A leaning shape has the same base and height as a rectangle. Does it hold less space?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Find the area of a parallelogram as base × height
Find the area of a triangle as half of base × height
A parallelogram is a four sided shape with two pairs of parallel sides. It looks like a rectangle that leans. Does it hold less space than a rectangle with the same base and height?
One side of the shape is its base. The height is the distance straight up from the base to the top side. It is measured at a right angle to the base. A dashed line shows it.
Draw it first
Turn a parallelogram into a rectangle
1 Draw a parallelogram. Its base is 6 cm.
2 Draw the height from the top corner straight down to the base. It is 4 cm. The slanted side is 5 cm, which is not the height.
3 Cut off the triangle at one end and slide it to the other end. You make a rectangle 6 cm by 4 cm.
The rectangle has the same area as the parallelogram, because nothing was added or lost. The rectangle is 6 cm long and 4 cm wide. So its area is 6 × 4 = 24 cm².
Pushing a frame over is different. Its sides keep their length, so the height falls and the space inside shrinks.
The area of a parallelogram is base × height. Use the height, not the slanted side. The slanted side is longer than the height, so it gives an answer that is too big.
Worked example
A parallelogram has a base of 8 cm and a height of 5 cm. What is its area?
The base is 8 cm. The height is the dashed line, 5 cm.
The area is base × height.
8 × 5 = 40 cm².
Answer: 40 cm²
The height is the dashed line, at a right angle to the base.
Now a triangle. Make a copy of any triangle and turn the copy half way round. Fit the two together along a side. They make a parallelogram.
Two copies of the triangle make a parallelogram with base 8 cm and height 5 cm.
The parallelogram is 8 × 5 = 40 cm². The triangle is only half of it. So the area of a triangle is half of base × height.
Worked example
A triangle has a base of 8 cm and a height of 5 cm. What is its area?
Two copies make a parallelogram with base 8 cm and height 5 cm.
That parallelogram has an area of 8 × 5 = 40 cm².
The triangle is half of it: 40 ÷ 2 = 20 cm².
Answer: 20 cm²
A right angled triangle has two sides at the right angle. These can be the base and the height. The third side is slanted, so you do not use it.
Worked example
A right angled triangle has sides of 6 cm and 8 cm at the right angle. The slanted side is 10 cm. What is its area?
The two sides at the right angle are the base and the height: 8 cm and 6 cm.
The slanted side, 10 cm, is not used.
Base × height = 8 × 6 = 48 cm².
The triangle is half of that: 48 ÷ 2 = 24 cm².
Check: a rectangle 8 cm by 6 cm has an area of 48 cm². The triangle is half of it, 24 cm².
Answer: 24 cm²
The base and the height are the two sides at the right angle.
See it three ways
With things
Cut a parallelogram from paper. Cut a triangle off one end and slide it to the other end.
As a picture
In symbols
Parallelogram: area = base × height. Triangle: area = half of base × height.
Try it
Two triangles make a parallelogram
You need: a sheet of paper, scissors, a pencil.
Draw a triangle of any shape on the paper. Cut it out.
Trace round it on a second sheet. Cut out the copy.
Turn the copy half way round, flat on the table. Do not flip it over. Fit one of its sides to the same side of the first triangle.
Look at the new shape.
What you will see: The two triangles fit together without a gap and make a parallelogram. Each triangle is half of it. This works for every triangle you try.
If you cannot try it, this is what happens.
A wrong idea many people have
To find the area of a parallelogram, multiply the base by the slanted side.
Why it feels right: The slanted side is a real side of the shape. It often has a length written beside it.
The test that breaks it: Cut the triangle off and slide it across. The rectangle is 6 cm by 4 cm, so the area is 24 cm². Using the slanted side gives 6 × 5 = 30, which is more squares than the shape holds.
What is true: Use the height, the dashed line that meets the base at a right angle.
