| Property / formula | Summary |
|---|---|
| Angle sum | Sum of interior angles of any triangle is \( 180^\circ \). |
| 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) | \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \) |
| Heron's formula | For sides \( a, b, c \) and semiperimeter \( s = \frac{a + b + c}{2} \), area \( = \sqrt{s(s - a)(s - b)(s - c)} \). |
| Right triangle (Pythagoras) | In a right triangle with hypotenuse \( c \) and legs \( a, b \): \( a^2 + b^2 = c^2 \). |
| Equilateral triangle | All sides equal; each angle = \( 60^\circ \); area \( = \frac{\sqrt{3}}{4}a^2 \), height \( h = \frac{\sqrt{3}}{2}a \). |
| 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) | \( \text{Area} = r \times s \), where \( s \) is semiperimeter (\( r = \frac{\text{Area}}{s} \)). |
| Circumradius (R) | For triangle with sides \( a, b, c \) and opposite angles \( A, B, C \): \( a = 2R \sin A \); also area \( = \frac{abc}{4R} \). |
| Result | Key idea |
|---|---|
| Basic formulas | Circumference \( = 2\pi r \), area \( = \pi r^2 \). |
| 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 (\( 90^\circ \)). |
| Cyclic quadrilateral | Opposite angles of a cyclic quadrilateral sum to \( 180^\circ \). |
| 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 | \( \sin\theta \) | \( \cos\theta \) | \( \tan\theta \) | \( \text{cosec}\;\theta \) | \( \sec\theta \) | \( \cot\theta \) |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Not defined | 1 | Not defined |
| 30° | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{3}} \) | 2 | \( \frac{2}{\sqrt{3}} \) | \( \sqrt{3} \) |
| 45° | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{\sqrt{2}} \) | 1 | \( \sqrt{2} \) | \( \sqrt{2} \) | 1 |
| 60° | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) | \( \frac{2}{\sqrt{3}} \) | 2 | \( \frac{1}{\sqrt{3}} \) |
| 90° | 1 | 0 | Not defined | 1 | Not defined | 0 |
| Identity | Expression |
|---|---|
| Pythagorean | \( \sin^2\theta + \cos^2\theta = 1 \); \( 1 + \tan^2\theta = \sec^2\theta \); \( 1 + \cot^2\theta = \text{cosec}^2\theta \). |
| Reciprocal | \( \sin\theta = \frac{1}{\text{cosec}\;\theta} \); \( \cos\theta = \frac{1}{\sec\theta} \); \( \tan\theta = \frac{1}{\cot\theta} \). |
| Quotient | \( \tan\theta = \frac{\sin\theta}{\cos\theta} \); \( \cot\theta = \frac{\cos\theta}{\sin\theta} \). |
| Complementary angles | \( \sin(90^\circ - \theta) = \cos\theta \); \( \cos(90^\circ - \theta) = \sin\theta \); \( \tan(90^\circ - \theta) = \cot\theta \); \( \sec(90^\circ - \theta) = \text{cosec}\;\theta \). |
| Sum & difference (sin) | \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \). |
| Sum & difference (cos) | \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \). |
| Double angle | \( \sin 2\theta = 2 \sin\theta \cos\theta \); \( \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \). |
| 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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