The area of the trapezoid and composite shapes

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

• The two parallel sides are the trapezoid’s bases: $\displaystyle a$ and $\displaystyle b$.
• The other two sides are its legs.
• The height $\displaystyle h$ of the trapezoid is perpendicular to both bases and joins them.

In the drawing below we put a copy of the trapezoid beside it, turned through half a turn. Together they make a parallelogram:

• The bottom side of the parallelogram is $\displaystyle a+b$: the base $\displaystyle a$ of the trapezoid and the base $\displaystyle b$ of the copy.
• The height of the parallelogram is $\displaystyle h$, the height of the trapezoid.
• The area of the parallelogram is $\displaystyle (a+b)\cdot h$, and the trapezoid is half of it:

• In words: the sum of the bases, times the height, divided by 2.

In the trapezoid $\displaystyle ABCD$ the lengths are in cm:
The bases are $\displaystyle {AB}={9}$ and $\displaystyle {DC}={5}$.
The height is $\displaystyle {CF}={4}$.
The example below calculates the area of the trapezoid by the formula, and then checks it by cutting.

• By the formula: $\displaystyle {\frac{(9+5)\cdot4}{2}}={\frac{14\cdot4}{2}}={28}$ cm².
• By cutting: the heights $\displaystyle DE$ and $\displaystyle CF$ cut the trapezoid into the rectangle $\displaystyle EFCD$ and two right triangles.
• The rectangle: $\displaystyle {EF}={DC}={5}$, so its area is $\displaystyle {5\cdot4}={20}$ cm².
• The triangles: both have the height 4, and their bases, $\displaystyle AE$ and $\displaystyle FB$, are together $\displaystyle {9-5}={4}$. So the two together have an area of $\displaystyle {\frac{4\cdot4}{2}}={8}$ cm².
• Together: $\displaystyle {20+8}={28}$ cm² – the same area.

Calculate the area of the trapezoid.

The lengths are in cm.

The area of a composite shape is found in pieces: cut it into shapes whose area you can calculate – rectangles, triangles, trapezoids – and add the areas.

Example: the front of a house. The lengths in the drawing are in metres.

• A dashed line cuts the front into a rectangle and a triangle.
• The rectangle: $\displaystyle {8\cdot5}={40}$ m².
• The triangle: its base is 8 and its height 3, so its area is $\displaystyle {\frac{8\cdot3}{2}}={12}$ m².
• The area of the front: $\displaystyle {40+12}={52}$ m².

Sometimes it is easier to complete the shape to a large one, and subtract what was added.

Example: a yard.
A yard is a square with a side of 10 m.
In the yard there is a flower bed, a rectangle 4 m by 3 m.
The whole yard, except the flower bed, is paved.

• The area of the whole yard: $\displaystyle {10\cdot10}={100}$ m².
• The area of the flower bed: $\displaystyle {4\cdot3}={12}$ m².
• The paved area: $\displaystyle {100-12}={88}$ m².

Calculate the area.

An arrow. The lengths in the drawing are in cm.

A picture frame is a rectangle 30 cm by 20 cm.
The picture inside it is a rectangle 24 cm by 14 cm.
Calculate the area of the frame, without the picture.

The vertices of the triangle $\displaystyle ABC$ are at corners of the squares, and each square is one unit of area. Its sides slant, so a base and a height are hard to measure. The example below puts a rectangle around the triangle, and subtracts from it what lies outside the triangle.

• The dashed rectangle: 5 squares by 4, so its area is $\displaystyle {5\cdot4}={20}$.
• Between the triangle and the rectangle three right triangles are left. Their legs lie on the lines of the squares, so their lengths can be counted:
  • beside $\displaystyle AB$: 5 and 1, the area $\displaystyle {\frac{5\cdot1}{2}}={2.5}$;
  • beside $\displaystyle BC$: 3 and 3, the area $\displaystyle {\frac{3\cdot3}{2}}={4.5}$;
  • beside $\displaystyle CA$: 2 and 4, the area $\displaystyle {\frac{2\cdot4}{2}}={4}$.
• The area of the triangle: $\displaystyle {20-2.5-4.5-4}={9}$ squares.

Each square is one unit of area.
Calculate the area of the shape.