Mensuration (2D)

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high importance~2 Q in Tier 123 formulas⚡ 12 shortcuts5 subtopics

Overview Learn (5)Formulas (23)Examples (4)Practice (62)

Subtopic 4 of 5·← Circles, sectors and rings·Percentage change, similarity and re-bent shapes →

Regular polygons and inscribed figures

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Regular hexagon (side $a$ ): area $\frac{3\sqrt3}{2}a^2$ — six equilateral triangles of side $a$ .
Regular octagon: area $2(1+\sqrt2)\,a^2$ .
Any regular $n$ -gon: area $=\frac12\times\text{perimeter}\times\text{apothem}$ .

Standard inscriptions (most-tested):

Largest square in a circle: diagonal $=$ diameter.
Largest circle in a square: diameter $=$ side of square.
Largest circle in an equilateral triangle: $r=\frac{a}{2\sqrt3}$ .
Largest triangle in a semicircle is right-angled.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 12 practice Q

Detailed notes

The regular-polygon toolkit

For a regular polygon of $n$ sides (side $a$ ):

Exterior angle $= \frac{360^\circ}{n}$ ; interior angle $= 180^\circ -$ exterior.
Interior angle sum $= (n-2) \times 180^\circ$ (any polygon).
Diagonals $= \frac{n(n-3)}{2}$ .
Area $= n \times$ (area of the $n$ congruent isosceles triangles from the centre) $= \frac{1}{2} \times P \times \text{apothem}$ , where $P = na$ and the apothem is the centre-to-side distance.

Angle chain: interior $150^\circ$ → ext $30^\circ$ → $n = 12$ . Sum chain: sum $1080^\circ$ → $n = \frac{1080}{180} + 2 = 8$ . Diagonal chain: $d = 20$ → $n(n-3) = 40$ → $n = 8$ .

The hexagon (the exam's favourite)

A regular hexagon is six equilateral triangles: side = radius of the circumscribed circle.

Area $= 6 \times \frac{\sqrt3}{4}a^2 = \frac{3\sqrt3}{2}a^2$ — know as "6 equilateral pieces", faster than the formula.
Long diagonal $= 2a$ (through the centre, two radii).
Short diagonal $= a\sqrt3$ (skips one vertex).
Hexagon side = radius: an equilateral triangle built on alternate vertices has side $a\sqrt3$ .

How the questions are built

Given an interior/exterior angle → find $n$ (or the other angle).
Given the sum of interior angles → find $n$ , then each exterior angle $\frac{360^\circ}{n}$ .
Given diagonals → solve the quadratic-ish $n(n-3) = 2d$ by testing factors.
Hexagon area from side, or side from area — pure equilateral work.
Apothem questions: $K = \frac12 \times$ perimeter $\times$ apothem, often with $\sqrt3$ apothems for hexagons.

The table worth memorising

$n$	Name	Interior angle	Diagonals
5	Pentagon	$108^\circ$	5
6	Hexagon	$120^\circ$	9
7	Heptagon	$\approx 128.57^\circ$	14
8	Octagon	$135^\circ$	20
9	Nonagon	$140^\circ$	27
10	Decagon	$144^\circ$	35
12	Dodecagon	$150^\circ$	54

Reading backwards from any column gives $n$ instantly: interior $144^\circ$ → decagon; 20 diagonals → octagon; interior $150^\circ$ → 12 sides. The exterior angle is always $\frac{360^\circ}{n}$ and the interior angles of ANY $n$ -gon (regular or not) sum to $(n-2) \times 180^\circ$ — only the "each angle" phrasing requires regularity.

Quick revision

ext $= \frac{360}{n}^\circ$ ; int $= 180 -$ ext; sum $= (n-2) \times 180^\circ$ .
Diagonals $= \frac{n(n-3)}{2}$ ; solve by factor pairs.
$K = \frac12 \times P \times$ apothem.
Hexagon $= 6$ equilateral triangles: $K = \frac{3\sqrt3}{2}a^2$ ; long diagonal $2a$ ; short $a\sqrt3$ .

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Hexagon as six equilateral trianglesvery common2 practice Q

How to spot it:

A regular hexagon's side given — area asked; or area/long-diagonal given — side asked.

K = 6 \times \frac{\sqrt3}{4}a^2 = \frac{3\sqrt{3}}{2}a^2

Picture the 6 equilateral triangles of side $a$ .
Area $= 6 \times \frac{\sqrt3}{4}a^2 = \frac{3\sqrt3}{2}a^2$ ; keep $\sqrt3$ symbolic.
Reverse: $a^2 = \frac{2K}{3\sqrt3}$ .

Why: the centre joins to all vertices, and the hexagon's central angle of $60^\circ$ makes each piece equilateral.

Example: Find the area of a regular hexagon of side 6 cm.

