1. Basic Shapes and Their Properties
Triangle
- Area: \( A = \frac{1}{2} \times \text{base} \times \text{height} \)
- Perimeter: \( P = a + b + c \) (sum of all sides)
- Pythagorean Theorem (Right Triangle): \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
- Equilateral Triangle:
- Area: \( A = \frac{\sqrt{3}}{4} \times s^2 \)
- Height: \( h = \frac{\sqrt{3}}{2} \times s \)
Rectangle
- Area: \( A = \text{length} \times \text{width} \)
- Perimeter: \( P = 2(\text{length} + \text{width}) \)
Square
- Area: \( A = s^2 \), where \( s \) is the side length.
- Perimeter: \( P = 4s \)
Parallelogram
- Area: \( A = \text{base} \times \text{height} \)
- Perimeter: \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the opposite sides.
Trapezoid
- Area: \( A = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height} \)
- Perimeter: \( P = \text{base}_1 + \text{base}_2 + \text{leg}_1 + \text{leg}_2 \)
Circle
- Area: \( A = \pi r^2 \)
- Circumference: \( C = 2\pi r \)
- Arc Length (for a sector with angle \( \theta \)): \( L = \frac{\theta}{360} \times 2\pi r \)
- Area of a Sector (for angle \( \theta \)): \( A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2 \)
2. Polygons
Regular Polygon
- Area: \( A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \)
- Interior Angle: \( \theta = \frac{(n - 2) \times 180^\circ}{n} \), where \( n \) is the number of sides.
- Exterior Angle: \( \theta = \frac{360^\circ}{n} \)
Rectangular Prism
- Volume: \( V = \text{length} \times \text{width} \times \text{height} \)
- Surface Area: \( SA = 2(\text{lw} + \text{lh} + \text{wh}) \)
Cube
- Volume: \( V = s^3 \)
- Surface Area: \( SA = 6s^2 \)
Cylinder
- Volume: \( V = \pi r^2 h \)
- Surface Area: \( SA = 2\pi r^2 + 2\pi r h \)
Cone
- Volume: \( V = \frac{1}{3} \pi r^2 h \)
- Surface Area: \( SA = \pi r^2 + \pi r l \), where \( l \) is the slant height.
Sphere
- Volume: \( V = \frac{4}{3} \pi r^3 \)
- Surface Area: \( SA = 4\pi r^2 \)
4. Angles
- Sum of Interior Angles of a Polygon: \( S = (n - 2) \times 180^\circ \), where \( n \) is the number of sides.
- Straight Angle: \( 180^\circ \)
- Right Angle: \( 90^\circ \)
- Complementary Angles: Two angles that sum to \( 90^\circ \).
- Supplementary Angles: Two angles that sum to \( 180^\circ \).
5. Coordinate Geometry
- \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
- \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Slope of a Line
- \( m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}\)
Equation of a Line
- Slope-Intercept Form: \( y = mx + b \)
- Point-Slope Form: \( y - y_1 = m(x - x_1) \)
- Standard Form: \( Ax + By = C \)
- \( (x, y) \to (x + a, y + b) \)
- 90° rotation: \( (x, y) \to (-y, x) \)
- 180° rotation: \( (x, y) \to (-x, -y) \)
- 270° rotation: \( (x, y) \to (y, -x) \)
Reflection:
- Over the x-axis: \( (x, y) \to (x, -y) \)
- Over the y-axis: \( (x, y) \to (-x, y) \)
- \( (x, y) \to (kx, ky) \)
7. Similarity and Congruence
Similarity:
- Two figures are similar if their corresponding angles are congruent and corresponding sides are proportional.
- Ratio of Similarity: \( \frac{\text{Side}_1}{\text{Side}_2} = k \)
Congruence:
- Two figures are congruent if they have the same shape and size.
- \( \triangle ABC \cong \triangle DEF \) means all corresponding sides and angles are equal.
8. Trigonometry (Right Triangles)
- Sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
- Cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- Tangent: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
- Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)