GCSEMathsEdexcelRevision Notes4. Geometry & MeasuresArea & PerimeterProblem Solving with Areas

Problem Solving with Areas (Edexcel GCSE Maths)

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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 triangleIGCSE & GCSE Maths revision notes

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.

table row cell Total space area space end cell equals cell space open parentheses 12 space cross times space 4 close parentheses space plus 1 half open parentheses 7 space cross times space 5 close parentheses end cell row blank equals cell space 48 space plus space 1 half open parentheses 35 close parentheses end cell end table

Area = 65.5 cm2

Problem Solving with Areas

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.

table row cell Total space Area space end cell equals cell space Area space straight A space plus space Area space straight B end cell row blank equals cell space open parentheses 3.2 space cross times space 3.6 close parentheses space plus space open parentheses 2.4 space cross times space 1.8 close parentheses end cell row blank equals cell space 11.52 space plus space 4.32 end cell row blank equals cell space 15.84 space straight m to the power of 2 space end exponent end cell end table

Find the cost for each of the two companies separately.

Company A:

table attributes columnalign right center left columnspacing 0px end attributes row cell Total space Cost space end cell equals cell space 15.84 space cross times space 5.50 end cell row blank equals cell space 87.12 end cell end table

Company B:

table row cell Total space Cost space end cell equals cell space £ 6 space cross times space first space 10 space straight m squared space plus space £ 4 space cross times space remaining end cell row blank equals cell space open parentheses 6 space cross times space 10 close parentheses space plus space 4 cross times open parentheses 15.84 space minus space 10 close parentheses space end cell row blank equals cell space 60 space plus space 23.36 space end cell row blank equals cell space 83.36 end cell end table

 

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

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.