021i - Area and Perimeter of Composite Shapes

Lesson 21 — Area and Perimeter of Composite Shapes

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

Learning Intentions

• To calculate the area of composite rectilinear shapes made from rectangles and triangles.
• To calculate the perimeter of a composite shape, including sides that are not directly labelled.

Success Criteria

I can:

1. Split a composite rectilinear shape into rectangles and triangles, and calculate its area.
2. Deduce unlabelled side lengths using the fact that opposite sides of a rectangle are equal.
3. Trace the complete outline of a composite shape to find its perimeter.
4. Explain why area and perimeter of the same shape are found by different processes.

Warmup

(6 minutes — retrieval relay, mini whiteboards)

1. A rectangle is cm by cm. Find its area and its perimeter.
2. A triangle has base cm and height cm. Find its area.
3. A rectangle’s total perimeter is m. One pair of opposite sides is m each. Find the length of the other pair.
4. True or false, with a reason: “The perimeter of a composite shape is just the sum of all the numbers written on the diagram.”

Answers: 1. Area , perimeter cm; 2. ; 3. , so m; 4. False — some outline sides are never labelled and must be deduced before they can be added.

Teacher note: Question 4 is the hook. Do not resolve it yet — Activity 2 answers it directly.

Activities

Activity 1 — Explicit Instruction: Area of Composite Rectilinear Shapes (12 min)

Recap the two Year 7 strategies in one line each: split and add the parts; or take a whole and subtract a hole. Today’s shapes need both, sometimes in the same problem.

I do — a garage floor plan. The floor is an overall rectangle m by m, with a m by m rectangle removed from the top-right corner.

Check by subtraction: ; notch ; ✓

We do — a T-shaped classroom. A horizontal bar cm wide and cm tall sits above a stem cm wide and cm tall, centred beneath it.

You do: Four composite shapes from a worksheet — an L-shape, a step-shape, a cross-shape and a T-shape — each requiring at least one missing length to be deduced before calculating.

Activity 2 — Explicit Instruction: the Perimeter of a Composite Shape (12 min)

Return to the warmup hook. Perimeter is found by tracing the full outline once, writing every segment — given or deduced — as you go, then summing them all.

I do — the same garage floor plan ( m by m rectangle, m by m notch removed from a corner). Trace clockwise from the bottom-left corner:

Notice: the bounding rectangle’s own perimeter is m — exactly the same! The two edges removed by the notch ( m m m) are exactly replaced by two new edges of the same total length elsewhere on the outline.

We do — the T-shaped classroom (bar , stem , centred). Trace clockwise from the top-left corner:

Stated generalisation (to be tested, not assumed, in Activity 3): removing a single rectangular notch from a corner of a rectangle always leaves the perimeter equal to that of the bounding rectangle.

You do: Find the perimeter of three shapes with unlabelled sides, deducing each missing length before summing.

Activity 3 — Inquiry: Does Cutting a Notch Always Leave the Perimeter Unchanged? (10 min)

Pairs.

A rectangular vegetable garden measures m by m. A rectangular herb bed, m by m, is paved over (removed from the growing area).

1. If the herb bed is cut from a corner of the garden, find the perimeter of the remaining growing area.
2. Now suppose the herb bed is cut from the middle of a long side instead, so it forms a notch that does not touch any corner. Find the new perimeter.
3. Compare your two answers. What do you notice? Explain why.

Socratic scaffolding:

Prompt	Purpose
Understand: what are you comparing?	Two different positions for the same-sized removed rectangle — corner vs middle of a side.
What do you already know from Activity 2?	Removing a notch from a corner does not change the perimeter.
Devise a plan	Trace the outline carefully for both cases, counting every segment, including any new inward walls.
Carry out: the corner case	m — matches the bounding rectangle, as expected.
Carry out: the middle case	Tracing gives m — four metres more.
Look back — why the difference?	A corner notch removes two edges and replaces them with two edges of the same total length. A middle notch removes one edge and replaces it with three (in, across, out) — always longer.
Generalise	The increase equals twice the notch’s depth (how far it cuts inward), regardless of how wide the notch is.

