National 5MathsSQAApplications of MathematicsRevision NotesGeometric SkillsPerimeter & AreaArea of Composite Shapes

Area of Composite Shapes (SQA National 5 Applications of Mathematics): Revision Note

Exam code: X844 75

Dan Finlay

Written by: Dan Finlay

Reviewed by: Roger B

Updated on

Area of composite shapes

How do I find the area of parts of circles?

  • Identify the radius of the full circle

    • You might have to halve the diameter

  • Calculate the area of the full circle

  • Divide the area by:

    • 2 if it is a semicircle

    • 4 if it is a quarter circle

Diagram of a semicircle and quadrant. Semicircle's area formula is πr²/2, labelled with diameter. Quadrant's area formula is πr²/4, labelled with radius.
Parts of circles

Examiner Tips and Tricks

You can put parts of circles together to make your calculation easier. For example, if the shape contains two equal semicircles, then you can just find the area of the full circle. This gives the same answer as halving it (to find the area of one semicircle) and then doubling it.

What is a composite shape?

  • Sometimes you will have a shape that is not a standard shape such as a rectangle, triangle or circle

    • These are called composite shapes

    • You can split the non-standard shape into standard shapes

How do I find the area of a composite shape?

  • Split the composite shape into standard shapes

    • This might be

      • two or more shapes joined together

      • a shape removed from a bigger shape

  • Find the areas of the standard shapes

  • Add or subtract these

    • Add if a shape is joined to another shape

    • Subtract if a shape is removed from another shape

Diagram showing rectangle and triangle subtracted to form a shape. Below, a rectangle, another rectangle, and a triangle are added together visually.
Example of splitting a composite shape into standard shapes

Examiner Tips and Tricks

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!

Worked Example

Kia has a garden in the shape of semicircle. Kia forms an isosceles triangle using the diameter of the semicircle and a third point on the edge of the semicircle, as shown in the diagram below.

Half-circle with inscribed isosceles triangle on a 7m base. Identical sides of triangle shown with equal marks, shaded semicircle area above.

Kia fills the triangle with grass and fills the remaining space with stones.

Calculate the area of the space that is filled with stones.

Answer:

Find the area of the semicircle

  • Halve the diameter to find the radius

7 divided by 2 equals 3.5

  • Use A equals pi r squared to find the area of the full circle

pi cross times 3.5 squared equals 38.484...

  • Halve the area

38.484... divided by 2 equals 19.242...

Find the area of the triangle

  • The perpendicular height is just the radius of the circle

h equals 3.5

  • Use A equals 1 half b h

1 half cross times 7 cross times 3.5 equals 12.25

Subtract the area of the triangle from the area of the semicircle

19.242... negative 12.25 equals 6.992...

Include the units

6.99 m2

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Roger B

    Reviewer: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.