Perimeter, circumference and area: shapes that relate back to a rectangle
Perimeter of 2D shapes, circumference of circles and the perimeter of half and quarter circles, and the area of parallelograms, trapeziums and kites
About three lessons of 45 to 60 minutes
Cut a corner off a rectangle: does the perimeter shrink?
It seems obvious that removing part of a shape should make its perimeter smaller. But cut a small rectangular notch out of one corner of a rectangle, and the total distance around the new L-shape is EXACTLY the same as the original rectangle's perimeter, not smaller. The two edges the notch removes are precisely replaced by two new edges of the same total length.
This unit is about spotting these kinds of relationships: a parallelogram's area is really just a rectangle's area in disguise (slide a triangle from one end to the other and it becomes a rectangle), and a semicircle's perimeter is half a circle's circumference PLUS a straight edge the full circle does not have. Every formula here connects back to a shape you already know.
- An L-shaped room, corner missingstill has the same perimeter as its full bounding rectangle
- A circle of radius 10 cmhas circumference 2 x pi x 10 = 20pi cm, about 62.8 cm
- A semicircle of radius 10 cmhas perimeter pi x 10 + 20, about 51.4 cm (arc plus the straight diameter)
- A parallelogram, base 12 cm, height 7 cmhas area 84 cm2, the same as a 12-by-7 rectangle
What students will be able to do
Students will find the perimeter of 2D shapes including regular polygons, kites and parallelograms (using equal-side properties) and L-shaped composite figures, find the circumference of a circle and the perimeter of a semicircle or quarter circle using pi ~ 3.14, and find the area of a parallelogram, trapezium and kite using formulae that relate back to the area of a rectangle.
- I can find the perimeter of a regular polygon, using its number of equal sides.
- I can find the perimeter of a kite or parallelogram, using its equal-side properties.
- I can find the perimeter of an L-shaped composite figure, using its overall dimensions.
- I can find the circumference of a circle, and the perimeter of a semicircle or quarter circle, using pi ~ 3.14.
- I can find the area of a parallelogram, a trapezium and a kite, using a formula related to the area of a rectangle.
Standards this unit teaches
- Phase 4 (Years 9-10): Measurement, MeasuringNew Zealand Curriculum (NZC), Mathematics and StatisticsMeasuring
Paraphrased (see the licence note in lib/content_nzsecondarymath2.ts) from the Ministry of Education's Tāhūrangi curriculum site, "NZC - Mathematics and Statistics Phase 4 (Years 9-10)" (official policy for all English-medium state and state-integrated schools from 1 January 2026), Measurement strand, "Measuring" section: during Year 9, students find the perimeter of 2D shapes, the circumference of circles, and the area of parallelograms, trapeziums and kites by relating each formula back to the area of a rectangle; they also derive and calculate the perimeter of half circles and quarter circles from the formula for a full circle. (Year 10 extends this to circle AREA, and to the surface area and volume of prisms, pyramids and cylinders, and is deliberately left out of this unit.) Source: https://newzealandcurriculum.tahurangi.education.govt.nz/nzc---mathematics-and-statistics-phase-4-years-9-10/5637291579.p (verified live 2026-07-14).
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 7 circles, area, volume and surface area teaching unitcircumference and area of a circle, the direct prerequisite for this unit's circle work
- Perimeter in the glossarya quick refresher on what perimeter means
- Circumference in the glossarythe distance around a circle
- Area in the glossarythe amount of surface a shape covers
Words to teach and display
- Regular polygon
- a shape with all sides and all angles equal, such as a regular pentagon or hexagon
- Composite figure
- a shape made by combining or cutting from simpler shapes, such as an L-shape made from a rectangle with a corner missing
- Semicircle
- half of a circle, cut along a diameter
- Trapezium
- a quadrilateral with exactly one pair of parallel sides
- Kite
- a quadrilateral with two pairs of equal adjacent sides (sides that meet at a vertex)
- Pi (pi)
- the constant found by dividing a circle's circumference by its diameter, approximately 3.14
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. Perimeter of 2D shapes
ConcreteA regular polygon's perimeter is simply the number of sides multiplied by the side length, because every side is the same length. A regular hexagon with side length 9 cm has perimeter 6 x 9 = 54 cm.
A kite has two PAIRS of equal adjacent sides (not opposite sides, like a parallelogram), but the perimeter formula ends up looking the same either way: perimeter = 2a + 2b, where a and b are the two different side lengths. The difference is in WHICH sides are equal, not in the arithmetic.
For an L-shaped composite figure, cutting a rectangular notch out of one corner does not change the total perimeter at all: the two edges the notch removes are exactly replaced by two new edges of the same total length. An L-shaped room 20 m by 15 m, with any size rectangular notch missing from one corner, still has perimeter 2(20 + 15) = 70 m, the same as the full rectangle.