Answers: 1. m. 2. m. 3. Cutting from a corner is perimeter-neutral; cutting from the middle of a side always increases the perimeter.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. An L-shaped room has overall dimensions m by m, with a m by m rectangle removed from one corner. Find its floor area.
2. Find the perimeter of the same L-shaped room.
3. A step-shape has an overall width of cm. Two of its three horizontal top segments measure cm and cm. Find the third.
4. Reasoning. Explain why splitting a composite shape in two different valid ways always gives the same total area.
5. A notch is cut from the middle of one side of a rectangle (not touching a corner). Compared to the original rectangle, does the perimeter increase, decrease, or stay the same? Explain.

Answers: 1. ; 2. Corner notch, so perimeter m; 3. cm; 4. The split lines lie inside the shape and neither add nor remove any of the region being measured — the pieces always cover exactly the original area, however it is divided; 5. Increases — the notch replaces one straight edge with three edges (two inward walls plus the notch’s own base), which together are longer than the edge removed.

Common Misconceptions

Misconception	How to pre-empt it
Perimeter is the sum of only the numbers already on the diagram.	Insist every side is either labelled, deduced, or explicitly assumed absent — trace with a finger before summing.
A corner-notch result (“perimeter unchanged”) is assumed to apply to any notch.	Activity 3 directly contrasts corner vs middle-of-side cuts to break this over-generalisation.
Using the split lines drawn for an area calculation as if they were part of the perimeter.	State explicitly: split lines are internal and are never part of the outline.
Missing a segment when tracing a many-sided outline.	Trace with a pencil, ticking off each segment as it is counted, always in one consistent direction.
Forgetting to halve a triangular part when finding area.	Require the formula, not just the numbers, on every working line.
Giving an area answer in linear units, or a perimeter answer in squared units.	Check units at the end of every line, not just the final answer.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cross (plus-sign) shape is formed from five congruent squares, each of side cm: one central square with four more attached to its sides. Find the perimeter of the whole cross.

Answer

Each of the five squares contributes exactly three of its four sides to the outer boundary (one side of each outer square touches the centre and is hidden). Tracing the full outline gives side-lengths in total:

E2 (AMC Junior style). A cm by cm rectangle has an identical square notch of side cm cut from each of its four corners. Find the perimeter of the resulting shape.

Answer

Every notch is a corner notch, and each one individually leaves the perimeter unchanged (Activity 2’s result). So the perimeter is simply that of the original rectangle:

(True however large or small the four notches are, as long as they don’t overlap.)

E3 (Challenge/Investigation). A T-shaped bracket has a horizontal bar of length cm and height cm, and a vertical stem of height cm, centred beneath the bar, with unknown width (where ). Show that the perimeter of the T-shape is the same no matter what is, and find that value.

Answer

Tracing the outline, the two “notch top” segments sum to regardless of position, and the stem’s base contributes exactly — these cancel:

The stem’s width never appears in the final total.

E4 (Challenge). A staircase garden bed is formed by removing three identical square notches, each of side m, evenly spaced along one edge of a m by m rectangular plot (none touching a corner). Find the total area and perimeter of the remaining plot.

Answer

Each middle-of-side notch adds m to the perimeter, regardless of its width.

Homework

1. An L-shaped patio has overall dimensions m by m, with a m by m rectangle removed from one corner. Find its area.
2. Find the perimeter of the patio in Q1.
3. A step-shape’s top edge is split into three unlabelled horizontal segments across an overall width of cm. Two segments measure cm and cm. Find the third.
4. A plus-shaped (cross) courtyard is made from five congruent squares of side m, arranged as in Enrichment E1. Find (a) its total area (b) its total perimeter.
5. A rectangle cm by cm has a rectangular notch cm by cm cut from the middle of one long side (not touching a corner). Find (a) the area remaining (b) the new perimeter.
6. Reasoning. A student claims: “Cutting any notch from a rectangle always keeps the perimeter the same, because you remove some edge and add some edge back.” Using your results from Q2 and Q5 as evidence, explain what is wrong with this claim.
7. Challenge. A rectangular sheet of metal cm by cm has an identical square notch of side cm cut from each of its four corners, so it can be folded into an open box. Find (a) the area of the resulting cross shape (b) its perimeter.

Answers: Q1 — . Q2 — corner notch, m. Q3 — cm. Q4 — (a) (b) m. Q5 — (a) (b) cm. Q6 — the claim only holds for corner notches, where two removed edges are exactly replaced; a middle-of-side notch (as in Q5) replaces one edge with three, always increasing the perimeter. Q7 — (a) (b) all four notches are corner notches, so perimeter is unchanged: cm.