Grade by Grade
E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Area of Composite Polygons

E2.4 A1About 45 minutes
Learning goals
  • I can split a composite polygon into rectangles, triangles, parallelograms and trapezoids to find its area.
  • I can find an area by subtracting a missing piece from a bigger shape.
  • I can choose a strategy and check my answer with a second strategy.

Words to know

composite polygon
A polygon made of two or more simpler shapes joined together, like an L-shape or a house shape.
decompose
To break a shape into smaller shapes whose areas you know.
square unit
A square that is 1 unit on each side, used to measure area, like 1 cm² or 1 m².

Let's start with a skating rink

Your neighbourhood is flooding an outdoor rink in the park this winter. It isn't a plain rectangle: it's shaped like an L so it can fit around the playground. How much area needs to be flooded?

No single formula works for an L. But you can decompose it into shapes you already know, find each area, and add. Or you can find the area of a bigger rectangle and subtract the missing piece.

An outdoor skating rink with wooden boards in a snowy park, beside a playground, at dusk.
Neighbourhood rinks are built to fit the space in the park. To find the area of an odd-shaped rink, decompose it into rectangles and other shapes you know. AI-generated image
ABAdd: 18 + 15 = 33Subtract: 48 − 15 = 338 wide, 6 tall8 wide, 6 tall
The same L-shaped polygon on a grid two ways. Left: split into rectangle A (3 by 6, area 18) and rectangle B (5 by 3, area 15), total 33 square units. Right: the 8 by 6 rectangle around it (48) minus the missing 5 by 3 corner (15) is 33 square units.
Worked example

Example 1: Add the parts

Find the area of the L-shape: it is 8 m wide and 6 m tall overall, with a 5 m by 3 m corner missing at the top right.

  1. Split with a line going straight down from the inside corner.
  2. Rectangle A: 3 × 6 = 18 m².
  3. Rectangle B: 5 × 3 = 15 m².
  4. Add: 18 + 15 = 33 m².

33 m²

Worked example

Example 2: Subtract the missing piece

Find the area of the same L-shape another way.

  1. Imagine the full rectangle: 8 × 6 = 48 m².
  2. The missing corner: 5 × 3 = 15 m².
  3. Subtract: 48 − 15 = 33 m².

33 m², the same answer. Using two strategies is a great way to check your work.

rectangletrapezoidEach square = 1 m²Rectangle: 9 × 3 = 27 m²Trapezoid: (9 + 4) × 3 ÷ 2 = 19.5 m²Total: 27 + 19.5 = 46.5 m²
A six-sided garden plot on a grid where each square is 1 m². It splits into a 9 m by 3 m rectangle on the bottom and a trapezoid on top with parallel sides of 9 m and 4 m and a height of 3 m.
Worked example

Example 3: Use a trapezoid

Find the area of the six-sided garden plot in the picture.

  1. Split it with a line across, where the sides start to slant.
  2. Bottom rectangle: 9 × 3 = 27 m².
  3. Top trapezoid: parallel sides 9 m and 4 m, height 3 m. (9 + 4) × 3 ÷ 2 = 39 ÷ 2 = 19.5 m².
  4. Total: 27 + 19.5 = 46.5 m².

46.5 m²

Just for fun

A composite shape is like a pizza: you can cut it into pieces any way you like, and there's still the same amount of pizza in total. (Sadly, finding its area won't make it any bigger.)

Strategy steps

  1. Look at the shape. Where could you draw lines to make rectangles, triangles, parallelograms or trapezoids?
  2. Label each part and find any missing lengths. (Often you subtract: if the whole side is 8 m and one part is 3 m, the other part is 5 m.)
  3. Find each area.
  4. Add the parts (or subtract a missing part from a bigger shape).
  5. Check with a different split if you can.

There's more than one way

Two people can split the same shape differently and both be right. If your answers match, you can be confident. If they don't, compare your work calmly to find the mistake. Explaining your strategy to someone else is part of thinking like a mathematician.

Hands-on activity

Floor plan detective

You need: a measuring tape, grid paper, a pencil and a ruler.

  1. Choose a room or a hallway in your home that is not a simple rectangle (for example, a room with a closet or a nook), or use a rug, a doormat or your bedroom.
  2. Measure each wall or edge to the nearest tenth of a metre (or to the nearest centimetre for a small object).
  3. Draw a floor plan on grid paper, letting each square stand for a convenient amount, like 1 m by 1 m or 10 cm by 10 cm. Write the measurements on it.
  4. Decompose your plan into rectangles (and triangles or trapezoids if needed) and find the total area.
  5. Find the area again a different way and check that your answers match.