Key words
parallelogram
a four sided shape with two pairs of parallel sides.
base
the side of a shape that you measure the height from.
height
the distance from the base straight up to the top, at a right angle to the base.
Back to your guess
The answer is A: No. It covers the same space, because the base and the height are the same.
Cut a triangle off one end of the leaning shape and slide it to the other end. You get the rectangle. Nothing was added or lost, so the area is the same.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
What is the area of this parallelogram?
See the answer
32 cm²
The area is base × height. The base is 8 cm and the height is 4 cm. So 8 × 4 = 32 cm².
If you chose A: You multiplied the base by the slanted side, 8 × 5. The slanted side is not the height. Use the dashed line. Go to M3.6.4.
If you chose B: You added the four sides, 8 + 5 + 8 + 5. That is the perimeter. The area is base × height. Go to M3.6.2.
A triangle has a base of 10 cm and a height of 6 cm. What is its area?
See the answer
30 cm²
Base × height = 10 × 6 = 60 cm². The triangle is half of that: 60 ÷ 2 = 30 cm².
If you chose A: You multiplied base × height and stopped. That is the area of a parallelogram. A triangle is half of it. Go to M3.6.4.
If you chose C: You added 10 + 6. Area comes from multiplying the base and the height, then halving. Go to M3.6.4.
This triangle has a right angle. Its sides at the right angle are 6 cm and 8 cm. The slanted side is 10 cm. What is its area?
See the answer
24 cm²
Use the two sides at the right angle: 6 × 8 = 48 cm². The triangle is half of that: 48 ÷ 2 = 24 cm².
If you chose A: You used the slanted side, 10, with 8 and halved. The base and height are the two sides at the right angle, 6 and 8. Go to M3.6.4.
If you chose B: You multiplied 6 × 8 and stopped. The triangle is half of that rectangle. Go to M3.6.4.
A parallelogram has a base of 9 cm and an area of 36 cm². How long is its height?
See the answer
4 cm
Base × height = 36 cm², so the height is 36 ÷ 9 = 4 cm. Check: 9 × 4 = 36 cm².
If you chose B: The 5 cm side is slanted. The height is the dashed line at a right angle to the base. Go to M3.6.4.
If you chose C: You took 9 from 36. Area is base × height, so go back with a division. Go to M3.6.4.
A rectangle is 6 cm by 4 cm. A parallelogram has a base of 6 cm and a height of 4 cm. Which has the larger area?
See the answer
They have the same area.
Both have a base of 6 cm and a height of 4 cm. Each has an area of base × height, so they are equal.
If you chose A: Cut a triangle off the parallelogram and slide it to the other end. You make a rectangle that is 6 cm by 4 cm. So the two are equal. Go to M3.6.4.
If you chose C: It looks longer because it leans. But its height is still 4 cm. The slanted sides do not add area. Go to M3.6.4.
A triangle has an area of 24 cm² and a base of 8 cm. How long is its height?
See the answer
6 cm
Two copies make a parallelogram of 2 × 24 = 48 cm². Its base is 8 cm, so the height is 48 ÷ 8 = 6 cm.
If you chose B: You divided 24 by 8. But 24 is only half of base × height, so 8 × height = 48 cm². Go to M3.6.4.
If you chose C: You halved 24. A triangle is half of base × height. So double the area first, to 48. Then divide by 8. Go to M3.6.4.
A triangle has a base of 10 cm and a height of 4 cm. What is its area?
See the answer
20 cm²
Base × height = 10 × 4 = 40 cm². The triangle is half of that: 40 ÷ 2 = 20 cm².
From an earlier chapter
A rectangle is 11 cm long and 6 cm wide. What is its area?
See the answer
66 cm²
The area of a rectangle is length × width: 11 × 6 = 66 cm².
Keep it
Copy this into your exercise book.
Height is at a right angle to the base, never the slanted side. Parallelogram: base × height. Triangle: half of base × height.
Draw: A parallelogram with a dashed height and a small square at the right angle. Beside it, a triangle with its height.
