Worked examples
The question
Question: find the area of a 6 square with a quarter circle cut out of one corner.
1 The whole and the hole
The whole shape would be a 6 square.
A quarter circle is cut out of one corner.
That part is not there any more.
So we take it away.
Its center is the corner of the square.
Its radius is the side of the square: 6.
2 Area of each part
The whole square: side times side.
6 times 6 is 36.
whole = 6 × 6 = 36
The quarter circle is a quarter of a circle.
A circle is π times r squared.
So we take a quarter.
quarter = 14 × π × 62
Careful!
A quarter circle is one quarter of a circle, not one half.
Four of them make a whole circle.
Careful: not 12 × π × 62 ✗
6² means 6 times 6.
That is 36.
62 = 6 × 6 = 36
So it is a quarter of π times 36.
quarter = 14 × π × 36
A quarter of 36 means 36 divided by 4.
That is 9.
364 = 9
We write it as 9π.
quarter = 9π
3 Take away the hole
The area is the whole, take away the quarter circle: 36 − 9π.
We keep π in the answer.
That is the exact answer.
A = 36 − 9π
4 About how much
To see about how much, put 3.14 for π.
A ≈ 36 − 9 × 3.14
9 times 3.14 is 28.26.
A ≈ 36 − 28.26
36 minus 28.26 is 7.74.
So the area is about 7.74 square units.
A ≈ 7.74
5 Check
Let's check: put the quarter circle back.
That is about 28.26.
We should get the whole square.
7.74 + 28.26 = 36
We get 36, the whole square.
So the area is right!
Let's look back.
First: the whole and the hole.
Then: area of each part.
Then: take away the hole.
Then: about how much.
Last: check.
▶ Watch it solved
The question
Question: find the area of this shape: a 10 by 4 rectangle with a quarter circle on the end.
1 Split into parts
This shape is made of simple shapes.
A dashed line splits off the round end.
The rectangle is 10 by 4.
The round end is a quarter circle: a quarter of a circle.
Its center is the corner of the rectangle.
Its radius is the end of the rectangle: 4.
2 Area of each part
The rectangle: length times width.
10 times 4 is 40.
rectangle = 10 × 4 = 40
The quarter circle is a quarter of a circle.
A circle is π times r squared.
So we take a quarter.
quarter = 14 × π × 42
Careful!
A quarter circle is one quarter of a circle, not one half.
Four of them make a whole circle.
Careful: not 12 × π × 42 ✗
4² means 4 times 4.
That is 16.
42 = 4 × 4 = 16
So it is a quarter of π times 16.
quarter = 14 × π × 16
A quarter of 16 means 16 divided by 4.
That is 4.
164 = 4
We write it as 4π.
quarter = 4π
3 Add the parts
The whole area is the two parts added: 40 + 4π.
We keep π in the answer.
That is the exact answer.
A = 40 + 4π
4 About how much
To see about how much, put 3.14 for π.
A ≈ 40 + 4 × 3.14
4 times 3.14 is 12.56.
A ≈ 40 + 12.56
40 plus 12.56 is 52.56.
So the area is about 52.56 square units.
A ≈ 52.56
5 Check
Let's check it makes sense.
The whole shape fits in a box 14 by 4: 56.
Let's check it makes sense.
The area must be more than the rectangle alone: 40.
And less than the box: 56.
40 < 52.56 < 56
52.56 is between them.
So the answer makes sense!
Let's look back.
First: split into parts.
Then: area of each part.
Then: add the parts.
Then: about how much.
Last: check.
▶ Watch it solved
The question
Question: find the area of this shape: a 10 by 6 rectangle with a triangle on top, 4 high, and a semicircle on the end.
1 Split into parts
This shape is made of three simple shapes.
Dashed lines split them.
The rectangle is 10 by 6.
The triangle sits on top.
Its base is 10, its height 4.
The round end is a semicircle: half of a circle.
2 Area of each part
The rectangle: length times width.
10 times 6 is 60.
rectangle = 10 × 6 = 60
The triangle: half of base times height.
triangle = 12 × 10 × 4
10 times 4 is 40.
triangle = 12 × 40
Half of 40 means 40 divided by 2.
That is 20.
402 = 20
So the triangle is 20 square units.
triangle = 20
The semicircle sits on the end of the rectangle.
So its diameter is 6.
The radius is half of it: 6 divided by 2 is 3.
r = 62 = 3
The semicircle is half of a circle.
A circle is π times r squared.
So we take half.
semicircle = 12 × π × 32
3² means 3 times 3.
That is 9.
32 = 3 × 3 = 9
So it is half of π times 9.
semicircle = 12 × π × 9
Half of 9 is 4.5.
We write it as 4.5π.
semicircle = 4.5π
3 Add the parts
The whole area is the three parts added: 60 + 20 + 4.5π.
A = 60 + 20 + 4.5π
60 plus 20 is 80.
So the area is 80 + 4.5π.
We keep π in the answer.
That is the exact answer.
A = 80 + 4.5π
4 About how much
To see about how much, put 3.14 for π.
A ≈ 80 + 4.5 × 3.14
Forget the points: 45 times 314 is 14130.
Together, the numbers have 1 + 2 = 3 digits after the point.
So the answer has 3 too: 14.130, which is 14.13.
45 × 314 = 14130
1 + 2 = 3 digits after the point → 14.130
So we have 80 + 14.13.
A ≈ 80 + 14.13
80 plus 14.13 is 94.13.
So the area is about 94.13 square units.
A ≈ 94.13
5 Check
Let's check it makes sense.
The whole shape fits in a box 13 by 10: 130.
Let's check it makes sense.
The area must be more than the rectangle and the triangle: 80.
And less than the box: 130.
80 < 94.13 < 130
94.13 is between them.
So the answer makes sense!
Let's look back.
First: split into parts.
Then: area of each part.
Then: add the parts.
Then: about how much.
Last: check.
▶ Watch it solved
