Teaching unit · UK Year 8-9 (Key Stage 3, ages 12 to 14)

Area and perimeter: triangles, parallelograms, trapezia, circles and composite shapes

Deriving and applying the right formula for each 2D shape, including circles in terms of pi, and composite shapes built from rectangles

About three lessons of 45 to 60 minutes

Student view

Start here · hook

One shape, one formula: never guess, always derive

A rectangular lawn, a triangular garden bed, a circular pond and an L-shaped patio all need their area worked out before you can order turf, mulch or paving, but each shape needs a DIFFERENT formula. Guess wrong and you either run short mid-job or pay for material you never use.

This unit builds a formula for every common 2D shape (triangle, parallelogram, trapezium, circle), then shows how a composite shape, built from 2 or more simpler shapes, is worked out by carefully adding or subtracting the simpler areas you already know how to find.

A triangular garden bed: base 4 m, height 3 marea = 1/2 x 4 x 3 = 6 $m^{2}$
A circular pond with radius 3 marea = pi x $3^{2}$ = 9π ≈ 28.3 $m^{2}$
An L-shaped patio, a rectangle with a corner cut offarea = big rectangle - the missing corner
A parallelogram-shaped paving tilearea = base x height, exactly like a rectangle

Learning objective

What students will be able to do

Students will derive and apply the area formulas for triangles, parallelograms and trapezia, find the perimeter of triangles and parallelograms, find the circumference and area of a circle both in terms of pi and as a rounded decimal, and find the area and perimeter of composite shapes built from rectangles.

Success criteria

I can find the area of a triangle using 1/2 x base x height.
I can find the area of a parallelogram using base x height, and its perimeter using 2 x (a + b).
I can find the area of a trapezium using 1/2 x (a + b) x height.
I can find the circumference and area of a circle, in terms of pi and as a rounded decimal.
I can find the area and perimeter of a composite shape built from rectangles.

Curriculum anchor

Standards this unit teaches

KS3 Maths: Geometry and measuresUK National Curriculum (England)
Geometry and measures
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Geometry and measures" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)" (this unit builds the perimeter/area half; a separate unit, ukYear9VolumeOfPrisms in lib/teachingUnits_secondary.ts, builds the volume half); "calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes".

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Base and height: for a triangle or parallelogram, the base is any side, and the height is the PERPENDICULAR distance from that side to the opposite vertex or side, not a slanted side length
Trapezium: a quadrilateral with exactly 1 pair of parallel sides, usually labelled a and b, with a perpendicular height between them
Circumference: the distance around a circle, found with C = 2 x pi x r or C = pi x d
Composite (compound) shape: a shape made from 2 or more simpler shapes joined together or with a piece removed
Pi (π): the constant ratio of any circle's circumference to its diameter, an irrational number approximately 3.14159

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Area of triangles, parallelograms and trapezia

Concrete

Every one of these formulas comes from the rectangle you already know. A parallelogram is a rectangle 'pushed over': cut a triangle off one end and move it to the other, and the area (base x height) does not change. A triangle is exactly HALF of the parallelogram built on the same base and height. A trapezium is 2 triangles side by side, sharing the height, with different bases a and b.

The most important word in every one of these formulas is height, and it must always be the PERPENDICULAR distance, not a slanted side. A tall, leaning parallelogram and a short, upright one can have the very same area if their base and perpendicular height match.

A right-angled triangle makes the formula easiest to see: the 2 legs ARE the base and the perpendicular height directly, so area = 1/2 x 8 x 6 = 24 square units. The same formula works for any triangle, using the perpendicular height to the chosen base.

A rectangle is a parallelogram with right angles: its area (base x height) still comes from base x height, 7 x 4 = 28 square units, exactly the same formula that works for a leaning parallelogram.

Worked example

A triangle has a base of 10 cm and a height of 7 cm. A trapezium has parallel sides of 6 cm and 10 cm, and a height of 4 cm. Find both areas.

Triangle area = 1/2 x base x height = 0.5 x 10 x 7 = 35 cm².
Trapezium area = 1/2 x (a + b) x height = 0.5 x (6 + 10) x 4 = 0.5 x 16 x 4 = 32 cm².

Answer: The triangle has area 35 cm². The trapezium has area 32 cm².

Check for understanding, ask

A parallelogram has a base of 9 cm and a height of 5 cm. Find its area.
Why is the height of a triangle or parallelogram always a PERPENDICULAR distance, not a slanted side?

