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Perimeter area volume and unit conversions - Mathematics Year 7

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Introduction

Why can a sloping parallelogram have the same area as a rectangle? Choose the right quantity, convert dimensions consistently and explain why square and cubic conversions differ from length conversions.

You will complete six measure calculations and one L-shaped composite problem. Include a trapezium, a mixed-unit cuboid and a perimeter check. Start by deciding what 9 cm of base and 4 cm of perpendicular height tell you—and what they do not tell you about a sloping side.

Derive area formulas from geometric structure

Perimeter measures the boundary length of a plane shape; area measures the amount of plane region inside it. A rectangle with sides 8 cm and 3 cm has perimeter 2×(8+3)=22 cm and area 8×3=24 cm². These are different quantities and use different units. An area value does not determine a unique perimeter unless enough shape information is supplied.

For a parallelogram, the area is base times perpendicular height: $A = b h$ . Imagine cutting a triangular end region and moving it to fill the other side, forming a rectangle of the same base and height without changing area. The source diagram illustrates that idea. Base 9 cm and perpendicular height 4 cm give 36 cm². The sloping side could be longer than 4 cm; substituting it for the perpendicular height would generally overestimate area.

An acute triangle with base b and a dashed perpendicular height h. — The small right-angle mark identifies the height as perpendicular to the base. The source contains no9cmor4cmlabels;our numerical triangle example is separate.

Two congruent copies of a triangle can form a parallelogram, so a triangle has $A = \frac{1}{2} b h$ . Base 9 cm and perpendicular height 4 cm therefore give 18 cm². The height must be perpendicular to the chosen base; an arbitrary side length is not automatically the required height. The triangle’s orientation on the page does not change its area.

Trapezium area. Let the parallel sides be a and b, and their perpendicular separation be h. Two congruent copies form a parallelogram with base a+b and height h; one copy has $A = \frac{1}{2} (a + b) h$ . For parallel sides 7 cm and 11 cm with height 5 cm, the area is½×18×5=45 cm². Add the two parallel sides before halving and multiplying by height; do not substitute a non-parallel edge for one of them.

These derivations explain why the formulas work rather than merely naming a mnemonic. Sketch the chosen base, parallel sides and right-angle height marker in your own example. A diagram can communicate exact stated dimensions without being drawn to scale; measuring screen pixels would not improve the given data.

Convert dimensions and complete six calculations

Convert dimensions to a consistent unit before multiplying or adding. Since 1 m=100 cm, a square metre is 100 cm×100 cm=10000 cm², not 100 cm². A cubic metre is 100×100×100=1000000 cm³. These factors preserve the same physical quantity while changing its numerical unit representation.

Mixed-unit cuboid. An original box model has dimensions 0.4 m, 30 cm and 20 cm. First convert 0.4 m to 40 cm, then $V = 40 \times 30 \times 20 = 24000 {c m}^{3}$ . Alternatively use metres: 0.4×0.3×0.2=0.024 m³. Multiplying 0.024 by 1000000 recovers 24000 cm³. The two paths check the same volume; 0.4×30×20 without reconciling units would not give a valid single-unit result.

Volume measures three-dimensional space and uses cubic units. Capacity can express the same amount in litres when suitable: 1000 cm³=1 litre, so this model corresponds to 24 litres. Actual container usable capacity may differ because of wall thickness or fill limits; the cuboid calculation assumes the stated internal dimensions. An online model does not certify a real container measurement.

Six original measure calculations.

Stated task	Result	Key method
Rectangle 8 cm by 3 cm: perimeter	22 cm	2 times the sum of side lengths
Triangle base 9 cm, height 4 cm: area	18 cm²	Half base times perpendicular height
Parallelogram base 9 cm, height 4 cm: area	36 cm²	Base times perpendicular height
Trapezium parallel sides 7 cm/11 cm, height 5 cm: area	45 cm²	Half the parallel-side sum times height
Cuboid 0.4 m by 30 cm by 20 cm: volume	24000 cm³	Convert all lengths then multiply
Convert 2.5 m² to square centimetres	25000 cm²	Multiply by 10000

Explain the unit at every step. Converting an area requires the square of the length factor; converting a volume requires its cube. Changing a unit label without the correct factor changes the claimed physical quantity. Check the size of the result against the original dimensions as well as checking the arithmetic.

