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Centroids and Area Moments of Inertia for 2D Shapes

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    Shape with Area and Centroid Location Shown Rectangular Area Moments of Inertia Polar Area Moments of Inertia

    Rectangle

    A rectangle of length b and height h in the first quadrant of a Cartesian coordinate plane with axes labeled x' and y', with the lower left corner at the origin. The centroid is located at the coordinates x = b/2, y = h/2. Another coordinate system is located with its origin on the centroid, with axes labeled x and y.

    \(Area = bh\)

    \begin{align*} I_x &= \frac{1}{12} b h^3 \\[4pt] I_y &= \frac{1}{12} b^3 h \end{align*}

    		 \( J_z = \dfrac{1}{12} bh(b^2 + h^2) \)

    Right Triangle

    The first quadrant of a Cartesian coordinate plane with axes labeled x' and y'. A right triangle with its right angle at the origin of this plane lies with its base of length b along the x'-axis and its height of length h along the y'-axis. The centroid of the triangle, located h/3 units above and b/3 units to the right of this origin, is labeled C and forms the origin for a second Cartesian coordinate system with axes labled x and y.

    \(Area = \dfrac{1}{2} bh\)

    \begin{align*} I_x &= \frac{1}{36} bh^3 \\[4pt] I_y &= \frac{1}{36} b^3 h \end{align*}

    \begin{align*} I_{x'} &= \frac{1}{12} bh^3 \\[4pt] I_{y'} &= \frac{1}{12} b^3 h \end{align*}

    Triangle

    The first quadrant of a Cartesian coordinate plane with axes labeled x' and y'. One vertex of a triangle lies at the origin of this plane, with one side of the triangle, of length b, lying along the x'-axis. The vertex where the other two sides of the shape intersect is located h units above the x'-axis. The centroid of the triangle is located h/3 units above and b/2 units to the right of the origin. The centroid, labeled C, forms the origin of another Cartesian coordinate system with axes labeled x and y.

    \(Area = \dfrac{1}{2} bh\)

    \(I_x = \dfrac{1}{36} bh^3\)

    \(I_{x'} = \dfrac{1}{12} bh^3\)

    Circle

    A circular disk of radius r is centered at the origin of a Cartesian coordinate plane with axes labeled x and y. The circle's centroid C is coincident with this origin.

    \(Area = \pi r^2\)

    \begin{align*} I_x &= \frac{\pi}{4} r^4\\[4pt] I_y &= \frac{\pi}{4} r^4 \end{align*}

    		 \(J_z = \dfrac{\pi}{2} r^4\)

    Circular Annulus

    A disk of radius r_o is centered at the origin of a Cartesian coordinate plane with axes labeled x and y. A disk of smaller radius r_i, also centered at the origin, is removed from that larger disk. The centroid of the shape, labeled C, is coincident with this origin.

    \(Area = \pi (r_o^2 - r_i^2) \)

    \begin{align*} I_x &= \frac{\pi}{4} (r_o^4 - r_i^4) \\[4pt] I_y &= \frac{\pi}{4} (r_o^4 - r_i^4) \end{align*}

    		 \(J_z = \dfrac{\pi}{2} (r_o^4 - r_i^4)\)

    Semicircle

    A Cartesian coordinate plane with axes labeled x' and y has the straight edge of a semicircle of radius r lying along the x-axis, centered at the origin. The semicircle stretches upwards along the positive y-axis. The centroid C of the semicircle lies on the y-axis, a distance 4r/(3 pi) units above the origin. Point C forms the origin of another Cartesian coordinate system, with the x-axis stretching to the right and the y-axis shared with the existing y-axis.

    \(Area = \dfrac{\pi}{2} r^2\)

    \(I_x = \left( \dfrac{\pi}{8} - \dfrac{8}{9 \pi} \right) r^4 \)

    \(I_y = \dfrac{\pi}{8} r^4\)

    \(I_{x'} = \dfrac{\pi}{8} r^4\)

    		 \(J_z = \left( \dfrac{\pi}{4} - \dfrac{8}{9 \pi} \right) r^4\)

    Quarter Circle

    The first quadrant of a Cartesian coordinate plane with axes labeled x' and y'. The two sides of a quarter-circle of radius r, centered at the origin, lie along these axes. The centroid C of the quarter circle is located a distance of 4r/(3 pi) units above and 4r/(3 pi) units to the right of the origin. Point C forms the origin of another Cartesian coordinate system, with axes labeled x and y.

    \(Area = \dfrac{\pi}{4} r^2\)

    \begin{align*} I_x &= \left( \frac{\pi}{16} - \frac{4}{9 \pi} \right) r^4 \\[4pt] I_y &= \left( \frac{\pi}{16} - \frac{4}{9 \pi} \right) r^4 \end{align*}

    \begin{align*} I_{x'} &= \frac{\pi}{16} r^4 \\[4pt] I_{y'} &= \frac{\pi}{16} r^4 \end{align*}

    		 \( J_z = \left( \dfrac{\pi}{8} - \dfrac{8}{9 \pi} \right) r^4 \)

    Ellipse

    An ellipse lies with its centroid C at the origin of a Cartesian coordinate plane with axes labled x and y. Its semi-major axis, of length a, stretches along the x-axis and its semi-minor axis of length stretches along the y-axis.

    \(Area = \pi ab\)

    \begin{align*} I_x &= \frac{\pi}{4} ab^3 \\[4pt] I_y &= \frac{\pi}{4} a^3 b \end{align*}
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