Half a rectangle

A flag is a triangle cut from a rectangle of cloth. How much of the rectangle does it use?

About 40 minutes
Class 6
Mathematics and money

What you will learn: Find the area of a triangle as half the rectangle around it, and of a parallelogram as a rectangle with one end moved, using the height at a right angle to the base.

What should I know first?

If this feels new, try these first. It is fine to go back.

The 12-tile garden Class 5
Fence or grass? Class 6

Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓

Jump to: Check yourself · For grown-ups · For teachers

Safety first

Use child-safe scissors, and keep them closed when you pass them.

Keep it safe:

Use child-safe scissors and cut away from your fingers. Keep scissors closed when you pass them.
Keep paper scraps, counters and pins away from babies: nothing goes in the mouth.

You will need

Squared paper and a pencilEach square can stand for 1 cm².
Child-safe scissorsTo cut a rectangle into two triangles.
A ruler or a set squareTo draw the height at a right angle to the base.

Time: About 40 minutes.

What do you think?

Dhruv says: "A triangle with base 20 cm and height 30 cm covers 20 × 30 = 600 cm²." Is he right?

Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.

Yes, 600 cm²

Let's test it. Draw the triangle inside a 20 by 30 rectangle on squared paper. Does the triangle fill the whole rectangle?

No, less than that

How much less? Cut the rectangle and see whether the left-over pieces make a second triangle.

It depends on the shape

Try a tall thin triangle and a leaning one with the same base and height. Count their squares.

Not sure

That is fine. Draw the rectangle first, then the triangle inside it, and compare what is left over.

Need a hint?

Draw the rectangle around the triangle, with the same base and the same height.

The story

Dhruv is cutting triangle flags for the school fair from rectangles of cloth 20 cm wide and 30 cm tall. "Each flag is 20 by 30, so 600 square centimetres of cloth," he says. Tara holds a flag up against its rectangle. "Look at the two pieces left over. Put them together and they make another flag." Tanmay frowns. "So is a flag 600 or not?"

The idea

Draw a rectangle around a triangle, using the triangle's base and its height. The triangle is exactly half of that rectangle.

So the area of a triangle is ½ × base × height. The two pieces left over always fit together to make a second copy of the triangle.

The height is the distance straight up from the base to the top point, at a right angle to the base. It is often not one of the sides.

A parallelogram is a leaning rectangle. Cut a triangle off one end and slide it to the other end: you get a rectangle with the same base and height. So its area is base × height.

Squared paper with a dashed rectangle 8 squares wide and 5 squares high. Inside it, a green triangle has the full 8-square base along the bottom and its top point on the top edge, 3 squares from the left. A red dashed line goes straight down from the top point: the height. The two white pieces left over together make a second triangle.

Formula cards

From counting to a formula

Area of a triangle

What it finds: The flat space inside a triangle.
The formula: Area = ½ × base × height
Why it works: Draw the rectangle around the triangle, with the same base and height. The triangle fills exactly half of it: the two left-over pieces make a second triangle.
When it works: For any triangle, as long as the height is measured at a right angle to the base. A slanted side is not the height.
Units: Square units, as for any area: cm² or m². Base and height must be in the same unit.
Use it backwards: If you know the area and the base: height = 2 × area ÷ base. A 20 cm² triangle with base 5 cm is 2 × 20 ÷ 5 = 8 cm high.

Area of a parallelogram

What it finds: The flat space inside a parallelogram (a leaning rectangle).
The formula: Area = base × height
Why it works: Cut the triangle off one end and slide it to the other end. The shape becomes a rectangle with the same base and height, and no space is lost.
When it works: Use the height at a right angle to the base, not the slanted side.
Units: Square units: cm² or m².
Use it backwards: A parallelogram of 35 cm² with base 7 cm has a height of 35 ÷ 7 = 5 cm.

Worked example

Cloth for one flag

A flag is a triangle with base 20 cm and height 30 cm. How much cloth does it use?

The rectangle around the flag is 20 cm by 30 cm: 20 × 30 = 600 cm².
The flag is half of that rectangle: 600 ÷ 2 = 300 cm².
Or in one go: ½ × 20 × 30 = 300 cm².
Check: the two pieces left over make a second flag, so one rectangle of cloth gives two flags.

