The area of a triangle is equal to $\frac{1}{2}$ times the length of the base times the height: \[ A = \frac{1}{2} a h_a. \] Note that $h_a$ is the height of the triangle relative to the side $a$.

The perimeter of the triangle is: \[ P = a + b + c. \]

Consider now a triangle with internal angles $\alpha$, $\beta$ and $\gamma$. The sum of the inner angles in any triangle is equal to two right angles: $\alpha+\beta+\gamma=180^\circ$.

The sine law is: \[ \frac{a}{\sin(\alpha)}=\frac{b}{\sin(\beta)}=\frac{c}{\sin(\gamma)}, \] where $\alpha$ is the angle opposite to $a$, $\beta$ is the angle opposite to $b$ and $\gamma$ is the angle opposite to $c$.

The cosine rules are: \[ \begin{align} a^2 & =b^2+c^2-2bc\cos(\alpha), \nl b^2 & =a^2+c^2-2ac\cos(\beta), \nl c^2 & =a^2+b^2-2ab\cos(\gamma). \end{align} \]

A sphere is described by the equation \[ x^2 + y^2 + z^2 = r^2. \]

Surface area: \[ A = 4\pi r^2. \]

Volume: \[ V = \frac{4}{3}\pi r^3. \]

The surface area of a cylinder consists of the top and bottom circular surfaces plus the area of the side of the cylinder: \[ A = 2 \left( \pi r^2 \right) + (2\pi r) h. \]

The volume is given by product of the area of the base times the height of the cylinder: \[ V = \left(\pi r^2 \right)h. \]

You open the hood of your car and see 2.0L written on top of the engine. The 2[L] refers to the total volume of the four pistons, which are cylindrical in shape. You look in the owner's manual and find out that the diameter of each piston (bore) is 87.5[mm] and the height of each piston (stroke) is 83.1[mm]. Verify that the total volume of the cylinder displacement of your engine is indeed 1998789[mm$^3$] $\approx 2$[L].

[ A formula for calculating the distance between two points on a sphere ]
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