Cut a triangle off one end of a parallelogram and move it to the other end: it becomes a rectangle with the same base and height. So the area of a parallelogram is base × perpendicular height.
Tiles, bricks in a wall, and the stripes of a pedestrian crossing seen from an angle are parallelograms. How much area does a slanted tile cover?
The area of a shape is how many unit squares it covers. A rectangle 6 cm long and 4 cm high has 4 rows of 6 squares: 6×4=24 square centimetres, written 24 cm².
A parallelogram has two pairs of parallel sides, like a rectangle pushed over. Its base is one side, and its perpendicular height is the distance straight up from the base to the opposite side, at a right angle (marked ∟). The slanted side is longer than the height.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Pushed over
Picture the parallelogram as the rectangle pushed over. Does anything get added or taken away?
TryTry it on the board and compare the result with your prediction.
Cut and move
The dashed orange line is the perpendicular height. The cut goes along it.
Try it yourselfYour edits are saved in this browser.
Press **Cut and move**. What shape do you get, and what is its area? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Base times height
Here is why the formula works, then how to use it.
Worked exampleBefore each step, say what you think comes next.
A=6×4=24 cm2A rectangle: 4 rows of 6 unit squares.
parallelogram→rectangleCut along the height and move the triangle: the same base and height.
A=b×hSo a parallelogram's area is base × perpendicular height.
A=8×5=40 cm2Use the height at a right angle to the base, never the slanted side.
Now try one: base 8 cm, height 5 cm.
Read Ruby's working. Which length is at a right angle to the base?
A parallelogram has base 8 cm and perpendicular height 5 cm. What is its area, in cm²?
Ruby finds the area of a parallelogram. Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Move the top until it sits straight above the left end of the base.
Need a hint?
Which number times 8 makes 24?
Another hint
Set b to 8 first.
Try it yourselfYour edits are saved in this browser.
Slide the top until the parallelogram is a rectangle. Did the area change?
Use the board. This ticks when you get there.
Make a parallelogram with base 8 cm and area 24 cm².
Use the board. This ticks when you get there.
A parallelogram has base 12 m and perpendicular height 3.5 m. What is its area, in m²?
A parallelogram has area 30 cm² and base 6 cm. What is its perpendicular height, in cm?
PractiseA few on your own, to check you have it.
Parallelograms
Base 9 cm, perpendicular height 7 cm. Area, in cm²?
A parallelogram has base 10 cm, slanted side 6 cm and perpendicular height 4.5 cm. Area, in cm²?
Review: work out 4.5×6.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Area 18, slanted
Then find a different base and height for the same area.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Which pairs of numbers multiply to 18?
Another hint
Then move the top so it is not a rectangle.
Make a parallelogram that is not a rectangle, with area 18 cm².
Use the board. This ticks when you get there.
For teachers
Students cut a parallelogram along its height and move the triangle to make a rectangle, see that sliding the top keeps the area, and use A = bh with the perpendicular height (AC9M7M01, node 1). About 40 minutes.
I can explain why a parallelogram has the same area as a rectangle with the same base and height.
I can find the area of a parallelogram with base × perpendicular height.
I can tell the perpendicular height from a slanted side.
Before class
Do the Run board yourself first. Students need area of rectangles by counting rows of squares (Year 6) and the meaning of cm².
The one idea: a parallelogram rearranges into a rectangle with the same base and height, so A = bh. The common mistake is using the slanted side (Ruby's mistake); the board's dashed height with its right-angle mark is the fix.
Running the lesson
Predict: take a show of hands. Students who chose the parallelogram think longer sides mean more area.
Run: ask where the cut went, and why there.
Investigate: have students point to the height on the bug task's picture in their head: which length meets the base at a right angle?
Modify: for the missing height, ask what operation undoes × 6.
Make: collect different base and height pairs for 18.
The hard part (on purpose)
Choosing the perpendicular height when a slanted side is also labelled. Ask students to trace the right-angle mark with a finger before they multiply, rather than telling them which number to use.
When students get stuck
A student multiplies by the slanted side (Ruby's mistake). Ask: Which length meets the base at a right angle? Fix: Use the perpendicular height: the dashed line with the ∟.
A student adds base and height. Ask: How many rows of squares, and how many in each row? Fix: Area is rows × squares in a row: multiply.
A student writes cm for an area. Ask: Are we counting lengths or squares? Fix: Squares: cm².
Stretch and support
Stretch: Draw three different parallelograms on squared paper, each with area 24 cm², none of them a rectangle.
Support: Cut paper parallelograms and rearrange them; count squares on the board for small cases.
Where it leads
Next: Area of triangles (node 1), half of a parallelogram.
Where it leads: area of trapeziums and composite shapes (Year 8), and volume of prisms (the base times the height again).
What each step is for
pushed over: Committing to which shape covers more surfaces the belief that the slanted sides, being longer, add area.
cut and move: Cutting off the triangle and moving it shows the parallelogram rearranges into a rectangle with the same base and height.
base times height: Area = base × perpendicular height; Ruby's mistake uses the slanted side, which is longer than the height.
one change at a time: One change at a time: sliding the top to make a rectangle, building an area from a base, a decimal height, finding a missing height (backwards), then the edge case of a very slanted parallelogram.
parallelograms: Whole and decimal bases, units, and a slanted side to ignore, mixed; the review keeps decimal multiplication fresh.
area 18, slanted: Building a slanted parallelogram with a given area means choosing a base and height that multiply to it, and seeing the slant does not matter.
Now you can
Explain why a parallelogram has the same area as a rectangle with the same base and height.
Find the area of a parallelogram with base × perpendicular height.
Tell the perpendicular height from a slanted side.