CDS Mathematics · Mensuration 2D

Inscribed & Circumscribed Figures

One figure drawn inside another so that it just touches: squares and circles, a triangle's incircle and circumcircle, figures in a semicircle, and regular polygons.

All Mensuration 2D notes CDS Mathematics strategy

Why this matters

One of the two hardest pages here, with a third of it HARD. Every question turns on one shared length: the diagonal that is also a diameter, the radius that is also a half-side. Find that length and the rest is substitution.

Concept 1 of 5: Square in a circle, circle in a square

A square inside a circle has its diagonal along a diameter. A circle inside a square has its diameter equal to the side. Those two facts, and the triangle's version below, settle almost every 'largest square' or 'largest disc' question.

Definition

Square inscribed in a circle of radius $r$ : diagonal $2r$ , side $r\sqrt2$ , area $2r^2$ . Circle : square $= \pi : 2$ .
Circle inscribed in a square of side $s$ : radius $\dfrac{s}{2}$ , area $\dfrac{\pi s^2}{4}$ .
Equilateral triangle inscribed in a circle of radius $r$ : side $r\sqrt3$ , area $\dfrac{3\sqrt3}{4}r^2$ .
Rectangle inscribed in a circle: its diagonal is a diameter.
Joining the midpoints of a square's sides gives a square of half the area.

In a circle of radius r

\text{square} = 2r^2, \qquad \text{equilateral triangle} = \tfrac{3\sqrt3}{4}r^2

Worked example

A rectangle with sides in the ratio

3 : 4

is inscribed in a circle of radius

10

cm. Find its area.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — Elementary Mathematics · Q65Moderate

Example 1 · Mensuration 2D · Inscribed and Circumscribed Figures

Let x be the area of a square inscribed in a circle of radius r and y be the area of an equilateral triangle inscribed in the same circle. Which one of the following is correct ?

Drill 8 more on square in a circle, circle in a square

Concept 2 of 5: A triangle's incircle and circumcircle

The incircle's radius is the area divided by the semi-perimeter. The circumcircle's radius is

\dfrac{abc}{4\times\text{area}}

. For the two special triangles CDS likes, there are one-line versions.

Definition

Any triangle: $r = \dfrac{\Delta}{s}$ , $R = \dfrac{abc}{4\Delta}$ .
Right triangle with legs $a, b$ and hypotenuse $c$ : $r = \dfrac{a + b - c}{2}$ , $R = \dfrac{c}{2}$ .
Equilateral triangle of side $a$ : $r = \dfrac{a}{2\sqrt3}$ , $R = \dfrac{a}{\sqrt3}$ , so $R = 2r$ , and the height is $3r$ .
A circle through the apex of an equilateral triangle, touching the base at its midpoint, has the altitude as its diameter.

Inradius and circumradius

r = \frac{\Delta}{s}, \qquad R = \frac{abc}{4\Delta}

Worked example

Find the inradius and circumradius of a triangle with sides

7

24

and

25

cm.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — Elementary Mathematics · Q61Moderate

Example 2 · Mensuration 2D · Inscribed and Circumscribed Figures

What is the area of the circle (approximately) inscribed in a triangle with side lengths 12 cm, 16 cm and 20 cm ?

The inradius and the circumradius swap easily

For an equilateral triangle

\dfrac{a}{\sqrt3}

is the circumradius and

\dfrac{a}{2\sqrt3}

the inradius. Check with

R = 2r

: the circle through the corners is the bigger one.

Drill 8 more on a triangle's incircle and circumcircle

Concept 3 of 5: Figures in a semicircle or a quarter circle

Put the centre of the flat side at the origin. A square in a semicircle sits symmetrically on the diameter, so its top corner is at

\left(\dfrac s2, s\right)

and lies on the circle. A circle in a quarter circle sits on the bisector of the right angle.

Definition

Square in a semicircle of radius $r$ : $\dfrac{s^2}{4} + s^2 = r^2$ , so $s^2 = \dfrac{4r^2}{5}$ .
Largest triangle in a semicircle: base on the diameter, apex at the top, area $r^2$ . Any triangle on the diameter is right-angled at the arc.
Circle in a quarter circle of radius $R$ : its centre is $r\sqrt2$ from the corner, so $r\sqrt2 + r = R$ and $R : r = (\sqrt2 + 1) : 1$ .