A regular hexagon has side length 9 cm. A kite has two pairs of equal adjacent sides, 11 cm and 6 cm. Find both perimeters.
- Hexagon: 6 equal sides of 9 cm, so perimeter = 6 x 9 = 54 cm.
- Kite: perimeter = 2(11) + 2(6) = 22 + 12 = 34 cm.
Answer: The hexagon's perimeter is 54 cm, and the kite's perimeter is 34 cm.
- Why does a kite's perimeter formula (2a + 2b) look the same as a parallelogram's, even though the equal sides are in different positions?
- If a rectangular notch is cut from a corner of a 20 m by 15 m room, does the perimeter get smaller, stay the same, or get bigger?
2. Circumference of circles, half circles and quarter circles
PictorialA circle's circumference is found using C = 2 x pi x r, where r is the radius and pi is approximately 3.14. A circle with radius 10 cm has circumference 2 x pi x 10 = 20pi cm, about 62.8 cm.
A semicircle's perimeter is NOT simply half the circle's circumference: it also needs the straight diameter edge that closes the shape. Perimeter of a semicircle = (half the circumference) + (the diameter) = pi x r + 2r. A semicircle of radius 10 cm has perimeter pi x 10 + 20, about 31.4 + 20 = 51.4 cm.
A quarter circle's perimeter needs the curved arc (a quarter of the full circumference) PLUS two straight radii (not a diameter, since the two straight edges meet at the centre): perimeter = (pi x r) / 2 + 2r.
Find the circumference of a circle with radius 10 cm, and the perimeter of a semicircle with radius 10 cm. Use pi ~ 3.14.
- Circle: C = 2 x pi x r = 2 x pi x 10 = 20pi cm ~ 20 x 3.14 = 62.8 cm.
- Semicircle: arc = pi x r = pi x 10 ~ 31.4 cm; straight edge = 2 x 10 = 20 cm; perimeter ~ 31.4 + 20 = 51.4 cm.
Answer: The circle's circumference is 20pi cm, about 62.8 cm. The semicircle's perimeter is about 51.4 cm.
- Why is a semicircle's perimeter NOT just half of the full circle's circumference?
- What TWO straight edges does a quarter circle's perimeter need, that a semicircle's perimeter does not?
3. Area of parallelograms, trapeziums and kites
AbstractA parallelogram's area is base x height, exactly the same formula as a rectangle. Slide the triangle cut from one end of a parallelogram across to the other end, and it becomes a rectangle with the same base and height, so the area was always identical.
A trapezium (one pair of parallel sides, lengths a and b) has area = (1/2)(a + b)h, which is the AVERAGE of the two parallel sides multiplied by the height, matching a rectangle whose width is that average. A kite's area uses its two diagonals instead: area = (1/2) x d1 x d2, because the two diagonals form a rectangle exactly twice the kite's area.
All three formulae connect back to a rectangle: a parallelogram directly (base x height), a trapezium through an averaged width, and a kite through half of a surrounding rectangle formed by its diagonals.
Find the area of a parallelogram with base 12 cm and height 7 cm, and a trapezium with parallel sides 8 cm and 14 cm and height 6 cm.
- Parallelogram: A = base x height = 12 x 7 = 84 cm2.
- Trapezium: A = (1/2)(a + b)h = (1/2)(8 + 14) x 6 = (1/2)(22) x 6 = 11 x 6 = 66 cm2.
Answer: The parallelogram's area is 84 cm2, and the trapezium's area is 66 cm2.
- Why does a parallelogram's area formula end up being identical to a rectangle's (base x height)?
- In the trapezium formula (1/2)(a + b)h, what does averaging a and b represent?
Common misconceptions and how to address them
MisconceptionCutting a rectangular notch out of a corner of a rectangle makes the perimeter smaller.
Why it happens: Students assume removing area always removes perimeter too, without tracing the ACTUAL outline of the new L-shape, which gains two new edges exactly as long as the two edges it lost.
How to address it: Trace the full outline of the L-shape edge by edge and add up every side, or notice that the shape still spans the same overall width and height as the original rectangle, so its bounding perimeter (2 x width + 2 x height) is unchanged.
MisconceptionA semicircle's perimeter is exactly half of the full circle's circumference.
Why it happens: Students only account for the curved arc (which genuinely is half the circumference) and forget the straight diameter edge that closes the semicircle into a complete shape.
How to address it: Perimeter always means the FULL distance around the outside of a closed shape. A semicircle's outline has two parts: the curved arc (pi x r) AND the straight diameter (2r), both needed to close the shape.
