the base of a parallelogram
\bm{h}
is the height of a parallelogram

The height is measured perpendicular to the base.

Outcomes

6.MG.2

The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles, and parallelograms.

6.MG.2a

Develop the formula for determining the area of parallelograms and triangles using pictorial representations and concrete manipulatives (e.g., two-dimensional diagrams, grid paper).

6.MG.2b

Solve problems, including those in context, involving the perimeter and area of triangles, and parallelograms.

What is Mathspace

About Mathspace