Combined figures

The most distinctive class of board-exam problems in this chapter shows a complex diagram with shaded regions and asks for an area. The recipe is the same every time: decompose the shaded region into known pieces, then add (or subtract) their areas. The pieces are sectors, segments, triangles, rectangles, and circles.

The general strategy

1. Identify all the basic shapes present in the figure: full circles, semicircles, sectors, triangles, rectangles, squares.
2. Identify the shaded region in terms of these basic shapes , what to add and what to subtract.
3. Compute each basic shape's area using its formula.
4. Sum or subtract as the decomposition demands.
5. Check the answer's units and approximate magnitude.

Common configurations

Configuration A: Three semicircles on the sides of a right triangle. The classic lune of Hippocrates. Total shaded area = sum of semicircle areas on the legs minus the segments of the largest semicircle covered by the triangle.

Configuration B: Circle inscribed in a square. Shaded region (square minus circle) = $s^2 - \pi(s/2)^2 = s^2(1 - \pi/4)$, where $s$ is the side.

Configuration C: Square inscribed in a circle. Shaded region (circle minus square) = $\pi r^2 - 2r^2$ (because the inscribed square has diagonal $= 2r$, side $= r\sqrt 2$, area $= 2r^2$).

Configuration D: Four quarter-circles at the corners of a square. The shaded centre region equals $s^2 - \pi(s/2)^2$ if the four arcs are quarter-circles of radius $s/2$ from each corner.

Configuration E: Two overlapping circles (Venn-style lune). Compute via inclusion-exclusion: shaded = circle$_1$ + circle$_2$ – (intersection), where the intersection is two segments combined.

Examples and applications

Example 1. A square of side $14$ cm has a circle of diameter $14$ cm inscribed in it. Find the area of the region outside the circle but inside the square.

Square area $= 196$. Circle area $= 22/7 \cdot 49 = 154$. Shaded $= 196 - 154 = 42$ cm$^2$.

Example 2. A square of side $14$ cm has four quadrants of radius $7$ cm at its four corners. Find the area of the region inside the square but outside the quadrants.

Total quadrant area $= 4 \cdot (1/4)\pi(7)^2 = \pi \cdot 49 = 154$. Shaded $= 196 - 154 = 42$ cm$^2$. (Same answer as Example 1 , the four quadrants tile up to a full circle of radius $7$.)

Example 3. A square has side $4$ cm. A semicircle is drawn on each side as a diameter, all four bulging outward. Find the total area enclosed.

The four semicircles have radii $2$, so each has area $(1/2)\pi(4) = 2\pi$. Total $= 8\pi$. Square area $= 16$. Combined area $= 16 + 8\pi \approx 16 + 25.13 = 41.13$ cm$^2$.

Example 4. A flower bed is in the shape of a circle of diameter $14$ m, with a square garden of side $14$ m surrounding it. Find the area of the garden (not the flower bed).

Square area $= 196$, circle area $= 154$ (as Example 1). Garden $= 196 - 154 = 42$ m$^2$.

Example 5. A piece of land is in the shape of an equilateral triangle of side $14$ m. A goat is tied at one vertex with a rope of length $7$ m. Find the area the goat can graze.

At the corner of an equilateral triangle, the interior angle is $60^\circ$. The goat can graze a $60^\circ$ sector of radius $7$. Area $= (60/360)(22/7)(49) = (1/6)(22)(7) = 154/6 \approx 25.67$ m$^2$.

Example 6 (compound sector). A pendant is in the shape of a circle of diameter $32$ mm hanging from a chain. There is a hole of diameter $6$ mm at the centre. Find the area of the pendant (not the hole).

Outer area $= \pi(16)^2 = 256\pi$. Hole $= \pi(3)^2 = 9\pi$. Pendant $= 247\pi \approx 776.0$ mm$^2$ (with $\pi = 22/7$, $= 776.86$ mm$^2$).

Worked examples (with diagrams to imagine)

Example A. In a circle of radius $21$, the perimeter of a sector is $54$. Find the sector's area.

Perimeter: $2r + \ell = 54 \Rightarrow \ell = 54 - 42 = 12$. Sector area $= (1/2) r \ell = (1/2)(21)(12) = 126$ cm$^2$.

Example B. A horse is tethered at the corner of a $15$ m by $8$ m rectangular grass plot by a $7$-m rope. Find the area of the plot the horse can graze.

The corner of a rectangle has interior angle $90^\circ$, so the horse grazes a $90^\circ$ sector of radius $7$: area $= (1/4)(22/7)(49) = 38.5$ m$^2$.

Example C. A square is inscribed in a quarter circle of radius $5$ cm such that one of its corners is at the centre of the quarter circle. Find the largest possible area of the square. (Concept: $s\sqrt 2 = 5 \Rightarrow s = 5/\sqrt 2 \Rightarrow s^2 = 25/2 = 12.5$ cm$^2$.)

Example D. A circular park of radius $14$ m has a square flower bed of side $14$ m inscribed in it. Find the area of the park outside the bed.

Square diagonal $= 14$ if inscribed (impossible since the side equals the diameter , re-read). If the square has side $14$, the circumscribed circle has diameter $14\sqrt 2$. If instead the square is inscribed in the circle of radius $14$, then diagonal $= 28$, side $= 14\sqrt 2$, square area $= 392$. Park area $= 22/7 \cdot 196 = 616$. Outside-square area $= 616 - 392 = 224$ m$^2$.

Try it yourself

1. A square of side $20$ cm has a circle inscribed in it. Find the area of the square minus the circle.
2. A circular table of radius $7$ cm has a square inscribed in it. Find the area of the circle minus the square.
3. Four quadrants of radius $4$ are cut from the corners of a $14$-cm square. Find the remaining area.
4. A goat is tied to one corner of a $20$-m square plot by an $8$-m rope. Find the grazing area.
5. A semicircle of radius $14$ cm has a triangle inscribed with base $=$ diameter. Find the area of the triangle if its third vertex is on the arc.
6. A pizza of radius $21$ cm is divided into $8$ equal slices. Find the area and arc-length of one slice.
7. Two circles of radii $10$ and $6$ touch externally. Find the area of the region enclosed by both circles' tangent lines and arcs (you'll need to sketch).
8. A track is a rectangle of $100$ m by $50$ m with semicircular ends of diameter $50$ m. Find the total track area.
9. A flower bed is in the shape of a sector of a circle, radius $10$, central angle $90^\circ$. Find its perimeter and area.
10. A square has side $14$ cm. Four semicircles are drawn on its sides bulging outward. Find the total combined area.
11. A circle of radius $7$ has three equal sectors removed (each $60^\circ$). Find the remaining area.
12. A swimming pool of dimensions $20$ m $\times$ $10$ m has semicircular ends. Find the area of the pool.

Pitfalls / Insight

(1) Always decompose the figure before computing , never try to compute a complex shape's area in one stroke.

(2) Common trick: four quadrants of radius $r$ at the corners of a square tile up to one full circle of radius $r$. Use this to short-circuit calculations.

(3) Be careful about "inscribed" vs "circumscribed". A square inscribed in a circle has diagonal $=$ diameter. A circle inscribed in a square has diameter $=$ side.

(4) For "goat grazing" problems, the available angle depends on the geometry of the boundary. At an interior corner of a square, the angle is $90^\circ$; at an equilateral triangle corner, $60^\circ$; in an open field, $360^\circ$.

(5) Always state your final answer in proper units (cm$^2$, m$^2$, etc.) and use the value of $\pi$ specified.