O Level Geometry Cheatsheet

Cheatsheet Content

1. Basic Shapes & Properties Triangle: 3 sides, 3 angles. Sum of angles is $180^\circ$. Equilateral: All sides equal, all angles $60^\circ$. Isosceles: Two sides equal, two base angles equal. Scalene: All sides and angles different. Right-angled: One angle is $90^\circ$. Quadrilateral: 4 sides, 4 angles. Sum of angles is $360^\circ$. Square: All sides equal, all angles $90^\circ$. Diagonals equal and bisect at $90^\circ$. Rectangle: Opposite sides equal, all angles $90^\circ$. Diagonals equal and bisect each other. Rhombus: All sides equal, opposite angles equal. Diagonals bisect at $90^\circ$. Parallelogram: Opposite sides parallel and equal, opposite angles equal. Diagonals bisect each other. Trapezium: One pair of parallel sides. Kite: Two pairs of equal adjacent sides. Diagonals intersect at $90^\circ$. Polygon: Closed 2D shape with straight sides. Sum of interior angles: $(n-2) \times 180^\circ$, where $n$ is number of sides. Each interior angle of a regular polygon: $\frac{(n-2) \times 180^\circ}{n}$. Sum of exterior angles: $360^\circ$. Each exterior angle of a regular polygon: $\frac{360^\circ}{n}$. Interior angle + Exterior angle = $180^\circ$. Circle: Radius ($r$): Distance from center to circumference. Diameter ($d$): $2r$. Circumference ($C$): $2\pi r$ or $\pi d$. Arc Length ($L$): $\frac{\theta}{360^\circ} \times 2\pi r$ (where $\theta$ is angle in degrees). Chord: Line segment connecting two points on the circumference. Tangent: Line touching the circle at one point, perpendicular to radius at point of contact. 2. Angles Types of Angles: Acute: $0^\circ Right: $\theta = 90^\circ$. Obtuse: $90^\circ Straight: $\theta = 180^\circ$. Reflex: $180^\circ Angle Relationships: Complementary: Sum is $90^\circ$. Supplementary: Sum is $180^\circ$. Angles on a straight line: Sum is $180^\circ$. Angles at a point: Sum is $360^\circ$. Vertically opposite angles: Equal. Parallel Lines & Transversal: Alternate angles: Equal (Z-angles). Corresponding angles: Equal (F-angles). Interior angles: Sum is $180^\circ$ (C-angles). Angles in a Circle: Angle at center is twice angle at circumference subtended by same arc. Angles in the same segment are equal. Angle in a semicircle is $90^\circ$. Opposite angles of a cyclic quadrilateral sum to $180^\circ$. Tangent-radius property: Tangent is perpendicular to radius at point of contact. Alternate segment theorem: Angle between tangent and chord is equal to angle in the alternate segment. 3. Area Formulas Triangle: $A = \frac{1}{2} \times \text{base} \times \text{height}$. $A = \frac{1}{2}ab \sin C$ (using two sides and included angle). For right-angled triangle: $A = \frac{1}{2} \times \text{product of perpendicular sides}$. Square: $A = \text{side}^2$. Rectangle: $A = \text{length} \times \text{width}$. Parallelogram: $A = \text{base} \times \text{height}$. Trapezium: $A = \frac{1}{2} (\text{sum of parallel sides}) \times \text{height}$. Rhombus/Kite: $A = \frac{1}{2} \times d_1 \times d_2$ (where $d_1, d_2$ are diagonals). Circle: $A = \pi r^2$. Sector of a Circle: $A = \frac{\theta}{360^\circ} \times \pi r^2$ (where $\theta$ is angle in degrees). Annulus (Ring): $A = \pi (R^2 - r^2)$ (where $R$ is outer radius, $r$ is inner radius). 4. Volume & Surface Area (3D Shapes) Cuboid: Volume ($V$) = $l \times w \times h$. Surface Area ($SA$) = $2(lw + lh + wh)$. Cube: $V = s^3$. $SA = 6s^2$. Cylinder: $V = \pi r^2 h$. Curved $SA = 2\pi r h$. Total $SA = 2\pi r h + 2\pi r^2$. Cone: $V = \frac{1}{3} \pi r^2 h$. Curved $SA = \pi r l$ (where $l$ is slant height, $l = \sqrt{r^2+h^2}$). Total $SA = \pi r l + \pi r^2$. Sphere: $V = \frac{4}{3} \pi r^3$. $SA = 4\pi r^2$. Pyramid: $V = \frac{1}{3} \times \text{base area} \times \text{height}$. Prism: $V = \text{cross-sectional area} \times \text{length}$. 5. Trigonometry (Right-Angled Triangles) SOH CAH TOA: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$ Pythagoras Theorem: $a^2 + b^2 = c^2$ (where $c$ is hypotenuse). Special Angles: $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 30^\circ = \frac{1}{\sqrt{3}}$ $\sin 45^\circ = \frac{1}{\sqrt{2}}$, $\cos 45^\circ = \frac{1}{\sqrt{2}}$, $\tan 45^\circ = 1$ $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\tan 60^\circ = \sqrt{3}$ 6. Trigonometry (General Triangles) Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Cosine Rule: $c^2 = a^2 + b^2 - 2ab \cos C$ (and permutations)