9. Measurement

Middle Years

Lesson

A composite shape is one that is made from a number of smaller shapes.

We can use the properties of these regular shapes to to learn more about the composite shape. For example, knowing the total area of all the smaller shapes is the same as knowing the area of the whole composite shape.

When finding the perimeter of composite shapes there are two main approaches.

The first approach is to find the length of all the sides and add them together like we would for an irregular shape. We can do this by using the lengths we are given to find any missing lengths.

The other approach is less obvious and relies on some visualisation. We can see in the image below that the composite shape actually has the same perimeter as a rectangle.

So the perimeter of this composite shape can be calculated as:

Perimeter | $=$= | $2\times\left(8+13\right)$2×(8+13) |

$=$= | $2\times21$2×21 | |

$=$= | $42$42 |

Careful!

When using this method it is important to keep track of any sides that do not get moved.

An example of a shape that we need to be careful with is:

Notice that we moved the indented edge to complete the rectangle but we still need to count the two edges that weren't moved.

We can calculate the perimeter of this shape as:

Perimeter | $=$= | $2\times\left(5+11\right)+2+2$2×(5+11)+2+2 |

$=$= | $2\times16+4$2×16+4 | |

$=$= | $32+4$32+4 | |

$=$= | $36$36 |

Consider the composite shape.

Which basic shapes make up this composite shape?

Two rhombuses

AOne rhombus

BTwo trapeziums

COne trapezium minus one triangle

DTwo rhombuses

AOne rhombus

BTwo trapeziums

COne trapezium minus one triangle

DFind the perimeter of the composite shape.

Consider the composite shape.

Which basic shapes make up this composite shape?

Three semicircles and one triangle

AThree quarter circles and one triangle

BThree semicircles and one square

CThree quarter circles and one square

DThree semicircles and one triangle

AThree quarter circles and one triangle

BThree semicircles and one square

CThree quarter circles and one square

DFind the exact perimeter of the composite shape.

To calculate the area of a composite shape, we can use either of two methods:

- Addition method - Divide the composite shape into basic shapes, work out the area of each basic shape, then add them together.
- Subtraction method - Work out the area of the basic shape that encloses the composite shape, then subtract the areas of smaller basic shapes as necessary.

We may also need to use a combination of the above methods. We can also try and re-arrange or visualise the shape in a different way.

How could we re-visualise the following shape up to make our calculations easier?

To find the area, we could work out the area of each semi circle individually, or we could join them back together to make one complete circle. This way we only need to work out the area of one square and one circle. Notice that if we calculated the semi-circular areas separately we are actually halving the area of the circle and then adding the two halves back together.

Similarly, to find the perimeter, by putting the two semi-circles back together we can work out the circumference of the full circle, and then add the two sides of the square that are on the outside of the shape. It is very important that you don't accidentally double-count sides.

Consider the composite shape.

Which basic shapes make up this composite shape?

A rectangle minus two triangles

AOne rectangle and two trapeziums

BTwo parallelograms

CTwo trapeziums

DA rectangle minus two triangles

AOne rectangle and two trapeziums

BTwo parallelograms

CTwo trapeziums

DFind the area of the composite shape.

Find the area of the composite shape rounded to two decimal places.