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5.3: Square and Rectangle Area and Perimeter

  • Page ID
    2155
  • Compute edge and coverage measures of rectilinear quadrilaterals, given linear measures.

    Area and Perimeter of Rectangles

    To find the area of a rectangle, calculate \(A=bh\), where \(b\) is the base (width) and \(h\) is the height (length). The perimeter of a rectangle will always be \(P=2b+2h\).

    f-d_0c9a79962fc5723da8c0215d936990f9fea47b63d4511708daa10731+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    If a rectangle is a square, with sides of length s, then perimeter is \(P_{square}=2s+2s=4s\) and area is \(A_{sqaure}=s\cdot s=s^2\).

    f-d_7ca9b8bf62b2e92936dd2e410390a2f666a3a0eef8c17ea45ba02cf8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    What if you were given a rectangle and the size of its base and height? How could you find the total distance around the rectangle and the amount of space it takes up?

    Example \(\PageIndex{1}\)

    The area of a square is \(75\text{ in}^2\). Find the perimeter.

    Solution

    To find the perimeter, we need to find the length of the sides.

    \(\begin{aligned} A&=s^2=75\text{ in}^2 \\ s&=\sqrt{75}=5\sqrt{3}\text{ in } \end{aligned}\)

    From this, \(P=4(5\sqrt{3})=20\sqrt{3}\text{ in }\)

    Example \(\PageIndex{2}\)

    Draw two different rectangles with an area of \(36\text{ cm^2 }\).

    Solution

    Think of all the different factors of 36. These can all be dimensions of the different rectangles.

    f-d_a010a8c55e067fc941596fc82902aae27eb7d1d59bbd8bfa870e5573+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Other possibilities could be \(6\times 6\), \(2\times 18\), and \(1\times 36\).

    Example \(\PageIndex{3}\)

    Find the area and perimeter of a rectangle with sides \(4\text{ cm }\) by \(9\text{ cm }\).

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    Figure \(\PageIndex{4}\)

    Solution

    The perimeter is \(4+9+4+9=26\text{ cm }\). The area is \(A=9\cdot 4=36\text{ cm}^2\).

    Example \(\PageIndex{4}\)

    Find the area and perimeter of a square with side \(5\text{ in }\).

    Solution

    The perimeter is \(4(5)=20\:in\) and the area is \(5^2=25\text{ in}^2\).

    Example \(\PageIndex{5}\)

    Find the area and perimeter of a rectangle with sides \(13\text{ m }\) and \(12\text{ m}^2\).

    Solution

    The perimeter is \(2(13)+2(12)=50\text{ m }\). The area is \(13(12)=156\text{ m}^2\).

    Review

    1. Find the area and perimeter of a square with sides of length \(12\text{ in }\).
    2. Find the area and perimeter of a rectangle with height of \(9\text{ cm }\) and base of \(16\text{ cm }\).
    3. Find the area and perimeter of a rectangle if the height is 8 and the base is 14.
    4. Find the area and perimeter of a square if the sides are \(18\text{ ft }\).
    5. If the area of a square is \(81\text{ ft}^2\), find the perimeter.
    6. If the perimeter of a square is \(24\text{ in }\), find the area.
    7. The perimeter of a rectangle is 32. Find two different dimensions that the rectangle could be.
    8. Draw two different rectangles that haven an area of \(90\text{ mm}^2\).
    9. True or false: For a rectangle, the bigger the perimeter, the bigger the area.
    10. Find the perimeter and area of a rectangle with sides \(17\text{ in }\) and \(21\text{ in }\).

    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.
    Area of a Rectangle To find the area '\(A\)' of a rectangle, calculate \(A = bh\), where \(b\) is the base (width) and h is the height (length).
    Perimeter of a Rectangle The perimeter '\(P\)' of a rectangle is equal to twice the base added to twice the height: \(P = 2b + 2h\).

    Additional Resources

    Interactive Element

    Video: Determine the Area of a Rectangle Involving Whole Numbers

    Activities: Area and Perimeter of Rectangles Discussion Questions

    Study Aids: Triangles and Quadrilaterals Study Guide

    Practice: Square and Rectangle Area and Perimeter

    Real World: Perimeter

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