One flag uses 300 cm² of cloth, half of Dhruv's 600.

A green triangle flag inside a dashed rectangle 4 squares wide and 6 squares high, each square 5 cm. Its top point is in the middle of the top edge, with the height drawn straight down.

Try it

Draw a rectangle.
On squared paper, draw a rectangle 8 squares wide and 6 squares high. Count its squares.
Cut it corner to corner.
Cut along a diagonal. Put one triangle on top of the other. Are they the same? What is the area of one?
Move the top point.
Draw a new triangle on the same 8-square base, with its top point 3 squares from the left on the top edge. Draw the height straight down.
Need a hint?
The height splits it into two right-angled triangles. Each one is half of a smaller rectangle. Add the two halves.
Lean it over.
Draw a parallelogram with base 8 squares and height 6 squares, leaning to the right. Cut the triangle off its left end and slide it to the right end.
Find the slanted side.
Measure the slanted side of your parallelogram with a ruler. Is it longer than the height? Which one does the area use?
Explain it.
Write one sentence on why a triangle is half of its rectangle, and one on why the slanted side is not the height.

What you make (paper or offline)

Your flag plan

A triangle flag inside a dashed rectangle 4 squares wide and 6 squares high, leaning to the left.

Design a flag with base 4 squares and height 6 squares on squared paper.

Work out its area. Then lean the top point to a new place. Does the area change?

How many flags can you cut from one rectangle of cloth 40 cm by 30 cm, if each flag has base 20 cm and height 30 cm?

Make it on paper. Keep it with you. Nothing to upload.

What did we find out?

Read this after you have tried it. Look back at your guess and your reason first.

A triangle is exactly half of the rectangle around it, with the same base and height: area = ½ × base × height.

The height is measured straight up from the base, at a right angle. A slanted side is not the height.

A parallelogram becomes a rectangle when you cut a triangle off one end and slide it to the other: area = base × height.

Think back

Why is the slanted side of a triangle not its height?

Why does a parallelogram not need the "½" that a triangle needs?

Investigate further

Find the area of a kite-shaped piece of paper by splitting it into two triangles.
Draw three different triangles with an area of 12 cm² on squared paper.

Can you use it here?

Find a triangle at home (a samosa, a sandwich cut corner to corner, a road sign). Measure its base and height and work out its area.

Ask someone at home

Ask someone at home: when a sandwich is cut corner to corner, how much of the sandwich is each half? Can you prove it?

No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.

For grown-upsTap a heading to open it. The safety notes are at the top of the page.

Before you start

Squared paper, a ruler and child-safe scissors.
Keep scraps, counters and pins away from babies: nothing goes in the mouth.

Questions that help

"Which rectangle goes round this triangle?"
"Is that line the height, or a slanted side?"
"What happens to the area if the top point moves along?"

What's really going on

Many children multiply base × height and forget the half. Cutting a rectangle in two shows why the half is there.

Children often use the slanted side as the height. Drawing the height with a set square fixes this.

Moving the top point while keeping the base and height shows that the area does not change.

Other ways to do it

No scissors? Fold a paper rectangle corner to corner and compare the halves.
Use a sandwich or a roti cut corner to corner.

For teachersLearning aims, classroom ideas and what to look for.

Learning aims

Find the area of a triangle as ½ × base × height and of a parallelogram as base × height.
Identify the perpendicular height and tell it apart from a slanted side.
Work backwards from an area to a missing height.

Subject links

Builds on "The 12-tile garden" (Class 5) and "Fence or grass?" (Class 6).
Leads to Heron's formula (Class 9) and areas of compound shapes.
Topic ladder 3.3 (area and volume).

These links are general. They have not been checked against a specific board syllabus.

In the classroom

Each pair cuts a paper rectangle into two triangles and proves they are equal.
Pin up triangles with the same base and height but different shapes; ask which is biggest, then count squares.

What to look for

Draws the height at a right angle without being asked.
Explains the ½ by pointing to the rectangle.
Uses cm² or m² for every area.