2. Circumference and area of circles

Pictorial

A circle is not built from straight edges, so it needs its own constant: pi (π), the ratio of any circle's circumference to its diameter, approximately 3.14159, and never-ending as a decimal. Circumference (the distance around) uses C = 2 x pi x r, and area (the space covered) uses A = pi x $r^{2}$ .

Answers can be given 2 honest ways: EXACTLY, as a whole number times pi (e.g. 6π cm), or APPROXIMATELY, as a rounded decimal from a calculator's real pi button (e.g. 18.8 cm to 1 decimal place). Neither is wrong, but mixing them up, rounding pi to '3' by hand, or forgetting units, all cause real errors.

Worked example

A circular pond has a radius of 5 m. Find its circumference and area, both in terms of pi and rounded to 1 decimal place.

Circumference = 2 x pi x r = 2 x pi x 5 = 10π m exactly, or approximately 31.4 m.
Area = pi x $r^{2}$ = pi x $5^{2}$ = pi x 25 = 25π $m^{2}$ exactly, or approximately 78.5 $m^{2}$ .

Answer: Circumference = 10π m ≈ 31.4 m. Area = 25π $m^{2}$ ≈ 78.5 $m^{2}$ .

Check for understanding, ask

A circle has a diameter of 14 cm. Find its circumference, in terms of pi and rounded to 1 decimal place.
Why is circumference measured in plain units (m) but area measured in square units ( $m^{2}$ )?

3. Composite (compound) shapes

Abstract

A composite shape is built from simpler shapes you already know how to handle. There are 2 common strategies: ADD the areas of 2 or more separate pieces, or SUBTRACT a missing piece from a larger whole shape.

For an L-shape cut from a rectangle's corner, the perimeter has a neat property worth noticing: it is always exactly 2 x (width + height) of the ORIGINAL rectangle, unchanged by the cut, because the 2 edges the cut removes are always replaced by 2 edges of the identical total length.

A composite shape split into 2 rectangles, 4 x 4 = 16 square units and 4 x 3 = 12 square units. Total area = 16 + 12 = 28 square units, whichever way the shape is cut.

Worked example

A composite shape is a rectangle 9 cm by 6 cm with a rectangular corner of 3 cm by 2 cm cut out. Find its area and perimeter.

Area = (whole rectangle) - (cut-out corner) = (9 x 6) - (3 x 2) = 54 - 6 = 48 cm².
Perimeter = 2 x (9 + 6) = 2 x 15 = 30 cm, the same as the original rectangle's perimeter.

Answer: Area = 48 cm². Perimeter = 30 cm.

Check for understanding, ask

A composite shape is made from 2 non-overlapping rectangles, 5 cm by 4 cm and 3 cm by 2 cm. Find its total area.
Why does cutting a rectangular corner out of a rectangle leave the perimeter unchanged?

Watch for

Common misconceptions and how to address them

MisconceptionThe height of a triangle or parallelogram is one of its slanted sides.

Why it happens: Students measure whichever side is drawn as a straight line next to the base, instead of the perpendicular distance to the opposite vertex or side.

How to address it: Trace the RIGHT ANGLE from the base up to the opposite vertex (or side) with a set square or by eye. Only that perpendicular distance is the height; a slanted side is longer and gives the wrong area if used instead.

MisconceptionCircumference and area use the same formula, just with different letters.

Why it happens: Both start with '2 x pi' or 'pi x', and students blur the 2 formulas together under exam pressure.

How to address it: Anchor each formula to its UNIT: circumference is a LENGTH (plain units, C = 2πr), area is a SPACE (square units, A = π $r^{2}$ ). If the answer needs a ^2, it must be the area formula.

MisconceptionPi should be rounded to 3 (or 3.14) before doing the calculation, always.

Why it happens: Early number work rounds early to keep arithmetic simple, so students apply the same habit to pi.

How to address it: Keep the exact form (a whole number times π) as the FIRST answer whenever asked for it; only round to a decimal, using a real pi button or 3.14159, at the very last step when a decimal answer is specifically requested.

MisconceptionA composite shape's perimeter is just the original rectangle's perimeter plus the notch's perimeter.

Why it happens: Students add every edge they can see, including the notch's own edges, rather than tracing only the OUTER boundary of the final shape.