Math talk: "Which way of splitting the shape was easier? Why?"

Your turn

1

An L-shape is 10 m wide and 8 m tall overall, with a 4 m by 3 m corner missing. Find its area.

2

A house-shaped pentagon is a 5 cm by 4 cm rectangle with a triangle on top. The triangle's base is 5 cm and its height is 2 cm. Find the area.

3

A polygon has vertices at (0, 0), (6, 0), (6, 2), (2, 2), (2, 5), (0, 5). Find its area in square units.

4

A T-shape has a top bar 9 cm by 2 cm and a stem 3 cm by 5 cm under its middle. Find its area.

5

A 8 cm by 5 cm rectangle has a triangular notch cut out of one side. The triangle has a base of 4 cm and a height of 2 cm. Find the area that's left.

6

A shape is an 8 m by 3 m rectangle with a trapezoid on top. The trapezoid has parallel sides of 8 m and 4 m and a height of 2 m. Find the total area.

More practice: warm-up

1

What does decompose mean?

2

Find the area of a 4 cm by 3 cm rectangle.

3

Find the area of a triangle with a base of 6 cm and a height of 4 cm.

4

A shape is split into two parts with areas of 15 m² and 20 m². What is the total area?

5

A 50 cm² card has an 8 cm² hole punched in it. What area of card is left?

More practice: on your own

1

A pentagon has vertices at (−3, 0), (3, 0), (3, 4), (0, 6), (−3, 4). Find its area in square units.

2

A bedroom is a 5 m by 4 m rectangle with a closet 2 m by 1.5 m sticking out of one wall. Find the total floor area.

3

A picture frame is 30 cm by 20 cm on the outside. The opening for the picture is 24 cm by 14 cm. Find the area of the frame itself.

4

A six-sided sign is a 10 cm by 4 cm rectangle with a triangle on each short end. Each triangle has a base of 4 cm and a height of 2 cm. Find the area.

5

Two students found the area of the same L-shape. One added two rectangles and the other subtracted a corner from a big rectangle. Should they get the same answer? Why?

Challenge

1

An arrow is made of a rectangle 6 cm long and 2 cm wide with a triangle at the end. The triangle's base is 6 cm and its height is 3 cm. Find the area of the arrow.

2

A polygon has vertices at (0, 0), (8, 0), (8, 2), (5, 5), (0, 5). Find its area two different ways.

3

A backyard is 12 m by 8 m. It has a square patio 3 m by 3 m and a triangular flower bed with a base of 4 m and a height of 3 m. The rest is grass. How much area is grass?

Check your understanding

Quiz

1.

An L-shape is made of a 4 m by 6 m rectangle and a 3 m by 2 m rectangle. What is its area?

2.

A 10 cm by 10 cm square has a 3 cm by 3 cm square cut from one corner. What area is left?

3.

A house shape is a 6 m by 4 m rectangle with a triangle on top (base 6 m, height 3 m). What is its area?

4.

Which is a good strategy for finding the area of a composite polygon?

5.

A shape is a rectangle with a trapezoid on top. The rectangle is 20 m² and the trapezoid is 7.5 m². What is the total area?

6.

A picture frame is 20 cm by 15 cm outside, with a 16 cm by 11 cm opening. What is the area of the frame?

For parents and teachers

Big idea: E2.4 includes finding the area of composite polygons by decomposing them into shapes with known areas. Children use rectangles, triangles, parallelograms, trapezoids, rhombuses and kites, and they either add the parts or subtract a missing piece.

Common difficulties:

  • Missing lengths. Many problems need a subtraction first (an 8 m side minus a 3 m part is 5 m). Encourage labelling every length on the sketch.
  • Overlapping parts. Make sure the parts don't overlap and don't leave gaps.
  • Area vs perimeter. Area is the surface inside (square units); perimeter is the distance around.

Social-emotional learning (A1): There is often more than one way to split a shape. Encourage your child to try a second way to check, and to explain their strategy. A wrong first try is part of learning.

How to help: Measure a room, a rug or a garden bed together and work out its area. Ask "Is there another way to split it?"