MisconceptionThe area of a trapezium is found using only ONE of its parallel sides, the same way as a parallelogram.
Why it happens: Students over-apply the parallelogram formula (base x height) to a trapezium, without noticing a trapezium's two parallel sides are DIFFERENT lengths, so a single base does not describe it.
How to address it: A trapezium's area formula averages its two different parallel sides first, (1/2)(a + b), before multiplying by the height, because the trapezium's width effectively changes between its two parallel sides.
Guided practice (with answers)
1. Find the perimeter of a regular pentagon with side length 7 cm.
Answer: 35 cm, because 5 x 7 = 35.
2. Find the perimeter of a parallelogram with sides 9 cm and 5 cm.
Answer: 28 cm, because 2(9) + 2(5) = 18 + 10 = 28.
3. An L-shaped room has an overall length of 20 m and an overall width of 15 m, with a 6 m by 4 m section missing from one corner. Find its perimeter.
Answer: 70 m, because the perimeter equals the bounding rectangle's: 2(20 + 15) = 70.
4. Find the circumference of a circle with radius 5 cm. Use pi ~ 3.14.
Answer: 10pi cm ~ 31.4 cm, because 2 x pi x 5 = 10pi, and 10 x 3.14 = 31.4.
5. Find the area of a kite with diagonals 10 cm and 8 cm.
Answer: 40 cm2, because (1/2) x 10 x 8 = 40.
6. Find the area of a trapezium with parallel sides 5 cm and 9 cm, and height 4 cm.
Answer: 28 cm2, because (1/2)(5 + 9) x 4 = (1/2)(14) x 4 = 7 x 4 = 28.
Independent practice worksheets
Practise perimeter, circumference, and area, with computed, never-wrong answer keys.
Differentiation
- For regular polygons, write out the multiplication as 'number of sides x side length' every time, rather than trying to recall a different formula per shape name.
- For L-shapes, trace the outline with a finger or pencil, labelling every edge (including the ones that must be calculated), before adding.
- For circles, keep a reference card showing C = 2 x pi x r for a full circle, then the semicircle and quarter-circle perimeters built on top of it.
- For trapeziums, always compute (a + b) first as its own step, before multiplying by height and halving.
- Investigate: for an L-shape, does the size of the missing rectangular notch (as long as it stays within the overall dimensions) ever change the perimeter? Explain why or why not using the outline argument.
- Derive the semicircle and quarter-circle perimeter formulae from the full-circle circumference formula, starting from C = 2 x pi x r, the way the curriculum itself asks students to.
- Pose a composite problem: find the total perimeter of a shape made from a rectangle with a semicircle attached to one side (a rectangle top with a semicircular roof).
- Compare a kite's area formula ((1/2) x d1 x d2) to a rhombus's (a rhombus is a special kite): explain why the SAME formula works for both.
Assessment: exit ticket
A three-question exit ticket sampling a regular polygon, a quarter circle, and a parallelogram.
1. Find the perimeter of a regular octagon with side length 6 cm.
Answer: 48 cm, because 8 x 6 = 48.
2. Find the perimeter of a quarter circle with radius 8 cm. Use pi ~ 3.14.
Answer: About 28.56 cm, because the arc is (pi x 8) / 2 ~ 12.56 cm, and the two straight radii total 2 x 8 = 16 cm: 12.56 + 16 = 28.56.
3. Find the area of a parallelogram with base 15 cm and height 9 cm.
Answer: 135 cm2, because 15 x 9 = 135.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 perimeter of 2D shapes including L-shapes (section 1), Lesson 2 circumference and the perimeter of half/quarter circles (section 2), Lesson 3 area of parallelograms, trapeziums and kites, plus the exit ticket (section 3).
- This is the site's third teaching unit anchored to the New Zealand Curriculum (NZC). See the licence note at the top of lib/content_nzsecondarymath2.ts: the NZC's Tāhūrangi content is licensed CC BY-NC 4.0 (NonCommercial), so this unit paraphrases the curriculum rather than quoting it, with the source URL cited for attribution.
- Scope note: this unit deliberately covers Year 9's measurement practices (perimeter, circumference, half/quarter-circle perimeter, and the area of parallelograms/trapeziums/kites) and leaves Year 10's continuation (circle AREA, and surface area and volume of prisms, pyramids and cylinders) for a later unit, matching how the source page itself splits Year 9 from Year 10. This unit never computes a circle's area.
- Language to keep repeating: every formula here relates back to a rectangle in some way, base x height for a parallelogram, an averaged width for a trapezium, half a surrounding rectangle for a kite, so encourage students to explain the connection, not just recall the formula.
- Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.