Problem Solving with Areas

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Adding & Subtracting Areas

What do we mean by an awkward shape (a compound shape)?

Sometimes the shape we want to find the area of isn’t one of the standard shapes in Area – Formulae
However, the area may be found by using a combination of standard shapes
These are often called Compound Shapes and you may see Compound Area mentioned too

Finding the area of an awkward shape (compound area)

When you are asked to find the area of an “awkward” shape, split the shape into standard shapes first, and then add them together

Examiner Tip

Take a moment to think about how to split up the shape into the easiest shapes possible – there will probably be more than one way to do it!
Occasionally it may be easier to add an extra shape to the diagram and subtract the area of the extra shape from the new bigger shape
- For example, for this shape you might complete the rectangle by putting a triangle in the top left corner
- Then the area of the whole shape is the rectangle minus the triangle

Worked example

Find the area of the pentagon shown in the diagram below.

Pukoyl43_edexcel-3-5-2-adding-and-subtracting-areas-we-diagram

Separate the diagram into two shapes that you are familiar with and know the formulae for the areas of.

This pentagon can easily be split into a rectangle and a triangle.
Use the values given to find the length of the base and the height of the triangle and add these to the diagram.

edexcel-3-5-2-adding-and-subtracting-areas-we-solution-1

The total area will be the area of the rectangle + the area of the triangle.

$\begin{array}{rcl} Total area & = & (12 \times 4) + \frac{1}{2} (7 \times 5) \\ = & 48 + \frac{1}{2} (35) \end{array}$

Area = 65.5 cm²

What is problem solving?

Problem solving, as far as GCSE Mathematics is concerned, usually has two key features:

A question is given as a real-life scenario (eg. Mary is painting a bedroom in her house …)
There is normally more than one topic of maths you will need in order to answer the question (eg. Area and Percentages)

Problem solving with areas

Area is a commonly used topic of maths in the real world
Laying a carpet, painting a house, designing a sports field, building a patio or decking all involve area
Also, doing each of these things has a cost – so a lot of area problems also involve calculations with money

How to solve problems

The key to getting started on problem-solving questions is to not focus only on what the question asks you to find out but thinking about what you can do with the information given
Often this will lead you to think of something else you can do and then eventually you may be able to see your way to answering the original question
These questions could appear on either a non-calculator paper or calculator paper, depending on how awkward they decide to make the numbers involved!

Examiner Tip

Even if you never get to a final answer always try to do some maths with the information from the question
- You are likely to score some extra marks!

Worked example

John wants a new carpet for the lounge in his house.

A sketch of his lounge is given below.

edexcel-3-5-2-problem-solving-with-areas-we-diagram

He gets quote from two local companies, Company A and Company B.

The amount they charge for laying a carpet is given below.

Company A: Fixed price of £5.50 per square metre
Company B: £6 per square metre for the first ten square metres, then £4 per square metre for anything over that.

Which company should John choose in order to keep the cost of laying the carpet to a minimum?

Although this question doesn't specifically tell you you need to find the area, it is implied as the costs both use 'square metre'.

The shape of the lounge is a compound shape consisting of two rectangles. Split the area into these two rectangles and find the missing distances by subtracting the smaller length (2.4 m) from the longer one (6 m).

6 - 2.4 = 3.6

Lounge-Floor-Area, downloadable IGCSE & GCSE Maths revision notes

Find the area of the lounge by adding the two areas together.

$\begin{array}{rcl} Total Area & = & Area A + Area B \\ = & (3.2 \times 3.6) + (2.4 \times 1.8) \\ = & 11.52 + 4.32 \\ = & 15.84 m^{2} \end{array}$

Find the cost for each of the two companies separately.

Company A:

$\begin{array}{rcl} Total Cost & = & 15.84 \times 5.50 \\ = & 87.12 \end{array}$

Company B:

$\begin{array}{rcl} Total Cost & = & £ 6 \times first 10 m^{2} + £ 4 \times remaining \\ = & (6 \times 10) + 4 \times (15.84 - 10) \\ = & 60 + 23.36 \\ = & 83.36 \end{array}$

John should choose Company B as it will cost £3.76 less than Company A.

Problem Solving with Areas (CIE IGCSE Maths: Extended)

Revision Note

Adding & Subtracting Areas

What do we mean by an awkward shape (a compound shape)?

Finding the area of an awkward shape (compound area)

Examiner Tip

Worked example