Square in a semicircle

s^2 = \tfrac45 r^2

Worked example

A square is inscribed in a semicircle of radius

5

cm with one side on the diameter. Find its area.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q55Moderate

Example 3 · Mensuration 2D · Inscribed and Circumscribed Figures

What is the ratio of the area of a square inscribed in a semicircle of radius

r

to the area of square inscribed in a circle of radius

r

Drill 3 more on figures in a semicircle or a quarter circle

Concept 4 of 5: A square or rectangle wedged into a corner

When a square sits in the corner of a right triangle, or a rectangle's far corner touches a circle, put the corner at the origin. The far vertex then has simple coordinates, and 'it lies on the hypotenuse' or 'it lies on the circle' is one equation.

Definition

Square in the right angle of a right triangle with legs $a, b$ : its far corner $(s, s)$ lies on the hypotenuse, so $s = \dfrac{ab}{a + b}$ .
Square standing on the hypotenuse $c$ with altitude $h$ : $s = \dfrac{ch}{c + h}$ , which is smaller.
Largest square in an equilateral triangle of side $a$ stands on a side: $s = \dfrac{a\sqrt3}{2 + \sqrt3} = a(2\sqrt3 - 3)$ .
A point $(p, q)$ measured from the corner of a square lies on its incircle of radius $t$ when $(p - t)^2 + (q - t)^2 = t^2$ .

Square in the right angle

s = \frac{ab}{a + b}

Worked example

A square has one corner at the right angle of a right triangle with legs

12

cm and

6

cm, and the opposite corner on the hypotenuse. Find its side.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2024 · CDS (I) 2024 — Elementary Mathematics · Q13Hard

Example 4 · Mensuration 2D · Inscribed and Circumscribed Figures

ABC is a right-angled triangle, right-angled at B such that AB = 6 cm and BC = 8 cm. What is the perimeter of the square inscribed in the triangle ABC with maximum area ?

Side or perimeter?

These stems often ask for the perimeter or the area of the square, and the side itself is an option. Finish the question.

Drill 4 more on a square or rectangle wedged into a corner

Concept 5 of 5: Regular polygons — hexagon and octagon

A regular hexagon is six equilateral triangles. A regular octagon is a square with its corners cut off as isosceles right triangles. Both facts turn a polygon question into triangle work.

Definition

Hexagon of side $a$ : area $6\times\dfrac{\sqrt3}{4}a^2 = \dfrac{3\sqrt3}{2}a^2$ .
Cutting the corners off an equilateral triangle of side $3s$ leaves a hexagon of side $s$ : triangle : hexagon $= 9 : 6 = 3 : 2$ .
Octagon from a square of side $a$ : cut legs $x$ with $a - 2x = x\sqrt2$ ; the octagon's side is $a(\sqrt2 - 1)$ .
Regular $n$ -gon of side $a$ : inradius $\dfrac a2\cot\dfrac{180^\circ}{n}$ .

Hexagon and octagon

A_{\text{hex}} = \tfrac{3\sqrt3}{2}a^2, \qquad \text{octagon side} = a(\sqrt2 - 1)

Worked example

A regular octagon is made by cutting the corners off a square of side

10

cm. Find the octagon's side.

(\sqrt2 = 1.414)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q76Hard

Example 5 · Mensuration 2D · Inscribed and Circumscribed Figures

The corners of a square of side 'a' are cut away so as to form a regular octagon. What is the side of the octagon ?

Drill 4 more on regular polygons — hexagon and octagon

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (5)

Square in a circle, circle in a square
In a circle of radius r
$\text{square} = 2r^2, \qquad \text{equilateral triangle} = \tfrac{3\sqrt3}{4}r^2$
A triangle's incircle and circumcircle
Inradius and circumradius
$r = \frac{\Delta}{s}, \qquad R = \frac{abc}{4\Delta}$
Figures in a semicircle or a quarter circle
Square in a semicircle
$s^2 = \tfrac45 r^2$
A square or rectangle wedged into a corner
Square in the right angle
$s = \frac{ab}{a + b}$
Regular polygons — hexagon and octagon
Hexagon and octagon
$A_{\text{hex}} = \tfrac{3\sqrt3}{2}a^2, \qquad \text{octagon side} = a(\sqrt2 - 1)$

Watch out for (2)

The inradius and the circumradius swap easily
→ A triangle's incircle and circumcircle
Side or perimeter?
→ A square or rectangle wedged into a corner

Test yourself on Mensuration 2D

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Mensuration 2D chapter test Drill all 32 questions