016i - Area of Parallelograms

Lesson 16 — Area of Parallelograms

Strand: Measurement | Descriptor: AC9M7M01 | Duration: 45 minutes

Learning Intentions

• To understand why the area of a parallelogram equals base times perpendicular height.
• To calculate areas of parallelograms using the established formula and appropriate units.

Success Criteria

I can:

1. Explain, using a cut-and-rearrange argument, why for a parallelogram.
2. Identify the perpendicular height corresponding to a chosen base.
3. Calculate the area of a parallelogram, including when a distractor slant length is given.
4. Find a missing base or height from a given area.

Warmup

(6 minutes — retrieval and physical demonstration)

Retrieval (3 min). Find the area of each triangle:

1. cm, cm
2. m, m
3. , cm — find

(Answers: ; ; cm.)

Demonstration (3 min). Hold up a paper parallelogram. Ask: is this a rectangle? What is different? Could we make it a rectangle without changing its area?

Activities

Activity 1 — Deriving the Formula by Dissection (12 min)

Every student needs a paper parallelogram and scissors.

The cut-and-slide argument:

1. Draw the perpendicular height from one vertex down to the base.
2. Cut along that line, removing a right-angled triangle.
3. Slide the triangle across to the other end.
4. The shape is now a rectangle with the same base and the same height — and, crucially, the same area, because nothing was added or removed.

Emphasise: is the perpendicular height, not the slanted side. This is the same trap as in Lesson 15.

I do: A parallelogram has base cm, perpendicular height cm, and slant side cm.

The cm is a distractor — it plays no part.

We do: m, m; cm, cm; cm, cm (slant cm given).

You do: Eight parallelograms, at least three with distractor slant lengths and one requiring unit conversion.

Activity 2 — Connecting Triangle and Parallelogram (8 min)

Explicit link, using two congruent triangles.

Two congruent triangles placed together edge to edge form a parallelogram. Therefore:

This is the same formula from Lesson 15, now seen from the other direction. Emphasise that these are one idea, not two separate facts to memorise:

Shape	Formula
Rectangle
Parallelogram
Triangle

Quick practice — mixed:

1. Parallelogram, cm, cm.
2. Triangle, cm, cm.
3. Rectangle, cm by cm.

(Answers: , , . Ask why two of them match.)

Activity 3 — Inquiry: the Shearing Parallelogram (10 min)

Pairs, dynamic geometry software or a linkage made from strips and split pins.

Build a parallelogram with base cm and height cm. Now “push” the top edge sideways, keeping the base fixed and the top edge on the same horizontal line.

1. What happens to the area? Explain.
2. What happens to the perimeter?
3. What happens to the slant side length?
4. How far can you push before the shape stops being useful?

Socratic scaffolding:

Prompt	Purpose
Understand: what stays fixed?	The base ( cm) and the perpendicular height ( cm).
Which of these appear in the area formula?	Both — and nothing else does.
So what happens to the area?	It stays at .
What is changing, then?	The slant sides get longer as the shape leans further.
Consequence for perimeter?	Perimeter grows without limit, while area stays constant.
Looking back — connect to Lesson 15.	This is the same phenomenon as sliding a triangle’s apex. Area depends on base and perpendicular height only.
Extend	For a fixed base and height, which parallelogram has the smallest perimeter? (The rectangle — no lean at all.)

Checks for Understanding

(5 minutes — exit ticket)

1. Find the area of a parallelogram with base cm and perpendicular height cm.
2. A parallelogram has area and base m. Find its height.
3. A parallelogram has base cm, perpendicular height cm and slant side cm. Find its area.
4. Reasoning. Explain, using the cut-and-slide argument, why a parallelogram and a rectangle with the same base and height have equal areas.
5. A triangle and a parallelogram both have base cm and height cm. How do their areas compare?

Answers: 1. ; 2. m; 3. (the cm is a distractor); 4. Cutting a right-angled triangle from one end and sliding it to the other converts the parallelogram into a rectangle with identical base and height, and rearranging does not change area; 5. The parallelogram is , twice the triangle’s .

Common Misconceptions

Misconception	How to pre-empt it
Using the slant side as the height.	Include a distractor slant length in most practice questions, and require students to mark the right-angle symbol before calculating.
Halving the parallelogram’s area (over-applying the triangle formula).	Use the two-congruent-triangles image: a parallelogram is two triangles, so no halving.
Believing a “leaning” parallelogram has less area than a rectangle of the same base and height.	Activity 3 confronts this directly.
Multiplying two adjacent side lengths, as though it were a rectangle.	Ask: which of these two numbers is perpendicular to the base? Only that one is the height.
Forgetting squared units.	Mark units in every model answer and require them on exit tickets.
Using the wrong height when the base is switched.	Each base has its own perpendicular height. Show a parallelogram labelled both ways, and confirm both give the same area.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A parallelogram has area . A second parallelogram has twice the base and three times the height. What is its area?

Answer

E2 (AMC Junior style). A parallelogram has base cm and area . Taking the adjacent side of cm as the base instead, what is the corresponding perpendicular height?

Answer

The area is unchanged, so:

E3 (Challenge). A parallelogram has whole-number base and height, and area . How many different (base, height) pairs are possible?

Answer

We need . The divisors of are — 9 ordered pairs.

E4 (Challenge). A rectangle measures cm by cm. A parallelogram is drawn inside it with the same base ( cm) and its top edge lying along the rectangle’s top edge. What fraction of the rectangle does the parallelogram occupy?

Answer

The parallelogram has base cm and perpendicular height cm (the rectangle’s width), so its area is — the whole rectangle. It occupies the full area, though the two shapes are not congruent, because parts overhang and parts are left empty in equal measure.

Homework

1. Find the area: (a) cm, cm (b) m, m (c) cm, cm (d) cm, cm.
2. Find the missing dimension: (a) , cm (b) , m (c) , cm.
3. A parallelogram has base cm, perpendicular height cm and slant side cm. Find its area and explain which measurement you did not use, and why.
4. A parallelogram-shaped paving stone has base cm and height cm. How many stones are needed to cover ? (Convert carefully.)
5. Find the total area of a shape made from one parallelogram ( cm, cm) with a triangle ( cm, cm) sitting on top of it.
6. Reasoning. A student says “a parallelogram with a longer perimeter must have a bigger area.” Give a counterexample.
7. Challenge. A parallelogram has area . Its perpendicular height is of its base. Find both dimensions.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) cm (b) m (c) cm. Q3 — ; the cm slant is not perpendicular to the base. Q4 — each stone ; ; stones. Q5 — . Q7 — , so , giving cm and cm.