Year 9MeasurementMeasuringYear 9 Maths

Circles, and every area formula as a rectangle

What this lesson covers

The constant π is found by dividing a circle’s circumference by its diameter.
For a circle of radius r, the circumference is 2πr.

Key idea

Two halves, and one idea holding them together

Circles first, then the area of some awkward shapes.
Every area formula here comes back to a rectangle.
So there is a lot less to remember than it looks.

Worked example

Pi is something you can measure

Take any circle. Measure around it, and measure across it.
Divide the first by the second. You get 3.14.
Do it with a different circle. Still 3.14. Every time.

Circle	Measurements	Around ÷ across
Circle A	around 31.4 cm, across 10 cm	3.14
Circle B	around 15.7 cm, across 5 cm	3.14
Any circle	around ÷ across	always 3.14...

Key idea

That number is pi, and it is a fact about circles

Nobody chose it. It is what circles do.
Pi is the circumference divided by the diameter.
Which means the circumference is pi times the diameter.

Key idea

Two ways to write the same formula

C = pi d if you were given the diameter.
The diameter is twice the radius, so C = 2 pi r.
Same formula. Use whichever matches what you were told.

Version	Formula	Why
Using the diameter	C = πd	because π is C ÷ d
Using the radius	C = 2πr	because d = 2r
They are the same	pick whichever you were given

Worked example

Two quick ones

Radius 5: two times pi times five, about 31.4 cm.
Diameter 12: pi times twelve, about 37.7 cm.
Watch which one you were given. That is the only trap.

Given	Working	Answer
r = 5 cm	2 × π × 5	about 31.4 cm
d = 12 cm	π × 12	about 37.7 cm

Worked example

Now walk around a semicircle

Radius 6. Start at one corner and go round the curve.
That is half the circumference, about 18.85 cm.
But you are not home yet. You have to walk back along the flat.

Step	Working	Distance
Radius 6 cm
Round the curve	half of 2πr, so π × 6	about 18.85 cm
Back along the flat	the diameter, 12 cm
Total perimeter	18.85 + 12	about 30.85 cm

Key idea

So it is not half the circumference

Half the circumference is 18.85, and that is only the curve.
Cutting the circle in half created an edge that was not there.
Perimeter means all the way round. Add the diameter.

Claim	Value	Verdict
Half the circumference	18.85 cm	only the curve
The cut made a new edge	the straight 12 cm
Perimeter means	all the way round	so both

Worked example

A quarter circle gets two straight edges

Radius 4. The curve is a quarter of the circumference, 6.28.
Then back along one radius, and up the other. Four each.
About 14.28 cm. Walk the outline and count what you cross.

Step	Working	Distance
Radius 4 cm
The curve	a quarter of 2πr	about 6.28 cm
Two straight edges	both radii, 4 cm each	8 cm
Total	6.28 + 8	about 14.28 cm

Key idea

Second half. Start with the easiest shape there is

A rectangle 8 by 3. Area is base times height. 24.
You have known that for years.
Every formula in the rest of this lesson comes back to it.

Worked example

A parallelogram is a rectangle, pushed over

Cut the triangle off one end and slide it to the other.
What you are left with is a rectangle. Same base, same height.
So the area is base times height. Nothing new.

Step	What happens	Result
Cut the triangle off one end	slide it to the other
You now have a rectangle	same base, same height
So the area	base × height	8 × 3 = 24 cm²
Careful	height means perpendicular height	not the slanted side

Key idea

One warning about that height

The height is the perpendicular height. Straight up.
Not the slanted side, which is longer.
Using the slant is the most common error in the topic.

Worked example

A trapezium is a rectangle whose width is the average

Parallel sides of 10 and 6. It is wide at one end, narrow at the other.
On average, it is 8 wide. And the height is 4.
So the area is 8 times 4, which is 32.

Step	Reasoning	Result
Parallel sides 10 and 6	the average is 8
Height 4
Area	8 × 4	32 cm²
Written as a formula	(a + b)/2 × h	the average, times the height

Worked example

And a kite fills half its rectangle

Draw the rectangle that just encloses the kite.
Its sides are the two diagonals: 10 and 6, so 60.
The kite is exactly half of it. Thirty.

Step	Reasoning	Result
Diagonals 10 and 6	draw the rectangle round it	60 cm²
The kite fills	exactly half of it
Area	(10 × 6)/2	30 cm²

Worked example

And when there is no formula at all

An L-shaped room is not any of these shapes.
For perimeter, it does not matter. Add up every side.
Perimeter never needs a formula. It is just the walk.

Shape	Method	Answer
An L-shaped room	sides 10, 4, 6, 3, 4, 7
Perimeter	add every side	34 m
No formula needed	just walk all the way round

Summary

What to take away

Seven things, and the third one is the mark people drop.


π	is circumference divided by diameter, for every circle
Circumference	is πd, which is the same as 2πr
A semicircle perimeter	is the curve plus the diameter
A quarter circle	is the curve plus two radii
Parallelogram	base × perpendicular height, like a rectangle
Trapezium	the average of the parallel sides, times the height
Kite	half of the rectangle that encloses it

Your turn

Pause here and try all four.
Answers are in the description.

	Question
1	Find the circumference of a circle with radius 7 cm.
2	Find the perimeter of a semicircle with radius 5 cm.
3	Trapezium: parallel sides 9 and 5, height 6. Area?
4	A parallelogram has base 12 cm and height 5 cm. Area?

Show answer

1. About 44.0 cm, from 2 × π × 7. 2. About 25.7 cm. The curve is π × 5, which is about 15.7, and then add the diameter of 10. If you answered 15.7 you forgot the straight edge. 3. 42 cm². The average of 9 and 5 is 7, and 7 × 6 is 42. 4. 60 cm². Base times perpendicular height, exactly like a rectangle.

Worked example

The answers

Two pi r, with r as seven.
On the semicircle, add the diameter.
Average 9 and 5 to get 7, then times 6.

	Answer
1	About 44.0 cm
2	About 25.7 cm
3	42 cm²
4	60 cm²

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