How to address it: Trace the outline with a finger, one continuous loop, and only count each edge on that outer path once. For a simple rectangular corner cut, the outer path always keeps the same total length as the original rectangle.

Do it together

Guided practice (with answers)

1. A triangle has a base of 12 cm and a height of 5 cm. Find its area.
Answer: 30 cm², because 0.5 x 12 x 5 = 30.
2. A parallelogram has sides of 8 cm and 5 cm. Find its perimeter.
Answer: 26 cm, because 2 x (8 + 5) = 26.
3. A trapezium has parallel sides of 4 cm and 8 cm, and a height of 3 cm. Find its area.
Answer: 18 cm², because 0.5 x (4 + 8) x 3 = 0.5 x 12 x 3 = 18.
4. A circle has a radius of 4 cm. Find its area in terms of pi.
Answer: 16π cm², because pi x $4^{2}$ = 16π.
5. A circle has a radius of 10 cm. Find its circumference rounded to 1 decimal place.
Answer: 62.8 cm, because 2 x pi x 10 = 62.83..., which rounds to 62.8.
6. A composite shape is a rectangle 7 cm by 5 cm with a 2 cm by 2 cm corner cut out. Find its perimeter.
Answer: 24 cm, because 2 x (7 + 5) = 24, unchanged by the corner cut.

On their own

Independent practice worksheets

Practise area and perimeter with computed, never-wrong answer keys.

Year 8 area & perimeter of triangles, parallelograms and trapezia worksheetthe core formulas from section 1 Year 9 circumference & area of circles worksheetin terms of pi and as a rounded decimal, the core skill from section 2 Year 9 area & perimeter of composite shapes worksheetadding and subtracting rectangle areas, the core skill from section 3

Reach every student

Differentiation

Support

Provide a formula reference card (triangle, parallelogram, trapezium, circle) to keep visible until each formula is memorised.
Physically mark the perpendicular height with a set square before measuring, rather than trusting the eye.
For composite shapes, colour each simpler piece a different colour before calculating any area.
Practise the exact-pi form (e.g. 6π) before introducing the decimal-rounding step, so the 2 skills do not blur together.

Extension

Derive the area of a parallelogram from a rectangle by physically cutting and rearranging a paper shape, then explain the reasoning in words.
Find the area of a composite shape built from a rectangle AND a semicircle (half of pi x $r^{2}$ ), combining both skills from this unit.
Investigate how doubling a circle's radius affects its circumference (doubles) versus its area (quadruples), and explain why using the formulas.
Work backward from a given area to find a missing base or height, requiring a division rather than a multiplication.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling triangle/parallelogram area, circle area, and a composite shape.

1. A triangle has a base of 14 cm and a height of 6 cm. Find its area.
Answer: 42 cm², because 0.5 x 14 x 6 = 42.
2. A circle has a radius of 6 cm. Find its area in terms of pi.
Answer: 36π cm², because pi x $6^{2}$ = 36π.
3. A composite shape is a rectangle 10 cm by 8 cm with a 3 cm by 3 cm corner cut out. Find its area and perimeter.
Answer: Area = 71 cm² (80 - 9). Perimeter = 36 cm (2 x (10 + 8)), unchanged by the cut.

For the teacher

Teacher notes and timings

Rough timing across 3 lessons: Lesson 1 triangles/parallelograms/trapezia (section 1), Lesson 2 circles (section 2), Lesson 3 composite shapes plus the exit ticket (section 3).
Every circle answer in the matching worksheets is given BOTH exactly (a whole number times pi, built from the same numbers in the prompt) and as a genuine decimal from the real Math.PI constant rounded to 1 decimal place, never a hand-typed 3.14 approximation, matching this site's 'never wrong' answer-key policy.
The composite-shape perimeter fact in section 3 (cutting a rectangular corner never changes the perimeter) is a genuine, provable geometric result, not a coincidence of the chosen numbers: the 2 edges removed by the cut are always replaced by 2 edges of the same total length. It is worth proving on the board with a worked example before stating it as a rule.
Language to keep repeating: height means PERPENDICULAR distance, never a slanted side; circumference is a length, area is a space (square units); keep pi exact until a decimal is specifically asked for.
Present and print both work: build the array-split composite-shape figure live on the board with the class, then print the 3 linked worksheets for independent practice.

All teaching units Make a worksheet