022i - Decomposing Irregular Shapes

Lesson 22 — Decomposing Irregular Shapes into Familiar Shapes

Strand: Measurement | Descriptor: AC9M8M01 | Duration: 45 minutes

Learning Intentions

• To decompose irregular polygons, including shapes with diagonal (chamfered) edges, into rectangles and triangles.
• To choose an efficient decomposition strategy for a given shape.

Success Criteria

I can:

1. Recognise a rectangle with a corner cut off along a diagonal, and find its area by subtracting a triangle.
2. Decompose an irregular polygon given only as a set of grid coordinates, without a pre-drawn split.
3. Compare two different valid decompositions of the same shape and confirm they give the same area.
4. Justify a choice of addition or subtraction as the more efficient method for a given shape.

Warmup

(6 minutes — retrieval and a new idea, mini whiteboards)

1. A rectangle is cm by cm. Find its area.
2. A right-angled triangle has legs cm and cm. Find its area.
3. The rectangle in Q1 has its top-right corner cut off along a diagonal, removing exactly the triangle from Q2. What is the area of the remaining five-sided shape (pentagon)?
4. Could you have found the same answer by splitting the pentagon into pieces and adding, instead of subtracting? Describe how, without calculating.

Answers: 1. ; 2. ; 3. ; 4. Yes — e.g. split it into a smaller rectangle and a trapezium, or two rectangles and a triangle — but subtraction is far simpler here.

Teacher note: This is new territory — Year 7 composite shapes were built entirely from right angles. Today’s shapes include a diagonal cut, which can only be handled by identifying a triangle to add or subtract.

Activities

Activity 1 — Explicit Instruction: Chamfered Corners (12 min)

The key idea: if a rectangle has a corner sliced off in a straight diagonal line, the missing piece is always a right-angled triangle. Find its two legs, calculate its area, and subtract.

I do — a pentagon-shaped deck. A rectangle m by m has its top-right corner cut off along a diagonal, removing a right triangle with legs m and m.

We do — a hexagon with two chamfered corners. A rectangle m by m has its top-left corner cut off (legs m and m) and its top-right corner cut off (legs m and m).

Why subtraction wins here: trying to add pieces for a chamfered shape forces you to split the remaining region into several awkward slivers. Subtracting one small triangle from the whole rectangle is almost always faster.

You do: An arrowhead pentagon, a hexagon with one inward triangular notch, and a rectangle with two chamfered corners on the same side — practise identifying and subtracting the correct triangle(s) each time.

Activity 2 — Guided Practice: Many Valid Decompositions (10 min)

Pairs, grid paper.

Same shape, two methods. A “home plate” pentagon has vertices at , , , , on a metre grid.

Method A — split and add. The lower part is a rectangle ; the upper part is a triangle with base (from to ) and height .

Method B — enclose and subtract. Enclose the whole pentagon in a rectangle , then subtract the two triangular corners left empty at top-left and top-right (each with legs and ).

Both methods agree. Now you try: given only the coordinates , , , , , on grid paper — no split lines are drawn for you. Plot the shape, choose your own decomposition, and compare your method (and your answer) with your partner’s.

Activity 3 — Inquiry: the Surveyed Block of Land (11 min)

Pairs.

A rectangular block of land measures m by m, except the back-right corner is cut off by a diagonal boundary fence, removing a right triangle with legs m and m from that corner.

1. Find the area of usable land, once the fence is accounted for.
2. A rectangular shed m by m is to be built somewhere within the usable land. Find the area of land remaining once the shed is built.
3. The owner wants to know: could the shed instead have been placed so that one of its corners touched the diagonal fence? Would that change your answer to Q2?

Socratic scaffolding:

Prompt	Purpose
Understand: what shape is the usable land?	A pentagon — a rectangle with one corner chamfered off.
Devise a plan for Q1	Whole rectangle minus the corner triangle.
Carry out	; triangle ; usable land .
Devise a plan for Q2	Usable land minus the shed’s footprint.
Carry out	Shed ; remaining .
Look back at Q3	Does the shed’s position affect its area? No — area depends only on the shed’s dimensions, not where it sits, as long as it fits entirely within the usable land.
Extend	What would change if the shed’s corner overlapped the diagonal fence? Then it would no longer be a simple rectangle-minus-rectangle problem — a new triangular piece would need to be found.

