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Mathematics LibreTexts

1.2.1: Parallelograms

  • Page ID
    39635
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    Lesson

    Let's investigate the features and areas of parallelograms.

    Exercise \(\PageIndex{1}\): Features of a Parallelogram

    Figures A, B, and C are parallelograms. Figures D, E, and F are not parallelograms.

    clipboard_e147e844790ac36ae5bbe837949b8142b.png
    Figure \(\PageIndex{1}\)

    Study the examples and non-examples. What do you notice about:

    1. the number of sides that a parallelogram has?
    2. opposite sides of a parallelogram?
    3. opposite angles of a parallelogram?

    Exercise \(\PageIndex{2}\): Area of a Parallelogram

    1. Find the area of the parallelogram and explain your reasoning.
    2. Change the parallelogram by dragging the green points at its vertices. Find its area and explain your reasoning.
    3. If you used the polygons on the side, how were they helpful? If you did not, could you use one or more of the polygons to show another way to find the area of the parallelogram?

    Exercise \(\PageIndex{3}\): Lots of Parallelograms

    Find the area of each parallelogram. Show your reasoning.

    clipboard_efc413d6f9308f7c0a160c1e89ae00818.png
    Figure \(\PageIndex{2}\)
    clipboard_ebea3bea20b4a32dcf92712417db26cb5.png
    Figure \(\PageIndex{3}\)

    Summary

    A parallelogram is a quadrilateral (it has four sides). The opposite sides of a parallelogram are parallel. It is also true that the opposite sides of a parallelogram have equal length, and the opposite angles of a parallelogram have equal measure.

    clipboard_e5df20abeed82ec670ad52ec496704063.png
    Figure \(\PageIndex{4}\): Two parallelograms with the angles and side lengths provided. On the left, Top and bottom sides = 5 units. Left and right sides = 4.24 units. Top left and bottom right angles = 135 degrees. Top right and bottom left angles = 45 degrees. On the right, Top and bottom sides = 9.34 units. Left and right sides = 4 units. Top left and bottom right angles = 27.2 degrees. Top right and bottom left angles = 152.8 degrees.

    There are several strategies for finding the area of a parallelogram.

    • We can decompose and rearrange a parallelogram to form a rectangle. Here are three ways:
    clipboard_ea20ebcb72c318d89764070d5d2b38241.png
    Figure \(\PageIndex{5}\): Three identical parallelograms on separate grids, each has a base of four units and a height of three units. First parallelogram, vertical dashed segment extending from the bottom left vertex to the opposite side, forming a triangle. An arrow extends from the triangle to the opposite side of the parallelogram to create a rectangle 4 units wide and 3 units high. Second parallelogram, vertical dashed segment extending from the top right vertex to the opposite side, forming a triangle. An arrow extends from the triangle to the opposite side of the parallelogram to create a rectangle 4 units wide and 3 units high. Third parallelogram, vertical dashed segment through the middle of the parallelogram. An arrow extends from the resulting shape to the opposite side of the parallelogram to create a rectangle 4 units wide and 3 units high.
    • We can enclose the parallelogram and then subtract the area of the two triangles in the corner.
    clipboard_ebae4d32f2736e8b6e28c18277d0a9bf1.png
    Figure \(\PageIndex{6}\): Two drawings of parallelograms on grids. On the left, a triangle in the bottom left and top right corner is in white and the center is colored blue. On the right, the image from the left is repeated, but the triangles in the corners are colored yellow. Arrows are drawn from the triangles to the right. The triangles two triangles are joined on the right to form a rectangle.

    Both of these ways will work for any parallelogram. However, for some parallelograms the process of decomposing and rearranging requires a lot more steps than if we enclose the parallelogram with a rectangle and subtract the combined area of the two triangles in the corners.

    clipboard_e3d0cdf1158ada8096e646db44f58dc8f.png
    Figure \(\PageIndex{7}\): A shaded parallelogram on a grid. Base of three units. Slanted sides that decline 6 vertical units over 9 horizontal units. Parallelogram decomposed by dashed segments into six equal right triangles. Each triangle has a vertical side of 2 units and horizontal side of 3 units. Arrows extend to the left from each of the lower 5 triangles. The resulting shape is a rectangle that is 6 units tall by 3 units wide.

    Glossary Entries

    Definition: Parallelogram

    A parallelogram is a type of quadrilateral that has two pairs of parallel sides.

    Here are two examples of parallelograms.

    clipboard_e63643e0f76082a51a15180251ba0759c.png
    Figure \(\PageIndex{8}\): Two parallelograms with the angles and side lengths provided. On the left, Top and bottom sides = 5 units. Left and right sides = 4.24 units. Top left and bottom right angles = 135 degrees. Top right and bottom left angles = 45 degrees. On the right, Top and bottom sides = 9.34 units. Left and right sides = 4 units. Top left and bottom right angles = 27.2 degrees. Top right and bottom left angles = 152.8 degrees.

    Definition: Quadrilateral

    A quadrilateral is a type of polygon that has 4 sides. A rectangle is an example of a quadrilateral. A pentagon is not a quadrilateral, because it has 5 sides.

    Practice

    Exercise \(\PageIndex{4}\)

    Select all of the parallelograms. For each figure that is not selected, explain how you know it is not a parallelogram.

    clipboard_e8bc4d9c72a5635f4e12a47c6d0cbf8fc.png
    Figure E is a right triangle.

    Exercise \(\PageIndex{5}\)

    1. Decompose and rearrange this parallelogram to make a rectangle.
    clipboard_e068aeb07fbb347d5efa34865eb0b4799.png
    Figure \(\PageIndex{10}\)
    1. What is the area of the parallelogram? Explain or your reasoning.

    Exercise \(\PageIndex{6}\)

    Find the area of the parallelogram.

    clipboard_ed5138a5704e59907f46bb435988033d9.png
    Figure \(\PageIndex{11}\)

    Exercise \(\PageIndex{7}\)

    Explain why this quadrilateral is not a parallelogram.

    clipboard_e1c60fd0c3b4c3af80d833be0eda7dfea.png
    Figure \(\PageIndex{12}\): A quadrilateral on a grid. Bottom side length of 8 units. Top side length of 4 units. The left side ascends 5 units while moving right 13 units, and the right side ascends 5 units while moving right 9 units.

    Exercise \(\PageIndex{8}\)

    Find the area of each shape. Show your reasoning.

    clipboard_e0c6057dfc1423563c59fa078564e59fd.png
    Figure \(\PageIndex{13}\): A shape with eight sides drawn on a grid. Four sides are straight sides and extend left, right, up and, down for 2 units each. The remaining sides are angled sides connecting each of the straight sides to the next. The shape is a total of 6 units tall and 6 units wide.
    clipboard_eecb185b31019de2345390b546580de0b.png
    Figure \(\PageIndex{14}\)

    (From Unit 1.1.3)

    Exercise \(\PageIndex{9}\)

    Find the area of the rectangle with each set of side lengths.

    1. \(5\) in and \(\frac{1}{3}\) in
    2. \(5\) in and \(\frac{4}{3}\) in
    3. \(\frac{5}{2}\) in and \(\frac{4}{3}\) in
    4. \(\frac{7}{6}\) in and \(\frac{6}{7}\) in

    (From Unit 1.1.1)