Technique

Cut a compound plane figure into standard pieces

Mensuration • Plane Figures • Composite and Shaded Regions

Statement

The area of a plane region bounded by straight edges and arcs is found by cutting it along drawn or added segments into rectangles, triangles, trapezia and circular sectors whose areas are known, then adding those areas and subtracting any piece that is a hole rather than part of the region.

How to use it

1Cut along the segments that are already drawn first, and add your own only where a piece is still not a standard shape.
2Pin every vertex before computing: use the marked right angles to lay out horizontal and vertical runs, and read each label as belonging to the segment it sits beside.
3Choose the formula from what the piece gives you — three known sides and no height is Heron's, a pair of parallel edges is a trapezium, a corner of a circle is a sector.
4Check the total against a crude bound, such as the figure's bounding rectangle, before choosing an option.

Worked example

A field is a

6

m by

4

m rectangle with a right triangle of legs

4

m and

3

m attached along one of its

4

m sides. Find its area.

Step-by-step

1
rectangle: $6 \times 4 = 24$ m $^{2}$
Why: The two pieces meet along the $4$ m side and do not overlap.
2
triangle: $\frac{1}{2} \times 4 \times 3 = 6$ m $^{2}$
3
total $= 24 + 6 = 30$ m $^{2}$

Answer

30

^{2}

Watch out

Every piece must be counted exactly once — overlapping cuts double-count, and a piece left out is lost silently. Take lengths only from the labels: these diagrams are routinely not to scale, so nothing may be measured off the drawing.

Builds on

Area of a triangle from base and height Perimeter and area of a rectangle

Also known as

compound figure area, composite plane figure, shaded region area, split the figure into rectangles and triangles, irregular area by decomposition

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)

2019 (I)

2018 (II)

2018 (I)

2017 (I)