Use Corbettmaths’ original trapezium derivation and examples to identify the parallel sides and perpendicular height. Then calculate our separate 7 cm/11 cm/5 cmcase and compare its two-copy explanation with the triangle/parallelogram models. Original: corbettmaths, Area of a Trapezium - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Compare composite methods and check dimensional units

An original L-shaped composite region. Start with an 8 cm by 6 cmrectangle. Remove a 3 cmwide, 2 cmhigh rectangle from its top-right corner. One precise boundary description is the vertices(0, 0), (8, 0), (8, 4), (5, 4), (5, 6), (0, 6), in centimetres, then back to the start. Draw this labelled schematic; the missing corner is not a hole in the middle.

The remaining area is 8×6−3×2=42 cm². A second decomposition splits it into a 5 cm by 6 cmleft rectangle and a 3 cm by 4 cmright rectangle, whose areas 30 and 12 also total 42 cm². The subtraction and addition methods describe the same non-overlapping region; do not count an overlap twice or include the removed corner.

For its perimeter, follow the actual boundary: 8+4+3+2+5+6=28 cm. It equals the outer rectangle’s perimeter in this special corner-removal construction because the two removed boundary lengths are replaced by equal inward lengths. This does not mean every cutout leaves perimeter unchanged: an interior hole or a different notch would need its own boundary accounting.

Unfamiliar mixed-unit triangle. A triangle has base 1.2 m and perpendicular height 75 cm. In centimetres, ½×120×75=4500 cm². In metres, ½×1.2×0.75=0.45 m². Multiply 0.45 by 10000 to check 4500. Comparing these paths reinforces why a square-unit conversion is not a simple factor 100.

Error clinic. A learner writes 2.5 m²=250 cm² using the length conversion once. One square metre contains 10000 unit-square centimetres, so the corrected answer is 25000 cm². Another uses the sloping side as a parallelogram height; mark the perpendicular distance before substituting. A third calls a volume 24000 cm²: the arithmetic may be right but the dimensional unit is not.

Choose a method to suit the information. A subtractive composite calculation is short when a clear corner is removed; adding rectangles can be safer when the pieces are directly given. Keep all six calculations, a labelled composite boundary, two area methods and an unfamiliar unit check. Return after a genuine break to one squared/cubed conversion and retain your actual later correction.

Interactive Tasks

Quiz: Test Your Knowledge

What is the perimeter of an 8 cm by 3 cmrectangle? (22 cm) (!24 cm²) (!11 cm) (!48 cm)

What area follows from triangle base 9 cm andperpendicular height 4 cm? (18 cm²) (!36 cm²) (!26 cm²) (!13 cm²)

What area follows from parallelogram base 9 cm andheight 4 cm? (36 cm²) (!18 cm²) (!26 cm²) (!13 cm²)

What is the stated trapezium area? (45 cm²) (!90 cm²) (!35 cm²) (!55 cm²)

How many square centimetres make one square metre? (10000) (!100) (!1000) (!1000000)

What is the stated mixed-unit cuboid volume? (24000 cm³) (!240 cm³) (!2400 cm³) (!240000 cm³)

What is 2.5 m² in square centimetres? (25000 cm²) (!250 cm²) (!2500 cm²) (!250000 cm²)

What is the remaining L-shaped area? (42 cm²) (!48 cm²) (!54 cm²) (!28 cm²)

What is the L-shaped boundary perimeter? (28 cm) (!42 cm) (!34 cm) (!20 cm)

What is the unfamiliar 1.2 m by 75 cmtriangle area? (0.45 m²) (!45 m²) (!0.9 m²) (!450 m²)

Memory Game

Perimeter	Length around a boundary
Area	Size of a plane region in square units
Volume	Space occupied in cubic units
Perpendicular	Meeting at a right angle
Trapezium	Here a quadrilateral with a pair of parallel sides
Conversion	Equivalent quantity expressed in another unit

