A combined figure is a geometrical shape that is the combination of many simple geometrical shapes.

To find the area of combined figures we will follow the steps:

**Step I:** First we divide the combined figure into its simple geometrical shapes.

**Step II: **Then calculate the area of these simple geometrical shapes separately,

**Step III:** Finally, to find the required area of the combined figure we need to add or subtract these areas.

Solved Examples on Area of combined figures:

**1.** Find the area of the shaded region of the adjoining figure. (Use π = \(\frac{22}{7}\))

JKLM is a square of side 7 cm. O is the centre of the
semicircle MNL.

**Solution:**

**Step I:** First we divide the combined figure into
its simple geometrical shapes.

The given combined shape is combination of a square and a semicircle.

**Step II:** Then calculate the area of
these simple geometrical shapes separately.

Area of the square JKLM = 7^{2} cm^{2}

=
49 cm^{2}

Area of the semicircle LNM = \(\frac{1}{2}\) π ∙ \((\frac{7}{2})^{2}\) cm^{2} , [Since,
diameter LM = 7 cm]

=
\(\frac{1}{2}\) ∙ \(\frac{22}{7}\) ∙ \(\frac{49}{4}\) cm^{2}

=
\(\frac{77}{4}\) cm^{2}

=
19.25 cm^{2}

**Step III:** Finally, add these areas up to get
the total area of the combined figure.

Therefore, the required area = 49 cm^{2} + 19.25 cm^{2}

=
68.25 cm^{2}.

**2.** In the adjoining figure, PQRS is a square of side 14 cm
and O is the centre of the circle touching all sides of the square.

Find the area of the shaded region.

**Solution:**

**Step I:** First we divide the combined figure into its simple geometrical shapes.

The given combined shape is combination of a square and a circle.

**Step II: **Then calculate the area of these simple geometrical shapes separately.

Area of the square PQRS = 14^{2} cm^{2}

= 196 cm^{2}

Area of the circle with centre O = π ∙ 7^{2} cm^{2}, [Since, diameter SR = 14 cm]

= \(\frac{22}{7}\) ∙ 49 cm^{2}

= 22 × 7 cm^{2}

= 154 cm^{2}

**Step III:** Finally, to find the required area of the combined figure we need to subtract the area of the circle from the area of the square.

Therefore, the required area = 196 cm^{2} - 154 cm^{2}

= 42 cm^{2}

**3.** In the adjoining figure alongside, there are four equal quadrants of circles each of radius 3.5 cm, their centres being P, Q, R and S.

Find the area of the shaded region.

**Solution:**

Step I: First we divide the combined figure into its simple geometrical shapes.

The given combined shape is combination of a square and four quadrants.

**Step II:**Then calculate the area of these simple geometrical shapes separately.

Area of the square PQRS = 7^{2} cm^{2}, [Since, side of the square = 7 cm]

= 49 cm^{2}

Area of the quadrant APB = \(\frac{1}{4}\) π ∙ r^{2} cm^{2}

= \(\frac{1}{4}\) ∙ \(\frac{22}{7}\) ∙ \((\frac{7}{2})^{2}\) cm^{2}, [Since, side of the square = 7 cm and radius of the quadrant = \(\frac{7}{2}\) cm]

= \(\frac{77}{8}\) cm^{2}

There are four quadrants and they have the same area.

So, total area of the four quadrants = 4 × \(\frac{77}{8}\) cm^{2}

= \(\frac{77}{2}\) cm^{2}

= \(\frac{77}{2}\) cm^{2}

**Step III:** Finally, to find the required area of the combined figure we need to subtract the area of the four quadrants from the area of the square.

Therefore, the required area = 49 cm^{2} - \(\frac{77}{2}\) cm^{2}

= \(\frac{21}{2}\) cm^{2}

= 10.5 cm^{2}

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