NDA Mathematics · Formula sheet

Properties of Triangle formulas

13 formulas and 12 common traps for NDA Mathematics Properties of Triangle, grouped by subtopic.

Sine & Cosine Rules — Solving Triangles

Triangle Notation & Basic Relations

Angle sum & semi-perimeter

A + B + C = \pi, \qquad s = \dfrac{a+b+c}{2}

The Sine Rule

Sine rule

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R

The Cosine Rule

Cosine rule

\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}

Determining the Nature of a Triangle

Right-angle tests

c^2 = a^2 + b^2 \iff C = 90^\circ \iff \cos^2 A + \cos^2 B + \cos^2 C = 1

Area of a Triangle

Area formulas

\Delta = \tfrac12 ab\sin C = \sqrt{s(s-a)(s-b)(s-c)} = \dfrac{abc}{4R} = rs

Angle Ratios ↔ Side Ratios

Sides proportional to sines

a : b : c = \sin A : \sin B : \sin C

Sine Rule in Geometric Configurations

Sine rule in a sub-triangle

\dfrac{AB}{\sin(\angle ADB)} = \dfrac{AD}{\sin(\angle ABD)}

Common traps

Side over sine of its OWN opposite angle

The sine rule pairs each side with the angle facing it:

\frac{a}{\sin A}

, never

\frac{a}{\sin B}

. Pairing a side with the wrong angle is the classic slip.

The ratio is $2R$ , not $R$

The common ratio

\frac{a}{\sin A}

equals the DIAMETER of the circumcircle,

2R

— not the radius

R

. So

a = 2R\sin A

; forgetting the factor 2 halves your circumradius.

The ambiguous SSA case can give TWO triangles

Given two sides and a non-included angle, the sine rule can yield a sine value that fits two angles (

\theta

and

180^\circ-\theta

), producing two valid triangles. Don't assume the answer is unique without checking the angle sum.

Put the opposite side as the one being squared on the left

For angle

C

, the formula has

c^2

(the side opposite

C

) isolated and

-2ab\cos C

. Mixing up which side is opposite the angle flips the sign of the cosine and the verdict on acute/obtuse.

A negative cosine means an OBTUSE angle

When

\cos C = \frac{a^2+b^2-c^2}{2ab}

comes out negative, the angle is obtuse (between

90^\circ

and

180^\circ

) — don't discard it as 'impossible'. It is acute only when the numerator

a^2+b^2-c^2 > 0

Test $a^2+b^2 = c^2$ on the LARGEST side only

The right-angle test

a^2 + b^2 = c^2

must use the largest side as

c

. Plugging a shorter side in for

c

will fail even for a genuine right triangle — the right angle always faces the longest side.

Area uses $\sin$ of the angle, not $\cos$

The two-sides-and-included-angle area is

\Delta = \tfrac12 ab\sin C

. Writing

\tfrac12 ab\cos C

is a classic slip —

\cos

belongs to the cosine rule, not the area.

Heron's $s$ is the SEMI-perimeter

\Delta = \sqrt{s(s-a)(s-b)(s-c)}

s = \frac{a+b+c}{2}

— half the perimeter. Using the full perimeter

a+b+c

gives a badly wrong area.

Triangle Identities — A+B+C = π, Half & Double Angle

Learn this subtopic in the notes

Consequences of A + B + C = π

Half-angle complement

\sin\dfrac{B+C}{2} = \cos\dfrac{A}{2}, \qquad \sin(B+C) = \sin A

The tan-Sum = tan-Product Identity

tan sum = tan product

\tan A + \tan B + \tan C = \tan A\,\tan B\,\tan C

cos 2A Sums & Right-Angle Detection

Right-angle signature

\sin^2 A + \sin^2 B + \sin^2 C = 2 \iff \text{right-angled}

Half-Angle Formulas & Sum-to-Product

Half-angle tangent

\tan\dfrac{A}{2} = \dfrac{r}{s-a} = \sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}

Common traps

$\sin(B+C) = +\sin A$ , but $\cos(B+C) = -\cos A$

Both

B+C

and

A

sum to

\pi

, so

\sin(B+C) = \sin A

(same sign) but

\cos(B+C) = -\cos A

(opposite sign). Forgetting the minus on the cosine is the standard error.

The sum equals the PRODUCT, only in a triangle

\tan A + \tan B + \tan C = \tan A\tan B\tan C

holds because

A+B+C=\pi

. It is NOT a general identity — don't apply it unless the three angles are a triangle's angles.

In-circle, Circumcircle & Regular Polygons

Learn this subtopic in the notes

Incircle, Circumcircle & the Central Angle

Inradius, circumradius, central angle

r = \dfrac{\Delta}{s}, \quad R = \dfrac{abc}{4\Delta}, \quad \angle BOC = 2\,\angle BAC

Regular Polygon Geometry

Regular n-gon inradius

r = \dfrac{s}{2}\cot\dfrac{\pi}{n}, \qquad \text{interior angle} = \dfrac{(n-2)180^\circ}{n}

Common traps

Inradius $r$ vs circumradius $R$ — different formulas

The INradius (inside, touching the sides) is

r = \frac{\Delta}{s}

; the CIRCUMradius (through the vertices) is

R = \frac{abc}{4\Delta}

. Swapping the two — using

\frac{\Delta}{s}

when the circle passes through the vertices — is the most common error here.

Central angle is TWICE the inscribed angle

An arc subtends

\angle BOC = 2\,\angle BAC

at the centre — twice, not half, the inscribed angle. Halving it instead of doubling reverses the relation.

Properties of Triangle formulas

Sine & Cosine Rules — Solving Triangles

Triangle Notation & Basic Relations

The Sine Rule

The Cosine Rule

Determining the Nature of a Triangle

Area of a Triangle

Angle Ratios ↔ Side Ratios

Sine Rule in Geometric Configurations

Common traps

Triangle Identities — A+B+C = π, Half & Double Angle

Consequences of A + B + C = π

The tan-Sum = tan-Product Identity

cos 2A Sums & Right-Angle Detection

Half-Angle Formulas & Sum-to-Product

Common traps

In-circle, Circumcircle & Regular Polygons

Incircle, Circumcircle & the Central Angle

Regular Polygon Geometry

Common traps

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