Study Resources: How to Calculate the Area of Complex Geometrical Figures

Students usually find it easy to find out the area of known geometrical figures involving common shapes like circles, triangles, rectangles, squares, and so on. But what if they come across complex geometrical figures like the one shown below?

Image 1- A complex geometrical figure

How would they find out the area of such complex figures, in particular? The following steps can help.

STEP 1: Observe closely

This one’s perhaps the most obvious step of the lot. However, it deserves mention in this frame because you need to do this religiously to keep the odds of silly mistakes low in the best possible manner.

So observe the figure as closely as possible, and note down the information given in the problem (like different side lengths and breadths) on your separate answer sheet. A simple step like that can go a long way indeed.

STEP 2: Break down the complex into known, simpler geometrical figures

Is it possible for you to break down the complex figure into known, simpler geometrical shapes?

Like if we take the above problem (the one depicted in Image 1) in consideration, we can break the problem into two simpler rectangles. Here’s how we do it:

Image 2: Complex geometrical figure broken into 2 simpler rectangular shapes

Pretty simple; isn’t it? You will have to try and find out known, simpler geometrical shapes (that are inclusive of shapes like triangle, circle, semi-circle, square, rectangle, etc.) inside the complex figure in the very same manner. If you can fulfill this step the right way, everything else becomes as easy as a piece of cake.

STEP 3: While breaking a complex figure into simpler shapes, ensure that the shapes DON’T overlap each other in any way whatsoever

If the simpler shapes overlap each other in your problem, you would not be able to get the correct solution to your problem. So make sure they don’t overlap each other in any possible way.

STEP 4: Find out the areas of the simpler figures INDIVIDUALLY based on the information given in the problem

The following information can come in handy for this purpose of yours:

Area of Square = Side2

Area of a Rectangle = Length X Breadth

Area of a circle = pi X radius2

Area of a triangle = ½ X Base X Height

Area of a semicircle = ½ X pi X radius2

You have to find out the area of the simpler figures individually and note them down separately on your answer sheet for the final calculation.

STEP 5: Find the sum of the areas of the simpler figures to get the correct result

The last step is the easiest of the lot. Just find out the sum of the individual areas done in step no. 4. The answer that you will get is going to be the total area of the composite figure, on the whole.

EXAMPLE:

We will use the following diagram of a complex geometrical figure for the sake of our explanation. The information is provided in the diagram itself. So that part won’t be a problem.

Note: The diagram is definitely NOT drawn to scale.

In our next step, we would break down the complex figure (shown above) into simpler geometrical shapes. It should look something like this -

So now you can easily detect the 2 simpler shapes formed in the complex figure depicted above. One is a semi-circle, and the other is a rectangle.

Anyway, let’s get to the solution now.

Given values:

CD=AB=6cm

AO=OD= 4cm (It is the radius; we will represent it by 'r' in our solution)

Calculation of Area of the Rectangle

Length= AD=BC= (AO+OD)= (4+4)cm= 8 cm

Width=CD=AB= 6cm

Area of the rectangle = Length x Width

                                  = (8 x 6) cm2

                                  = 48 cm2

Calculation of Area of the Semi-circle

Area of the circle= π x (r^2)

Area of the semi-circle = ½ x π x (r^2) cm2

                                     = ½ x π x (OD^2) cm2

                                     = ½ x π x (4^2) cm2

                                     = ½ x (22/7) x 16 cm2

                                     = 25.142 cm2 (correct to 2 units after decimal)

Total area of the composite figure

Total area= Area of the rectangle + Area of the semi-circle

                = (25.142 + 48) cm2

                = 73.142 cm2

If you want to refer to a more difficult problem in comparison to the one highlighted above, you may refer to this Valentine problem, in particular. With that, we’ll bring this article to an end for now. Hope you had a good and productive read.