Answers: 1. . 2. . 3. No — the subtraction only depends on the two areas, not on placement, provided the shed lies entirely within the usable land.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A rectangle cm by cm has its top-right corner cut off along a diagonal, removing a triangle with legs cm and cm. Find the area of the remaining shape.
2. A pentagon is a rectangle m by m with a triangular point (base m, height m) attached to one side. Find its total area.
3. Two students decompose the same hexagon differently. One gets , the other gets . What must be true, and what should they do?
4. Reasoning. Explain why “whole rectangle minus a corner triangle” is usually more efficient than splitting a chamfered shape into several pieces and adding.
5. A pentagon has vertices , , , , on a centimetre grid. Find its area.

Answers: 1. ; 2. ; 3. Area cannot depend on the method used, so one (or both) of them has made an error — they should re-check their splits for double-counting or a missed piece; 4. Subtraction requires locating and calculating only one small triangle, versus splitting the remaining irregular region into several awkward pieces; 5. Rectangle plus triangle base , height , area ; total .

Common Misconceptions

Misconception	How to pre-empt it
Assuming all composite shapes are rectilinear (right angles only).	Explicitly contrast Lesson 21’s notches with today’s diagonal cuts side by side.
Treating a chamfered corner as a rectangular notch — subtracting a rectangle instead of a triangle.	Insist students identify and label the two legs of the missing triangle before calculating anything.
Adding pieces for a chamfered shape rather than subtracting, leading to unnecessary complexity and arithmetic slips.	Model the “why subtraction wins” comparison explicitly in Activity 1.
Misreading a vertex on a coordinate grid, producing the wrong side length.	Require every coordinate to be labelled with its pair directly on the sketch before any lengths are found.
Believing only one decomposition is “correct” for a shape.	Activity 2 shows two valid methods reaching the same answer — normalise this explicitly.
Forgetting to halve the area of the triangular piece being added or removed.	Require the triangle formula, not just the numbers, on every working line.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A square of side cm has each of its four corners cut off along diagonals, each cut removing a right triangle with legs cm. Find the area of the resulting octagon.

Answer

E2 (AMC Junior style). An arrowhead-shaped flag is a rectangle cm by cm with a triangular point attached to one short end. The point’s base equals the rectangle’s height ( cm) and it extends cm beyond the rectangle. Find the flag’s total area.

Answer

E3 (Challenge — a preview of Pythagoras). A rectangular block of land m by m has its top-right corner cut off along a diagonal fence, removing a right triangle with legs m and m. Find (a) the area of the resulting pentagon (b) the length of the diagonal fence (c) the perimeter of the pentagon.

Answer

(a) .

(b) The legs and belong to a special right triangle whose hypotenuse is a whole number: ––. The fence is m long. (You’ll meet the rule behind this — Pythagoras’ theorem — later this year.)

(c) The two edges removed (totalling m) are replaced by the single m fence, so:

E4 (Challenge). A pentagon is formed by cutting a triangular corner from a cm by cm rectangle. The removed triangle has one leg of cm. If the pentagon’s area is , find the length of the other leg of the removed triangle.

Answer

Homework

1. A rectangle m by m has its top-right corner cut off diagonally, removing a triangle with legs m and m. Find the area of the remaining pentagon.
2. A pentagon is a rectangle cm by cm with a triangular point (base cm, height cm) attached to one short side. Find its total area.
3. A hexagon is formed by chamfering both top corners of a m by m rectangle: the left cut removes a triangle with legs m and m; the right cut removes a triangle with legs m and m. Find the hexagon’s area.
4. A shape’s vertices on a centimetre grid are , , , , . Find its area.
5. A square of side cm has each of its four corners cut off, each removing a right triangle with legs cm and cm. Find the area of the resulting octagon.
6. Reasoning. Explain why cutting a triangular corner from a rectangle always decreases the perimeter, while (from Lesson 21) cutting a rectangular notch from a corner never changes it.
7. Challenge. A rectangular sheet cm by cm has a right-triangular corner removed, with legs cm and cm (a Pythagorean triple, so the hypotenuse is cm). Find (a) the area of the remaining pentagon (b) its perimeter.

Answers: Q1 — . Q2 — . Q3 — . Q4 — rectangle plus triangle (base , height ) , total . Q5 — . Q6 — a rectangular notch replaces two edges with two new edges of exactly the same total length, but a diagonal cut replaces two edges (the legs) with a single straight edge (the hypotenuse), which by the triangle inequality is always shorter than the sum of the legs it replaces. Q7 — (a) (b) cm.