Geometry Cheatsheet

Cheatsheet Content

### Analytic Geometry #### Lines - **Slope-Intercept Form:** $y = mx + b$ - $m$: slope - $b$: y-intercept - **Point-Slope Form:** $y - y_1 = m(x - x_1)$ - **Standard Form:** $Ax + By = C$ - **Distance Formula:** $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ - **Midpoint Formula:** $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ #### Conic Sections - **Circle:** $(x - h)^2 + (y - k)^2 = r^2$ - Center: $(h, k)$, Radius: $r$ - **Parabola:** - Vertical: $(x - h)^2 = 4p(y - k)$ - Horizontal: $(y - k)^2 = 4p(x - h)$ - Vertex: $(h, k)$, Focus: $(h, k+p)$ or $(h+p, k)$ - **Ellipse:** $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ - Center: $(h, k)$, Major axis length: $2a$, Minor axis length: $2b$ - Foci: $c^2 = a^2 - b^2$ (if $a > b$) - **Hyperbola:** - Horizontal: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ - Vertical: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ - Center: $(h, k)$, Vertices: $(h \pm a, k)$ or $(h, k \pm a)$ - Foci: $c^2 = a^2 + b^2$ ### Plane Geometry #### Areas of Basic Shapes - **Triangle:** $A = \frac{1}{2}bh$ - **Square:** $A = s^2$ - **Rectangle:** $A = lw$ - **Parallelogram:** $A = bh$ - **Trapezoid:** $A = \frac{1}{2}(b_1 + b_2)h$ - **Circle:** $A = \pi r^2$ - **Sector of a Circle:** $A = \frac{\theta}{360^\circ} \pi r^2$ (where $\theta$ is in degrees) #### Polygons - **Sum of Interior Angles:** $(n - 2) \times 180^\circ$ (for an n-sided polygon) - **Regular Polygon Area:** $A = \frac{1}{2}Pa$ (where $P$ is perimeter, $a$ is apothem) #### Composite Figures - Break down complex shapes into simpler ones. - Calculate the area of each simple shape. - Add or subtract areas as needed. ### Solid Geometry #### Prisms - **Volume:** $V = B h$ (where $B$ is area of base) - **Surface Area (Right Prism):** $SA = 2B + Ph$ (where $P$ is perimeter of base) #### Cylinders - **Volume:** $V = \pi r^2 h$ - **Surface Area:** $SA = 2\pi r^2 + 2\pi r h$ #### Cones - **Volume:** $V = \frac{1}{3}\pi r^2 h$ - **Surface Area:** $SA = \pi r^2 + \pi r l$ (where $l$ is slant height, $l = \sqrt{r^2 + h^2}$) #### Pyramids - **Volume:** $V = \frac{1}{3}Bh$ (where $B$ is area of base) - **Surface Area (Regular Pyramid):** $SA = B + \frac{1}{2}Pl$ (where $P$ is perimeter of base, $l$ is slant height) #### Spheres - **Volume:** $V = \frac{4}{3}\pi r^3$ - **Surface Area:** $SA = 4\pi r^2$