Area of Triangle, Circumradius and Inradius | NIMCET

Quick answer: For a triangle with sides $a, b, c$ , semi-perimeter $s$ and area $\Delta$ : $\Delta = \frac{1}{2}bc\sin A = \sqrt{s(s - a)(s - b)(s - c)}$ , the circumradius is $R = \frac{abc}{4\Delta}$ , the inradius is $r = \frac{\Delta}{s}$ , and the exradius opposite A is $r_1 = \frac{\Delta}{s - a}$ . The half-angle formulas, such as $\tan\frac{A}{2} = \sqrt{\frac{(s - b)(s - c)}{s(s - a)}}$ , connect these quantities to the angles.

This article builds on the sine rule and cosine rule.

Half-angle formulas in a triangle

Formula
$\sin\frac{A}{2} = \sqrt{\frac{(s - b)(s - c)}{bc}}$
$\cos\frac{A}{2} = \sqrt{\frac{s(s - a)}{bc}}$
$\tan\frac{A}{2} = \sqrt{\frac{(s - b)(s - c)}{s(s - a)}} = \frac{\Delta}{s(s - a)}$

All roots are positive because each half-angle is acute. These come from the cosine rule together with $\cos A = 1 - 2\sin^2\frac{A}{2}$ (see the sub-multiple angle formulas).

Area formulas

Formula	Use when
$\Delta = \frac{1}{2}bc\sin A$	Two sides and the included angle are known
$\Delta = \sqrt{s(s - a)(s - b)(s - c)}$	All three sides are known (Heron's formula)
$\Delta = \frac{abc}{4R}$	Sides and circumradius are known
$\Delta = rs$	Inradius and semi-perimeter are known
$\Delta = 2R^2\sin A\sin B\sin C$	Angles and circumradius are known

Circumradius, inradius and exradii

Quantity	Formulas
Circumradius $R$	$R = \frac{a}{2\sin A} = \frac{abc}{4\Delta}$
Inradius $r$	$r = \frac{\Delta}{s} = (s - a)\tan\frac{A}{2} = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$
Exradius $r_1$ (opposite A)	$r_1 = \frac{\Delta}{s - a} = s\tan\frac{A}{2}$
Exradii $r_2, r_3$	$r_2 = \frac{\Delta}{s - b}$ , $r_3 = \frac{\Delta}{s - c}$

Useful relations:

$r_1 + r_2 + r_3 - r = 4R$

$\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{1}{r}$

Special triangles

Triangle	Key results
Equilateral, side $a$	$\Delta = \frac{\sqrt{3}}{4}a^2$ , $R = \frac{a}{\sqrt{3}}$ , $r = \frac{a}{2\sqrt{3}}$ , so $R = 2r$
Right-angled at C	$R = \frac{c}{2}$ (half the hypotenuse), $r = s - c = \frac{a + b - c}{2}$

Worked examples

Example 1: the 3-4-5 triangle

Find $\Delta$ , $R$ and $r$ for sides 3, 4, 5.

Step 1: It is right-angled, so $\Delta = \frac{1}{2} \cdot 3 \cdot 4 = 6$ and $s = 6$ .

Step 2: $R = \frac{5}{2} = 2.5$ and $r = \frac{\Delta}{s} = 1$ .

Check: $R = \frac{abc}{4\Delta} = \frac{60}{24} = 2.5$ ✓.

Example 2: Heron's formula

Find the area of the triangle with sides 13, 14, 15.

Step 1: $s = 21$ .

Step 2:

$\Delta = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84$

Step 3: Then $r = \frac{84}{21} = 4$ and $R = \frac{13 \cdot 14 \cdot 15}{4 \cdot 84} = \frac{2730}{336} = \frac{65}{8}$ .

Example 3: exradius

For the 13, 14, 15 triangle, find $r_1$ (opposite the side 13).

$r_1 = \frac{\Delta}{s - a} = \frac{84}{21 - 13} = \frac{84}{8} = 10.5$

Example 4: area from two sides and an angle

Two sides are 6 and 8 with included angle 30°. Then $\Delta = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 30^\circ = 12$ .

Summary table

Quantity	Fastest formula
Area	$\frac{1}{2}bc\sin A$ or Heron's formula
$R$	$\frac{abc}{4\Delta}$
$r$	$\frac{\Delta}{s}$
$r_1$	$\frac{\Delta}{s - a}$
$\tan\frac{A}{2}$	$\frac{\Delta}{s(s - a)}$

Exam tips

Compute $s$ , $s - a$ , $s - b$ and $s - c$ first; almost every formula here uses them.
Remember 13, 14, 15 gives area 84 and 3, 4, 5 gives area 6; both appear often.
For a right triangle, $R$ is half the hypotenuse, which saves time.
In an equilateral triangle, $R = 2r$ ; the incentre and circumcentre coincide.

See all topics in the NIMCET Trigonometry chapter.

Common mistakes

Using the perimeter instead of the semi-perimeter in Heron's formula.
Writing $r = \frac{\Delta}{s - a}$ ; that is the exradius $r_1$ , not the inradius.
Taking a negative square root in the half-angle formulas.
Forgetting that $\Delta = \frac{1}{2}bc\sin A$ needs the angle between sides $b$ and $c$ .

Frequently Asked Questions

What is the formula for the circumradius of a triangle?

R = abc/(4Δ), or R = a/(2 sin A).

What is the formula for the inradius of a triangle?

r = Δ/s, where s is the semi-perimeter.

What is Heron's formula?

Δ = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2.

What is the relation between R and r in an equilateral triangle?

R = 2r. For side a, R = a/√3 and r = a/(2√3).

What is the exradius of a triangle?

The exradius r₁ is the radius of the circle touching side a and the extensions of the other two sides. It equals Δ/(s − a).