Geometry Basics — Angles, Area, and Similar Triangles
The foundational facts of plane geometry — angle relationships, area and perimeter formulas, and the power of similar triangles.
Angles
An angle measures the rotation between two rays from a common point.
Units:
- Degrees: full rotation = 360°
- Radians: full rotation = 2π ≈ 6.28. One radian is the angle that subtends an arc equal in length to the radius.
Conversion: degrees × π/180 = radians. Radians × 180/π = degrees.
90° = π/2 rad
180° = π rad
270° = 3π/2 rad
360° = 2π rad
Angle types
- Acute: 0° < θ < 90°
- Right: θ = 90°
- Obtuse: 90° < θ < 180°
- Straight: θ = 180°
- Reflex: 180° < θ < 360°
Angle relationships
- Complementary: two angles summing to 90°
- Supplementary: two angles summing to 180°
- Vertically opposite: formed by two intersecting lines — equal
- Alternate interior angles: formed by a transversal crossing parallel lines — equal
- Corresponding angles: same position at each intersection with parallel lines — equal
- Co-interior (same-side interior): sum to 180°
Triangles
Three sides, three angles. The angles always sum to 180°.
Types by sides
- Equilateral: all sides equal, all angles 60°
- Isosceles: two sides equal, base angles equal
- Scalene: all sides different
Types by angles
- Acute: all angles < 90°
- Right: one angle = 90°
- Obtuse: one angle > 90°
Key triangle facts
- Exterior angle = sum of the two non-adjacent interior angles
- The longest side is opposite the largest angle
- Triangle inequality: sum of any two sides > third side
Pythagoras’ theorem
For a right triangle with legs a, b and hypotenuse c:
a² + b² = c²
Common Pythagorean triples (worth memorising):
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Multiples also work: 6, 8, 10 or 9, 12, 15 (both multiples of 3,4,5).
Similar Triangles
Two triangles are similar if they have the same angles (and hence proportional sides). Similar ≠ congruent — they have the same shape but not necessarily the same size.
Conditions for similarity (AA, SAS, SSS):
- AA: two angles match → third must too (angles sum to 180°)
- SAS: two sides proportional and included angle equal
- SSS: all three sides proportional
The power of similar triangles
If triangles ABC and DEF are similar with ratio k:
DE/AB = EF/BC = FD/CA = k
Area(DEF)/Area(ABC) = k²
Similar triangles let you calculate unknown lengths by setting up ratios — the basis of all indirect measurement.
Example: A 2m pole casts a 3m shadow. A tree casts a 21m shadow at the same time. Height of tree = 2 × (21/3) = 14m. The triangles formed by sun, pole/tree, and shadow are similar.
Area Formulas
| Shape | Formula |
|---|---|
| Rectangle | A = l × w |
| Triangle | A = ½ × base × height |
| Parallelogram | A = base × height |
| Trapezium | A = ½(a + b) × h |
| Circle | A = πr² |
| Sector (angle θ in rad) | A = ½r²θ |
Triangle area from sides (Heron’s formula):
s = (a + b + c)/2 (semi-perimeter)
A = √(s(s−a)(s−b)(s−c))
Perimeter and Circumference
| Shape | Formula |
|---|---|
| Rectangle | P = 2(l + w) |
| Triangle | P = a + b + c |
| Circle (circumference) | C = 2πr = πd |
| Arc length (angle θ in rad) | L = rθ |
3D Geometry
| Shape | Volume | Surface Area |
|---|---|---|
| Cube (side a) | a³ | 6a² |
| Cuboid | lwh | 2(lw + lh + wh) |
| Sphere | 4/3 πr³ | 4πr² |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | 1/3 πr²h | πr² + πrl (l = slant height) |
Volume scales as the cube of linear dimension. Double the radius of a sphere → 8× the volume. This is why large animals have disproportionately thick bones, and why scaling up machines isn’t as simple as scaling up a diagram.
The Geometry of Circles
- Chord: line segment joining two points on the circle
- Arc: portion of the circumference
- Tangent: line touching the circle at exactly one point — perpendicular to the radius at that point
- Secant: line intersecting the circle at two points
Key circle theorems:
- Angle at the centre = twice angle at the circumference (subtended by same arc)
- Angles in the same segment are equal
- Angle in a semicircle = 90° (Thales’ theorem)
- Opposite angles of a cyclic quadrilateral sum to 180°
- Tangent from external point: two tangents to the same circle are equal in length
The Bridge to Trigonometry
Similar triangles explain why trigonometry works. In any right triangle with a given angle θ, the ratio of opposite to hypotenuse is always the same — regardless of the size of the triangle. All such triangles are similar to each other. That fixed ratio is sin(θ). Trig ratios are just names for the ratios of sides in similar right triangles.