Geometry equations

2D Shapes (Area & Perimeter)

• Square:

  • Area ($A$) = $s^{2}$ (where $s$ is the side length)

  • Perimeter ($P$) = $4s$

• Rectangle:

  • Area ($A$) = $l \times w$ (length times width)

  • Perimeter ($P$) = $2l + 2w$

• Triangle:

  • Area ($A$) = $\frac{1}{2}bh$ (base times height divided by two)

  • Perimeter ($P$) = $a + b + c$ (sum of all three sides)

• Circle:

  • Area ($A$) = $\pi r^2$ ($r$ is the radius)

  • Circumference ($C$) = $2\pi r$ or $\pi d$ ($d$ is the diameter)

• Parallelogram:

  • Area ($A$) = $bh$ (base times height)

  • Perimeter ($P$) = $2a + 2b$

• Trapezoid:

  • Area ($A$) = $\frac{1}{2}h(b_1 + b_2)$ ($h$ is height, $b_{1}$ and $b_{2}$ are parallel bases)

  • Perimeter ($P$) = $a + b + c + d$ (sum of all sides)

3D Shapes (Volume & Surface Area)

• Cube:

  • Volume ($V$) = $s^{3}$

  • Surface Area ($SA$) = $6s^{2}$

• Rectangular Prism (Cuboid):

  • Volume ($V$) = $l \times w \times h$

  • Surface Area ($SA$) = $2lw + 2lh + 2wh$

• Cylinder:

  • Volume ($V$) = $\pi r^2 h$

  • Surface Area ($SA$) = $2\pi r h + 2\pi r^2$

• Cone:

  • Volume ($V$) = $\frac{1}{3}\pi r^2 h$

  • Surface Area ($SA$) = $\pi r(l + r)$ ($l$ is the slant height)

• Sphere:

  • Volume ($V$) = $\frac{4}{3}\pi r^3$

  • Surface Area ($SA$) = $4\pi r^2$

Coordinate & Triangle Formulas

• Pythagorean Theorem: $a^2 + b^2 = c^2$ (for right triangles, where $c$ is the hypotenuse)

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ (distance between two points)

• Midpoint Formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

• Slope of a Line: $m = \frac{y_2 - y_1}{x_2 - x_1}$