Master Geometry
& Trigonometry
Six units from basic angles to sine, cosine, and the unit circle — with interactive diagrams, full worked examples, and a 20-question graded exam.
Angles & lines
An angle is formed when two rays share a common endpoint called the vertex. Angles are measured in degrees (°). Understanding angles is the foundation of all geometry.
Less than a right angle. Sharp, like the tip of a pencil.
A perfect corner. Marked with a small square in diagrams.
Greater than a right angle but less than a straight line.
A straight line. Two rays pointing in opposite directions.
Two angles that together form a right angle.
Two angles that together form a straight line.
Angle relationships with parallel lines
When a transversal crosses two parallel lines, it creates special angle pairs. Knowing these saves enormous calculation time.
On opposite sides of the transversal, between the parallel lines. Always equal.
Same position at each intersection. Always equal when lines are parallel.
Same side of transversal, between parallel lines. Always supplementary.
Formed by two intersecting lines. Always equal regardless of parallel lines.
67° + x = 180°x = 180° − 67° = 113°x = 113°48°180° − 48° = 132° (and its vertical = 132°)48° and four of 132°(n−2) × 180°(6−2) × 180° = 720°720° ÷ 6 = 120°120°Triangles
A triangle has three sides and three angles. The angles always sum to 180° — no matter the shape or size. This single rule unlocks almost every triangle problem.
All sides equal, all angles 60°. Perfect symmetry.
Base angles are always equal. Used constantly in proofs.
No equal sides, no equal angles. Most general case.
Foundation of trigonometry and Pythagoras. The most useful triangle type.
Pythagoras' theorem
In any right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
c² = 3² + 4² = 9 + 16 = 25c = √25 = 55. This is a 3-4-5 Pythagorean triple.a² = c² − b² = 13² − 5² = 169 − 25 = 144a = √144 = 1212. Another triple: 5-12-13.7² + 24² = 49 + 576 = 62525² = 625Triangle area and congruence
Height must be perpendicular to the base, even if outside the triangle.
If all three sides match, triangles are congruent (same shape and size).
Two sides and the angle between them determine the triangle uniquely.
Same shape, different size. Ratios of corresponding sides are equal.
A = ½ × 10 × 630 cm²Shapes, area & perimeter
Every polygon has formulas for its perimeter (distance around) and area (space inside). Memorising the key ones — and understanding where they come from — is essential.
Length times width. Perimeter is twice the sum of both dimensions.
A rectangle with all sides equal. Area is side squared.
Half of base times perpendicular height. Perimeter is all three sides.
Same formula as a rectangle but height is perpendicular, not the slant side.
Average of the two parallel sides, multiplied by the height between them.
n is the number of sides. Divide by n for each interior angle.
A = ½ × (8 + 12) × 5A = ½ × 20 × 5 = 50 cm²50 cm²(5−2) × 180° = 540°540° ÷ 5 = 108°108°10 × 6 = 60 cm²½ × 10 × 4 = 20 cm²60 + 20 = 80 cm²Circles
A circle is the set of all points equidistant from a centre point. That distance is the radius (r). The diameter is twice the radius. All circle formulas flow from π (pi ≈ 3.14159…).
The perimeter of a circle. Use 2πr when you know the radius, πd for diameter.
Pi times radius squared. Radius must be squared — a common mistake is using diameter.
A fraction of the full circumference. θ is the central angle in degrees.
A fraction of the full circle area. Think of it as a slice of pie.
A diameter is the longest possible chord — passing through the centre.
Always perpendicular to the radius at the point of contact.
C = 2π × 7 = 14π ≈ 43.98 cmA = π × 7² = 49π ≈ 153.94 cm²43.98 cm, A ≈ 153.94 cm²72/360 = 1/5(1/5) × 2π × 10 = 4π ≈ 12.57 cm12.57 cm120/360 = 1/3(1/3) × π × 6² = 12π ≈ 37.70 cm²37.70 cm²Trigonometry
Trigonometry connects angles to side lengths in right triangles. The three core ratios — sine, cosine, and tangent — work for any angle and are the foundation of all advanced mathematics.
Sin θ = Opposite/Hypotenuse · Cos θ = Adjacent/Hypotenuse · Tan θ = Opposite/Adjacent
Opposite side divided by hypotenuse. Ranges from −1 to 1.
Adjacent side divided by hypotenuse. Also ranges from −1 to 1.
Opposite divided by adjacent. Equals sin/cos. Can be any value.
Use inverse trig to find the angle when you know the side ratio.
Exact values — must memorise
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
sinsin 35° = opp / 12opp = 12 × sin 35° = 12 × 0.5736 ≈ 6.88 cm6.88 cmtantan θ = 5/8 = 0.625θ = tan⁻¹(0.625) ≈ 32.0°32.0°c = √(1²+1²) = √2sin 45° = 1/√2 = √2/2 cos 45° = 1/√2 = √2/2√2/2 ≈ 0.7071a/sin A = b/sin B8/sin 40° = b/sin 75°b = 8 × sin 75° / sin 40° = 8 × 0.9659 / 0.6428 ≈ 12.02 cm12.02 cmGeometric proofs & reasoning
A proof is a logical argument that shows a geometric statement is true. Every step must be justified by a theorem, definition, or previously proven fact.
Every proof lists what is given, states what must be proved, then shows step-by-step reasoning.
A statement that has been proved and can now be used to justify other statements.
Foundational rules we accept as obviously true, like "a straight line is 180°".
A result so close to a proved theorem that little extra work is needed.
∠A + ∠B + ∠C = 180° (angle sum of triangle)∠C + d = 180°d = ∠A + ∠B20-question graded test
Answer all 20 questions then submit. You will see your score, grade, and a full explanation for every question.