Geometry (Euclidean)
Essential formulas and concepts in geometry (euclidean)
Pythagorean Theorem
\(c^2 = a^2 + b^2\)
Variables:
- \(a and b are the lengths of the legs (perpendicular sides) of a right triangle, and c is the length of the hypotenuse (side opposite the right angle).\)
Description/Usage:
Relates the side lengths of a right-angled triangle. It is used to find the third side given the other two.
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Sum of Interior Angles (Polygon)
\(S_{\text{angles}} = (n - 2)\times 180^\circ\)
Variables:
- \(n is the number of sides of a polygon.\)
Description/Usage:
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Heron's Formula (Triangle Area by Sides)
\(A = \sqrt{s(s-a)(s-b)(s-c)}\)
Variables:
Description/Usage:
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Volume of a Rectangular Prism (Cuboid)
\(V = L \times W \times H\)
Variables:
- \(L, W, H are the length, width, and height of the rectangular solid (for a cube, all three are equal to a).\)
Description/Usage:
Computes the volume (space inside) of a box-shaped object. Visualized as counting unit cubes that fill a rectangular box.
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Volume of a Cylinder
\(V = \pi r^2 h\)
Variables:
- \(r is the base radius, h is the height of the cylinder.\)
Description/Usage:
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Volume of a Cone/Pyramid
\(V = \frac{1}{3} A_{\text{base}} \, h\)
Variables:
Description/Usage:
The volume of any pyramid (base with triangular sides meeting at a point) is one-third the base area times height. Visual proof involves stacking three congruent pyramids to fill a prism.
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Volume of a Sphere
\(V = \frac{4}{3}\pi r^3\)
Variables:
- \(r is the radius of the sphere.\)
Description/Usage:
Gives the volume inside a sphere (ball).
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