| Property / formula | Summary |
|---|---|
| Angle sum | Sum of interior angles of any triangle is 180°. |
| Exterior angle | An exterior angle equals the sum of the two interior opposite angles. |
| Triangle inequality | Sum of any two sides is greater than the third side. |
| Area (base–height) | Area = ½ × base × height. |
| Heron's formula | For sides a, b, c and semiperimeter s = (a + b + c)/2, area = √[s(s − a)(s − b)(s − c)]. |
| Right triangle (Pythagoras) | In a right triangle with hypotenuse c and legs a, b: a² + b² = c². |
| Equilateral triangle | All sides equal; each angle = 60°; area = (√3/4)a². |
| Isosceles triangle | Two equal sides; base angles opposite equal sides are equal. |
| Centroid | Medians intersect at centroid, which divides each median in ratio 2:1 from vertex. |
| Inradius (r) | Area = r × s, where s is semiperimeter. |
| Circumradius (R) | For triangle with sides a, b, c and opposite angles A, B, C: a = 2R sin A, etc.; also area = abc / (4R). |
| Result | Key idea |
|---|---|
| Basic formulas | Circumference = 2πr, area = πr². |
| Angle at centre vs circumference | Angle at centre is twice the angle at the circumference standing on the same arc. |
| Angles in same segment | Angles at the circumference standing on the same chord/arc are equal. |
| Semi-circle | Angle in a semicircle is a right angle (90°). |
| Cyclic quadrilateral | Opposite angles of a cyclic quadrilateral sum to 180°. |
| Tangent–radius | A tangent to a circle is perpendicular to the radius at the point of contact. |
| Two tangents from a point | From an external point, tangents drawn to a circle are equal in length. |
| Chord distance | Equal chords are equidistant from centre; chord nearer centre is longer. |
| Angle | sin θ | cos θ | tan θ | cosec θ | sec θ | cot θ |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Not defined | 1 | Not defined |
| 30° | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° | 1 | 0 | Not defined | 1 | Not defined | 0 |
| Identity | Expression |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. |
| Reciprocal | sin θ = 1/cosec θ; cos θ = 1/sec θ; tan θ = 1/cot θ. |
| Quotient | tan θ = sin θ / cos θ; cot θ = cos θ / sin θ. |
| Complementary angles | sin(90° − θ) = cos θ; cos(90° − θ) = sin θ; tan(90° − θ) = cot θ; sec(90° − θ) = cosec θ. |
| Sum & difference (sin) | sin(A ± B) = sin A cos B ± cos A sin B. |
| Sum & difference (cos) | cos(A ± B) = cos A cos B ∓ sin A sin B. |
| Double angle | sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. |
| Product–sum | 2 sin A cos B = sin(A + B) + sin(A − B); 2 cos A cos B = cos(A + B) + cos(A − B). |
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