A composite shape is made of shapes you know. Cut it into pieces and add their areas, or see it as a big rectangle with pieces taken away; the area is the same either way. The perimeter is the distance round the outside.
An L-shaped bedroom, a house front with a pointed roof, a photo frame: none is a single rectangle, but each is easy once you cut it up.
A composite shape is made of simple shapes joined together. To find its area:
Cut it into rectangles, triangles or other shapes you know, find each area and add them; or
see it as a big rectangle and take away the pieces that are missing.
Read every length you need from the diagram: a missing side is often the difference of two others. The perimeter is the total length round the outside, in units of length (m), not area (m²).
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
An L-shaped room
Sketch it, then decide: cut it up, or take a piece away?
TryTry it on the board and compare the result with your prediction.
Cut it up
Each button shows a different way to see the same shape.
Try it yourselfYour edits are saved in this browser.
Try a cut, then the take-away. Do they all give the same area? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Lengths and pieces
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
an L 10 m by 7 m, a 6×4 corner missingA different L, as an example.
pieces: 4×7 and 6×3Cut into shapes you know, with each piece's own lengths.
28+18=46m2Add the pieces.
10×7−6×4=70−24=46m2Or: the big rectangle, take away the missing corner.
10+3+6+4+4+7=34mThe perimeter adds every side, in metres.
Now find the step.
Read Ella's working. How tall is each piece?
The L-shape is 8 m along the bottom and 3 m along the top. How long is the step across in the middle?
Ella finds the area of the L-shape. Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
Other shapes
Need a hint?
The rectangle around the U is 10 m by 8 m.
Another hint
The gap is 4 m wide and 5 m deep.
Need a hint?
Cut where the roof meets the walls.
Another hint
The roof is a triangle: half of base times height.
Try it yourselfYour edits are saved in this browser.
Find the area of the U-shape by taking away from a rectangle.
Use the board. This ticks when you get there.
The front of a house: walls 4 m tall, the roof's top 7 m up. Cut it into a rectangle and a triangle.
Use the board. This ticks when you get there.
Two sides of the L are marked ?. Work them out, then find the perimeter in metres.
PractiseA few on your own, to check you have it.
Composite shapes
A shape is a rectangle 10 cm by 4 cm with a triangle on top (base 10 cm, height 3 cm). What is its area in cm²?
A photo frame is 30 cm by 20 cm on the outside, with a 24 cm by 14 cm hole for the photo. What is the frame's area in cm²?
An L-shape fits exactly inside an 8 m by 6 m rectangle (one corner cut out). What is its perimeter in metres?
Review: what is the area of a triangle with base 6 cm and height 5 cm, in cm²?
An L-shape is 10 m wide along the bottom and 7 m tall on the left. Its left part is 4 m wide; its right part is 6 m wide and only 3 m tall. What is its area in m²?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Design a garden
Then find your garden's perimeter, and say how many metres of edging it needs.
Try it yourselfYour edits are saved in this browser.
Need a hint?
The left part is always 3 × 6 = 18 m².
Another hint
The right part must be 38 − 18 = 20 m².
Change a and b so the L-shaped garden has an area of 38 m².
Use the board. This ticks when you get there.
For teachers
Students cut composite shapes into rectangles and triangles, or take pieces away from a rectangle, and find the area and the perimeter (AC9M8M01, node 1). About 40 minutes.
I can cut a composite shape into pieces I know and add their areas.
I can find an area by taking pieces away from a big rectangle.
I can work out a missing side and the perimeter.
Before class
Do the Run board yourself first. Students need the area of rectangles and triangles (Area unit).
The one idea: the area is the same however you cut it, as long as each piece uses its own lengths. The common mistake is giving a piece the whole shape's height (Ella's mistake).
The board's three buttons cut across, cut up and down, or take away from the surrounding rectangle; it names each piece and gives its area. The perimeter needs every side, so missing sides (the ? sides) come first.
Wider: floor plans of real rooms, shapes that need both adding and taking away.
Running the lesson
Predict: ask a 48 student what is missing from the rectangle.
Run: ask which way they would choose for this shape, and why.
Investigate: ask how tall each of Ella's pieces is.
Modify: ask why taking away is quicker for the U.
Make: ask for a second pair of lengths that also works.
The hard part (on purpose)
Using each piece's own lengths. Ask students to write the two lengths on each piece before they multiply.
When students get stuck
A student uses the whole height for every piece. Ask: How tall is this piece on its own? Fix: Write each piece's two lengths on it before you multiply.
A student cannot find a missing side. Ask: Which two sides together make the opposite side? Fix: A missing side is the difference of two others.
A student gives a perimeter in m². Ask: Are you counting squares or measuring a length? Fix: Perimeter is a length: m. Area counts squares: m².
Stretch and support
Stretch: Draw a floor plan of a room at home with at least six walls. Find its area two ways and its perimeter.
Support: Give the shape on grid paper so pieces can be counted as well as multiplied.
Where it leads
This starts AC9M8M01.
Where it leads: irregular shapes (next lesson), prisms with composite bases (Volume), and circles in composite shapes.
What each step is for
an L-shaped room: Committing to an area for an L-shape shows whether they split it or multiply the two outside lengths.
cut it up: Trying each way of cutting the same shape shows that every valid split, and take-away, gives the same area.
lengths and pieces: Missing lengths come from differences of others; each piece needs its own height (Ella uses the whole height twice); every method agrees.
other shapes: A U-shape is quickest by taking away, a house needs a triangle, and the perimeter needs missing sides first; units separate area from length.
composite shapes: Adding a triangle, taking away a frame and a perimeter shortcut, mixed; using the whole height for a piece is the planted slip; the triangle review keeps half base times height fresh.
design a garden: Choosing two lengths to hit a target area uses the decomposition in reverse.
Now you can
Cut a composite shape into pieces I know and add their areas.
Find an area by taking pieces away from a big rectangle.