$K = \frac{3\sqrt3}{2} \times 36 = 54\sqrt{3}$ sq cm (six $\frac{\sqrt3}{4} \times 36 = 9\sqrt3$ pieces).

Type 2: Area from perimeter and apothemcommon2 practice Q

How to spot it:

Perimeter and apothem (centre-to-side distance) given — or both derived — area asked.

K = \frac{1}{2} \times P \times \text{apothem}

Half of (perimeter × apothem) — one multiplication, no side counting.
For a hexagon the apothem is $\frac{\sqrt3}{2}a$ ; square-side shapes give integer apothems.
If the apothem arrives with $\sqrt3$ , expect a $\sqrt3$ in the answer.

Why: the polygon splits into $n$ triangles of base (side) and height (apothem); their areas total exactly that formula.

Example: A regular polygon has perimeter 72 cm and apothem 6 cm. Its area is:

$K = \frac12 \times 72 \times 6 = 216$ sq cm.

Type 3: Interior/exterior angles and the angle sumvery common2 practice Q

How to spot it:

An interior angle, exterior angle, or the angle sum of a regular polygon given — find $n$ or the other angle.

\text{ext} = \frac{360^\circ}{n},\quad \text{int} = 180^\circ - \text{ext},\quad \text{sum} = (n-2) \times 180^\circ

Whatever is given, route through the exterior angle: ext $= 180^\circ -$ int.
$n = \frac{360^\circ}{\text{ext}}$ ; from the sum: $n = \frac{\text{sum}}{180^\circ} + 2$ .
Answer the quantity actually asked (each angle vs total sum vs $n$ ).

Why: exterior angles always total $360^\circ$ for any convex polygon — the cleanest hub of the three routes.

Example: Find the measure of each interior angle of a regular octagon.

ext $= \frac{360}{8} = 45^\circ$ ; interior $= 180 - 45 = 135^\circ$ .

Type 4: Counting diagonalscommon2 practice Q

How to spot it:

'How many diagonals does a polygon of n sides have?' — or reversed: given the diagonal count, find $n$ .

d = \frac{n(n-3)}{2}

Forward: substitute $n$ ; the product $n(n-3)$ is always even.
Reverse: set $n(n-3) = 2d$ and test factor pairs of $2d$ that differ by 3.
Sanity: triangle 0, square 2, pentagon 5, hexagon 9, octagon 20, decagon 35.

Why: each of the $n$ vertices connects to $n-3$ non-adjacent vertices, and every diagonal gets counted twice.

Example: A polygon has 90 diagonals. The number of its sides is:

$n(n-3) = 180$ : try $15 \times 12 = 180$ → $n = 15$ .

Type 5: Hexagon diagonals and side relationsoccasional2 practice Q

How to spot it:

A hexagon's long/short diagonal related to its side, or a triangle built inside the hexagon.

\text{long diagonal} = 2a,\quad \text{short diagonal} = a\sqrt{3}

Long diagonal = two radii through the centre: $2a$ .
Short diagonal (skip one vertex) $= a\sqrt3$ — it is the side of the inscribed equilateral triangle.
Radius of the circumscribed circle = side $a$ — used for 'circle through hexagon corners' questions.

Why: the six $60^\circ$ centre angles make every such segment a chord whose length follows from an equilateral or 30-60-90 triangle.

Example: The length of the longest diagonal of a regular hexagon of side 8 cm is:

$2a = 16$ cm (it passes through the centre, spanning two radii = two sides).

Formulas

Regular hexagon

K=\frac{3\sqrt3}{2}a^2=6\times\frac{\sqrt3}{4}a^2

Regular octagon

K=2(1+\sqrt2)a^2

Square in a circle

\text{diagonal}=\text{diameter}=2R

Circle in a square

d_{\text{circle}}=a

Circle in an equilateral triangle

r=\frac{a}{2\sqrt3}

Shortcut tricks

⚡ Hexagon = six equilateral triangles

Multiply the equilateral area by 6: $\frac{3\sqrt3}{2}a^2$ . Given the perimeter, divide by 6 first.

Example: Find the area of a regular hexagon of perimeter 36 cm.

Side $=6$ : area $=\frac{3\sqrt3}{2}\times36=54\sqrt3$ sq cm.

⚡ Inscribed square → the diameter is the diagonal

All four corners touch the circle, so the square's diagonal passes through the centre and equals the diameter.

Example: A square is inscribed in a circle of radius $7\sqrt2$ cm. Find the side of the square.

Diagonal $=14\sqrt2\Rightarrow$ side $=\frac{14\sqrt2}{\sqrt2}=14$ cm.

Where students lose marks

Hexagon area computed with $\frac{\sqrt3}{4}a^2$ (one triangle) instead of six triangles.
Inscribed vs circumscribed confusion: circle inside the triangle vs triangle inside the circle.
Using the side where the diagonal (or vice versa) is required.

Practice sets — 12 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 12 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.