5.10: Area of a Parallelogram
- Page ID
- 4994
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Use \(A=bh\) to find area.
A parallelogram is a quadrilateral whose opposite sides are parallel.
To find the area of a parallelogram, make it into a rectangle.
From this, we see that the area of a parallelogram is the same as the area of a rectangle. The area of a parallelogram is \(A=bh\). The height of a parallelogram is always perpendicular to the base. This means that the sides are not the height.
What if you were given a parallelogram and the size of its base and height? How could you find the amount of space the parallelogram takes up?
Example \(\PageIndex{1}\)
Find the area of the parallelogram.
Solution
Area is \(15(6)=90\text{ un}^2\).
Example \(\PageIndex{2}\)
Find the area of the parallelogram with a base of 10 m and a height of 12 m.
Solution
Area is 10(12)=120\text{ m}^2\).
Example \(\PageIndex{3}\)
Find the area of the parallelogram.
Solution
\(A=15\cdot 8=120 \text{ in}^2\)
Example \(\PageIndex{4}\)
If the area of a parallelogram is \(56 \text{ units }^2\) and the base is 4 units, what is the height?
Solution
Solve for the height in \(A=bh\).
\(56\text{ units }=4h\)
\(14 \text{ units }=h\)
Example \(\PageIndex{5}\)
If the height of a parallelogram is 12 m and the area is \(60 m^2\), how wide is the base?
Solution
Solve for the base in \(A=bh\).
\(60 \text{ units } =12b\)
\(5 \text{ units } =b\)
Review
- Find the area of a parallelogram with height of 20 m and base of 18 m.
- Find the area of a parallelogram with height of 12 m and base of 15 m.
- Find the area of a parallelogram with height of 40 m and base of 33 m.
- Find the area of a parallelogram with height of 32 m and base of 21 m.
- Find the area of a parallelogram with height of 25 m and base of 10 m.
Find the area of the parallelogram.
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Figure \(\PageIndex{9}\) -
Figure \(\PageIndex{10}\) -
Figure \(\PageIndex{11}\) -
Figure \(\PageIndex{12}\) - If the area of a parallelogram is \(42\text{ units }^2\) and the base is \(6\text{ units }\), what is the height?
- If the area of a parallelogram is \(48\text{ units }^2\) and the height is \(6\text{ units }\), what is the base?
- If the base of a parallelogram is 9 units and the area is \(108\text{ units }^2\), what is the height?
- If the height of a parallelogram is 11 units and the area is \(27.5\text{ units }^2\), what is the base?
Vocabulary
| Term | Definition |
|---|---|
| area | The amount of space inside a figure. Area is measured in square units. |
| perimeter | The distance around a shape. The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet, inches, centimeters, etc), write units. |
| Parallelogram | A parallelogram is a quadrilateral with two pairs of parallel sides. |
| Area of a Parallelogram | The area of a parallelogram is equal to the base multiplied by the height: \(A = bh\). The height of a parallelogram is always perpendicular to the base (the sides are not the height). |
Additional Resources
Interactive Element
Video: Area of a Parallelogram (Whole Numbers)
Activities: Area of a Parallelogram Discussion Questions
Study Aids: Triangles and Quadrilaterals Study Guide
Practice: Area of a Parallelogram
Real World: Perimeter