Mathematics · Year 5 · Measuring the World · Spring Term

Perimeter of Rectilinear Shapes

Students will calculate the perimeter of rectilinear shapes, including composite shapes.

• National Curriculum Attainment Targets
• KS2: Mathematics - Measurement

About This Topic

Rectilinear shapes consist of horizontal and vertical straight lines only, making perimeter calculation straightforward by adding all outer side lengths. In Year 5, students extend this to composite rectilinear shapes, which combine multiple rectangles. They break down complex figures into simpler parts, identify missing lengths using opposite sides or totals, and calculate total perimeters accurately. This aligns with KS2 Measurement objectives, building on prior knowledge of 2D shapes and perimeter of rectangles.

Students explore key questions like finding missing sides, designing shapes with a specific perimeter such as 24 cm, and comparing perimeters of squares versus rectangles with equal areas. These tasks develop spatial reasoning, problem-solving, and justification skills. For instance, a square always has the smallest perimeter for a given area among rectangles, which encourages mathematical reasoning and prediction.

Active learning suits this topic well. When students physically construct shapes with string, straws, or grid paper, then measure and adjust perimeters, they grasp composite breakdowns intuitively. Collaborative design challenges reveal relationships between side lengths and totals, while hands-on comparisons make abstract comparisons concrete and foster discussion.

Key Questions

1. Explain how to find the perimeter of a shape with missing side lengths.
2. Design a rectilinear shape with a perimeter of 24 cm.
3. Compare the perimeter of a square with a rectangle that has the same area.

Learning Objectives

• Calculate the perimeter of rectilinear composite shapes by summing all exterior side lengths.
• Identify and calculate missing side lengths in rectilinear shapes using properties of parallel and equal sides.
• Design a rectilinear shape with a specific given perimeter, justifying the chosen side lengths.
• Compare the perimeters of a square and a rectangle that share the same area, explaining the relationship observed.

Before You Start

Perimeter of Rectangles

Why: Students need to understand the concept of perimeter and how to calculate it for a basic rectangle before moving to more complex shapes.

Properties of 2D Shapes

Why: Knowledge of basic shapes like squares and rectangles, including their properties (e.g., opposite sides are equal), is essential for calculating missing lengths.

Key Vocabulary

Rectilinear shape	A shape made up of only horizontal and vertical straight lines. All angles are right angles.
Composite shape	A shape made up of two or more simpler shapes, such as rectangles, joined together.
Perimeter	The total distance around the outside edge of a two-dimensional shape. It is found by adding the lengths of all the sides.
Adjacent sides	Sides of a shape that are next to each other and share a common vertex (corner).

Watch Out for These Misconceptions

Common Misconception

Perimeter of composite shapes requires adding all internal lines.

What to Teach Instead

Students must exclude internal sides that cancel out, focusing only on the outer path. Hands-on building with straws helps them trace the perimeter path physically, seeing internal edges disappear. Group dissection of shapes reinforces this through shared measurement.

Common Misconception

All opposite sides in rectilinear shapes are equal, even with missing lengths.

What to Teach Instead

Opposite sides are equal only if parallel and in rectangles; composites require careful breakdown. Puzzle activities with partial lengths prompt trial-and-error deduction, while peer teaching clarifies assumptions. Collaborative verification builds confidence in strategies.

Common Misconception

Rectangles with same area always have same perimeter.

What to Teach Instead

Longer thinner rectangles have larger perimeters than compact ones like squares. Sorting tasks with physical models let students measure and compare directly, revealing the pattern through data collection and class graphs.

Active Learning Ideas

Stations Rotation: Perimeter Challenges

Prepare four stations with rectilinear shapes on grid paper: simple rectangles, L-shapes, shapes with missing lengths, and composite figures. Students measure sides, calculate perimeters, and explain missing values using station worksheets. Groups rotate every 10 minutes, then share one key insight as a class.

45 min·Small Groups

Design Challenge: Fixed Perimeter Builds

Provide multilink cubes or straws and ask pairs to build rectilinear shapes with exactly 24 units perimeter. They sketch designs, measure to verify, and label side lengths. Pairs present one shape and justify how they achieved the target.

30 min·Pairs

Puzzle Pairs: Missing Lengths

Give pairs cards with rectilinear shapes showing some lengths and totals. They deduce missing sides by adding known parts or using opposites equal. Switch puzzles midway and discuss strategies whole class.

25 min·Pairs

Compare and Sort: Whole Class Relay

Display rectangles and squares with same areas on board. Teams race to calculate perimeters, sort shapes by perimeter size, and explain why the square has the smallest. Debrief patterns observed.

35 min·Whole Class

Real-World Connections

• Architects and builders use perimeter calculations when determining the amount of fencing needed for a property or the baseboards required for a room, ensuring accurate material orders for construction projects.
• Urban planners and landscape designers calculate perimeters for park layouts, garden beds, and pathways to efficiently allocate space and materials for public areas.
• Manufacturers of picture frames or window panes need to measure the perimeter of the glass or frame to ensure the correct size is cut and fitted precisely.

Assessment Ideas

Quick Check

Provide students with a worksheet featuring several rectilinear composite shapes, some with missing side lengths. Ask them to calculate the perimeter for each shape, showing their working. Check for accurate addition and correct identification of missing lengths.

Discussion Prompt

Present students with a square and a rectangle that have the same area (e.g., a 4x4 square and an 8x2 rectangle, both area 16). Ask: 'Which shape has the larger perimeter? How do you know?' Facilitate a discussion comparing their findings and reasoning.

Exit Ticket

Give each student a card with the instruction: 'Design a rectilinear shape with a perimeter of 20 cm. Draw it and label all side lengths.' Collect the drawings to assess their ability to create a shape meeting the specified perimeter.

Frequently Asked Questions

How do you teach perimeter of composite rectilinear shapes?

Break shapes into rectangles, label all sides, and sum outer edges only. Use grid paper for accuracy and colour-coding for parts. Start with simple L-shapes, progress to complex ones, and have students self-check by rebuilding with string to trace the path.

What activities help students find missing side lengths?

Provide shapes with some lengths and totals; students solve by subtraction or equal opposites. Card sorts and digital puzzles build fluency. Follow with design tasks where they create their own missing-length challenges for peers to solve, deepening understanding through creation.

How can active learning help students understand perimeter of rectilinear shapes?

Physical construction with everyday materials like straws or geoboards makes abstract calculations tangible, as students measure real perimeters and adjust shapes. Collaborative stations encourage explaining strategies, reducing errors from misconceptions. Design challenges link perimeter to real-world fencing, boosting engagement and retention through movement and talk.

Why compare perimeters of squares and rectangles with same area?

It shows squares minimise perimeter for fixed area, a key isoperimetric principle. Students predict, calculate, and test with models, developing reasoning. This connects to optimisation problems later, like enclosing gardens efficiently, and sparks curiosity about shape efficiency.