Calculating Volume and Surface Area: A Comprehensive Guide

Rectangles:

Area: \begin{equation} A = lw \end{equation}

where l is the length and w is the width.

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Squares:

Area: \begin{equation} A = s^2 \end{equation}

where s is the length of one side.

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Parallelograms:

Area: \begin{equation} A = bh \end{equation}

where b is the base and h is the height.

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Rhombus:

Area: \begin{equation} A = d_1d_2\frac{sin(\theta)}{2} \end{equation}

where \( d_1 \) and \(d_2 \) are the lengths of the diagonals and \(\theta \) is the angle between them.

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Triangles:

Area: \begin{equation} A = \frac{1}{2}bh \end{equation}

where b is the base and h is the height.

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Circles:

Area: \begin{equation} A = \pi r^2 \end{equation}

where r is the radius.

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Cylinders:

Volume: \begin{equation} V = \pi r^2 h \end{equation} Surface Area: \begin{equation} SA = 2\pi rh + 2\pi r^2 \end{equation}

where r is the radius and h is the height.

Cones:

Volume: \begin{equation} V = \frac{1}{3}\pi r^2 h \end{equation} Surface Area: \begin{equation} SA = \pi r^2 + \pi r\sqrt{r^2 + h^2} \end{equation}

where r is the radius and h is the height.

Spheres:

Volume: \begin{equation} V = \frac{4}{3}\pi r^3 \end{equation} Surface Area: \begin{equation} SA = 4\pi r^2 \end{equation} where r is the radius.

Prisms:

Volume: \begin{equation} V = Bh \end{equation} Surface Area: \begin{equation} SA = 2B + Ph \end{equation} where B is the area of the base, P is the perimeter of the base, and h is the height.

Pyramids:

Volume: \begin{equation} V = \frac{1}{3} Bh \end{equation} Surface Area: \begin{equation} SA = B + \frac{Pl}{2} \end{equation} where B is the area of the base, P is the perimeter of the base, l is the slant height,