Drag and Drop

22 cm	Rectangle boundary result
18 cm²	Triangle area result
36 cm²	Parallelogram area result
45 cm²	Trapezium area result
24000 cm³	Cuboid volume result

Crossword Puzzle

PERIMETER	Boundary length
AREA	Plane-region size
VOLUME	Three-dimensional space
PARALLEL	Lines in a plane with constant separation
HEIGHT	Perpendicular distance to a chosen base
CUBOID	Rectangular three-dimensional box

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Choose the quantity: Distinguish rectangle boundary length from area and cuboid volume.
Mark the correct height: Explain perpendicular height versus a sloping side.
Calculate cases 1 to 3: Complete the original rectangle, triangle and parallelogram examples with units.
Explain a squared factor: Build a 100 cm by 100 cmmodel of one square metre.

Standard

Complete all six measure tasks: Show each formula, conversion and dimensional unit.
Explain the trapezium derivation: Use two congruent copies and the sum of parallel sides.
Solve the L-shaped region: Retain both area methods and the full boundary sum.
Check cuboid units two ways: Compare centimetre and metre products for the same volume.

Advanced

Transfer to mixed-unit triangle: Obtain 4500 cm² and 0.45 m² by two independent paths.
Test a perimeter generalisation: Explain why this corner-removal case preserves perimeter without asserting every cutout does.
Correct dimension errors: Diagnose factor 100 forarea, slopingheight and square-volume-unit errors.
Retrieve a conversion later: Keep a genuine later squared/cubed unit task and explain your method choice.

Learning Check

Six measure calculations: Complete the six original table cases.
Formula reasoning: Explain triangle, parallelogram and trapezium formulas.
Mixed-unit volume: Verify the cuboid by two unit choices.
Composite area and boundary: Solve the stated top-right corner removal.
Unfamiliar conversion and first errors: Solve the mixed-unit triangle and correct dimension mistakes.
Evidence and method choice: Keep labelled working and genuine later retrieval.

Worked checks and assessment guidance

In order 22 cm, 18 cm², 36 cm², 45 cm², 24000 cm³, 25000 cm². Use perpendicular heights, parallel sides and consistent units; the length/area/volume distinctions are part of the answer.
Cut/rearrange a parallelogram to a same-base/height rectangle; two equal triangles make a parallelogram; two equal trapezia make basea+b, heighth. Thus bh, ½bh and½(a+b)h respectively.
40×30×20=24000 cm³; 0.4×0.3×0.2=.024 m³; factor 1000000 makes them equal. The model corresponds to 24 Lunderinternal-dimensionassumptions, not an observed real fill measurement.
48−6=42 cm² and 5×6+3×4=42 cm². Boundary 8+4+3+2+5+6=28 cm. The equality with outer perimeter is specific to this corner replacement, not every cutout.
½×120×75=4500 cm² and½×1.2×.75=.45 m². Multiply by 10000, not 100. A sloping side is not necessarily height, and volume requires cubic units.
Declare dimensions, units, assumptions and the actual boundary. Compare additive/subtractive composite methods and retain real later work; digital calculation does not certify physical instrument accuracy.

Learning evidence

Keep six calculations, a labelled L-shaped region, two area methods, the boundary sum and a real later unit-conversion retry.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, geometry, measures and statistics; fluency, reasoning and problem solving. Year 7 allocation is editorial within KS3. All numerical tasks, worked answers, coordinate models and example data are original. Actual Commons files retain the creator/licence credits below; original-publisher YouTube embeds supplement the complete written lesson. Reviewed 12 November 2026.

Parallelogram area.svg, en:User:Cburnett, CC BY-SA 3.0. Unchanged original; caption and learning activities are original.

Triangle (acute) bh.PNG, H Padleckas at English Wikibooks, Public domain. Unchanged original; caption and learning activities are original.

Cuboid.abc.png, Cuboid.png: en:User:Tomruen, CC BY-SA 4.0. Unchanged original; caption and learning activities are original.