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Wize High School Geometry Textbook (Common Core) > 2D Measurements

Area & Perimeter

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Perimeter and Area of 2D Figures

The perimeter of a 2D figure measures the length of the outside "edges" of the figure, (this is often measured in cm, m, km, inches, feet, etc.)


The area of a 2D figure measures the "space" within the figure (this is often measured in cm2, m2, km2, inch2, feet2, etc.)



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Formulas


If you are asked to find the area of a composite 2D figure, break the shape up into one of these basic shapes, then calculate the area of each smaller shape.
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Example: Composite 2D Figures

Calculate the perimeter and area of this composite figure. (The diagram may not be drawn to scale)


Label our diagram
Solving for the unknowns
xx represents half the circumference of a circle with radius 2 cm2\ cm:
x=2πr2=2π(2)2=4π2=2π\begin{aligned} x&=\dfrac{2\pi r}{2}\\[0.5em] &=\dfrac{2\pi\left(2\right)}{2}\\[0.5em] &=\dfrac{4\pi}{2}\\[0.5em] &=2\pi \end{aligned}

yy represents the hypotenuse of the right angle triangle:
y2=42+22y=42+22y=16+4y=20y≈4.47\begin{aligned} y^2&=4^2+2^2\\[0.5em] y&=\sqrt{4^2+2^2}\\[0.5em] y&=\sqrt{16+4}\\[0.5em] y&=\sqrt{20}\\[0.5em] y&\approx4.47 \end{aligned}
Perimeter
P=5+y+2+5+x=5+20+2+5+2π=12+20+2π≈22.76\begin{aligned} P&=5+y+2+5+x\\ &=5+\sqrt{20}+2+5+2\pi\\ &=12+\sqrt{20}+2\pi\\ &\approx22.76 \end{aligned}
So, the perimeter is approximately 22.76 cm

Area
A=semi-circle+rectangle+triangle=(πr22)+(l×w)+(b×h2)=(π×222)+(5×4)+(2×42)=(4π2)+20+82=2π+20+4≈30.28\begin{aligned} A&=\text{semi-circle}+\text{rectangle}+\text{triangle}\\[0.5em] &=\left(\dfrac{\pi r^2}{2}\right)+\left(l\times w\right)+\left(\dfrac{b\times h}{2}\right)\\[0.5em] &=\left(\dfrac{\pi \times2^2}{2}\right)+\left(5\times 4\right)+\left(\dfrac{2\times 4}{2}\right)\\[0.5em] &=\left(\dfrac{4\pi}{2}\right)+20+\dfrac{8}{2}\\[0.5em] &=2\pi+20+4\\[0.5em] &\approx 30.28 \end{aligned}
So, the area is approximately 30.28 cm2

Practice: Composite 2D Figures

Calculate the perimeter and area of this figure. (The diagram may not be drawn to scale)

Practice: Composite 2D Figures

Calculate the perimeter and area of the composite figure. (The diagram may not be drawn to scale)


Practice: Composite 2D Figures

A shaded regular hexagon (6-sided polygon) has a smaller regular hexagon carved out of it. Calculate the perimeter and area of the shaded shape. (The diagram may not be drawn to scale)