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Area of Compound Figures

Geometry is a branch of mathematics concerned with various 2-dimensional as well as 3-dimensional shapes. The area of a object or figure is the amount of space covered by that object or figure.
The area is measured in square units for example, cm$^2$, m$^2$, in$^2$ etc.(depends on given unit of measurement). To get the exact amount of space covered by a figure, usually we use a square to represent one unit. For example, if we divide a rectangle into 12 squares, so the area for that rectangle is 12 square units.

Area is size of a plane surfaces. In geometry we studied about area of various geometric regular and irregular figures like square, triangle, rectangle, ellipse, parallelogram, circle etc. Composite figures are made up of two or more well known figures. Compound figures that can be divided into various plane figures.
This section helps the students to explore how different shapes can be used to create composite figures and how to find the area of that. In this section we are going to deal with area of various compound figures.

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Definition

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In mathematics, area is the amount of space inside the boundaries of two dimensional figures such as square, circle, rectangle, parallelogram or triangle. Or we can say that area of plane objects refer to the number of square units the objects cover in it.

The area of figures measured in square units, it could be meter, centimeters, inches or yards etc. Compound figures are just a combination of known figures. To find the area of the composite figure, divide entire figure into simpler known figures.

Formula

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In geometry, there are several well known formulas used to calculate the area of different polygons or shapes such as square, circle or rectangle. But there is no any particular formula to determine the area of compound polygons.

To calculate area of compound figures, we must know formulas of some known figures  given below:

Figure
 Area
 Square  Side $\times$ Side
 Triangle  $\frac{Base \times Height}{2}$
 Circle  $\pi$(Radius)$^2$
 Parallelogram  Base $\times$ Height
 Rectangle  Length $\times$ Width


How to Draw?

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There is no any particular way to draw composite figures. As we know, compound figures are combination of two or more plane figures. Or we can say that compound figures can be drawn by joining different 2-Dimensional figures such as rectangle, circle or square. Here we will see how different shapes are used to create composite figures.

Below are some shapes which are the combination of different figures:
Combination of Different Figures

The first figure is combination of 2 rectangles and one square. Second figure is made by joining 2 squares and one triangle. And third one is the combination of one rectangle and one circle.

How to Find?

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To find the area of a compound figure follow the below steps:

Step 1: Split the given figure into known figures like square, circle or rectangle.

Step 2. Determine the area of each of those individual figures.

Step 3: Add together all the results obtained in step 2.

Step 4: Write your answer in square units.
Let us study with the help of an example how to find the area of a compound figure.

Example: Find the area of figure below:
Combination of Square with Right Angle Triangle
Solution:
Above figure is the combination of a square and a right triangle.

So divide given figure into square and a right triangle with the sides measurement.
Combination of Square with Right Angle Triangle Example

Determine the area of square and triangle.

Area of square = (Side)$^2$ = 15 $\times$ 15 = 225

Area of triangle = $\frac{1}{2}$ * Base * Height = $\frac{1}{2}$ $\times$ 5 $\times$ 9 = 22.5

Area of figure = 225 + 22.5 = 247.5

Area of given figure is 247.5 square unit.

Examples

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Some examples based on area of compound figures are given below;

Example 1: What is the area of below figure.
Combination of Rectangle with Right Angle Triangle
Solution:  Given figure is the combination of two figures, triangle and rectangle. So to find the area of whole figure, we need to find the area of triangle and rectangle first.
Combination of Rectangle with Right Angle Triangle Example
Rectangle area = length $\times$ width = 110 $\times$ 140 = 15400

Triangle area = $\frac{1}{2}$ * Base * Height = $\frac{1}{2}$ $\times$ 90 $\times$ 70 = 3150

Area of whole figure = Rectangle area + Triangle area = 15400 + 3150 = 18550

Therefore the area of given figure is 18550 cm$^2$.

Example 2: Find the area of the figure below:
Combination of Two Rectangle
Solution:
Divide given figure into well known shapes, say A , B and C.

This figure is the combination of two rectangles and one square.

Combination of Two Rectangle Example

Calculate the area of each figure.

Area of figure A (rectangle) = 21 $\times$ 13 = 273

Area of figure B (square) = 6 $\times$ 6 = 36

Area of figure C (rectangle) = 10 $\times$ 3 = 30

Area of given figure = 273 + 36 + 30 = 339

339 sq. unit is the area of given compounded figure.

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