AFCAT Numerical Ability · Area and Perimeter

Combined Figures: Inscribed Shapes and Shaded Regions

One figure drawn inside another, or a region cut from two of them: find the measure the two figures share, then add or subtract the pieces.

All Area and Perimeter notes

Why this matters

More than half the chapter's HARD questions are on this page, and nearly half of its questions come with a figure. Each one turns on a single link between the shapes, such as a square's diagonal being the circle's diameter. Miss that link and every option looks possible; find it and the rest is the earlier pages' formulas.

Concept 1 of 3: Figures inscribed in one another

When one figure sits exactly inside another, they share a length. A circle inside a square touches all four sides, so its diameter is the square's side. A square inside a circle has its corners on the circle, so its diagonal is the circle's diameter.

Definition

Circle in a square of side $a$ : radius $\tfrac a2$ . This is the largest circle that fits.
Square in a circle of radius $r$ : diagonal $2r$ , so area $\tfrac{(2r)^2}{2} = 2r^2$ .
Equilateral triangle of side $a$ in a circle: circumradius $R = \dfrac{a}{\sqrt 3}$ , so $a = R\sqrt 3$ . Its inscribed circle has radius $\dfrac{a}{2\sqrt 3}$ .
Regular hexagon in a circle: its side equals the radius, so its area is $\dfrac{3\sqrt 3}{2}R^2$ .
Regular $n$ -gon in a circle of radius $R$ : it is $n$ triangles with two sides $R$ and angle $\tfrac{360°}{n}$ between them, so area $= \tfrac12 nR^2 \sin\tfrac{360°}{n}$ . Its inscribed circle has radius $R\cos\tfrac{180°}{n}$ .
A rhombus with one corner at the centre and three on the circle has every side equal to the radius: it is two equilateral triangles.

Square and circle

\text{circle in square: } r = \tfrac a2;\qquad \text{square in circle: } d = 2r,\ A = 2r^2

Worked example

A circle has a radius of 7 cm. Find the area of the largest square that fits inside it, and of the smallest square that fits round it.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 1 · Area and Perimeter · Combined Figures

A square is inscribed in a circle of radius

5

cm. Find the area of the square:

Which length is shared

For a square inside a circle of radius 7, the diameter 14 is the square's diagonal, giving 98. Taking 14 as the side gives 196, the square outside the circle.

The circumradius is not half the side

An equilateral triangle of side

6\sqrt 3

has a circumradius of 6, from

R = \tfrac{a}{\sqrt 3}

. Half its side,

3\sqrt 3

, is neither its circumradius nor its inradius.

Drill 6 more on figures inscribed in one another

Concept 2 of 3: Shaded regions by subtraction

A shaded region is usually what is left when one figure is taken out of another. Find the outer area, find the area removed, and subtract. When the region is an overlap, cut it into triangles and sectors whose areas you know.

Definition

Outer minus inner: square less circle, rectangle less a quadrant, big circle less small circles.
Four quadrants at the corners of a square of side $2r$ make one full circle of radius $r$ .
Segment (between a chord and its arc) = sector − triangle. For a 90° angle at the centre, the triangle is $\tfrac12 r^2$ .
An overlap of two figures: join the centre to the crossing points and split the region into triangles and sectors.
If the figure is symmetric and only part is shaded, find the whole unshaded-to-shaded split and take the fraction the figure shows.

Shaded area

\text{shaded} = \text{outer} - \text{removed},\qquad \text{segment} = \text{sector} - \triangle

Worked example

A square of side 14 cm has a quadrant of radius 7 cm drawn at each corner. Find the area of the square not covered by the quadrants. (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Area and Perimeter · Combined Figures

A square park has side

20

m. A circular fountain of radius

3

m is built in the center. Find the remaining area: (use

\pi = 3.14

)

A segment is not a sector

With radius 14 and a right angle at the centre, the sector is 154 but the segment is only 56. The triangle between the radii must come off.

Read what is shaded

In a square with quadrants at its corners, the shaded part may be the quadrants (154 for side 14) or what is left between them (42). Both numbers are usually among the options.

Drill 6 more on shaded regions by subtraction

Concept 3 of 3: Building a figure from pieces

Some figures are several simple ones joined edge to edge: a rectangle with a semicircle on each end, an L or T made of rectangles, a boundary of half-circles. Cut the figure where the pieces meet, find each one, and add, making sure no piece is counted twice.

Definition

Rectangle with semicircular ends on the side $b$ : the two ends make one circle of diameter $b$ . Area $= lb + \pi\left(\tfrac b2\right)^2$ ; perimeter $= 2l + \pi b$ .
L and T shapes: split into rectangles that do not overlap, or take a small rectangle away from a big one.
Semicircles on a line $AC$ with $B$ between: the region inside the big one and outside the two small ones is $\tfrac{\pi}{8}(AC^2 - AB^2 - BC^2)$ .
Arc lengths add: two semicircles of diameter $\tfrac d2$ have the same total arc as one semicircle of diameter $d$ .

Rectangle with semicircular ends

A = lb + \pi\left(\tfrac b2\right)^2,\qquad P = 2l + \pi b

Worked example

A running ground is a rectangle 50 m by 21 m with a semicircle on each 21 m side. Find its area and its perimeter. (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Area and Perimeter · Combined Figures

A garden is in the form of a rectangle with semi-circular ends on either side as shown in the diagram. The length and breadth of the rectangle are 20 m and 14 m, respectively. The cost of levelling the plot at ₹25 per square metre is

Two semicircles, one circle

A 50 m by 21 m ground with round ends adds one circle of radius 10.5, not two. Adding two full circles doubles the curved part.

Do not count the overlap twice

Splitting a T shape into a 6 by 2 bar and a stem, measure the stem from below the bar. Taking its full height through the bar counts the shared square twice.

Drill 3 more on building a figure from pieces

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 (3)

Figures inscribed in one another
Square and circle
$\text{circle in square: } r = \tfrac a2;\qquad \text{square in circle: } d = 2r,\ A = 2r^2$
Shaded regions by subtraction
Shaded area
$\text{shaded} = \text{outer} - \text{removed},\qquad \text{segment} = \text{sector} - \triangle$
Building a figure from pieces
Rectangle with semicircular ends
$A = lb + \pi\left(\tfrac b2\right)^2,\qquad P = 2l + \pi b$

Watch out for (6)

Which length is shared
→ Figures inscribed in one another
The circumradius is not half the side
→ Figures inscribed in one another
A segment is not a sector
→ Shaded regions by subtraction
Read what is shaded
→ Shaded regions by subtraction
Two semicircles, one circle
→ Building a figure from pieces
Do not count the overlap twice
→ Building a figure from pieces

Drill the questions on this topic

18 questions from the bank, with cart and Word export.

Drill all 18 questions