Step by step
The question
Question: find the area of this shape: a 10 by 6 rectangle with a semicircle on one end.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Split into parts
This shape is made of simple shapes.
A dashed line splits off the round end.
The rectangle is 10 by 6.
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.
The semicircle sits on the end of the rectangle.
Its flat side is that end, 6.
So its diameter is 6.
The radius is half of it: 6 divided by 2 is 3.
The semicircle is half of a circle.
A circle is π times r squared.
So we take half.
3² means 3 times 3.
That is 9.
So it is half of π times 9.
Half of 9 is 4.5.
We write it as 4.5π.
3 Add the parts
The whole area is the two parts added: 60 + 4.5π.
We keep π in the answer.
That is the exact answer.
4 About how much
To see about how much, put 3.14 for π.
Forget the points: 45 times 314 is 14130.
4.5 is 45 tenths.
3.14 is 314 hundredths.
Tenths times hundredths make thousandths.
That is 3 digits after the point: 14.130, which is 14.13.
So we have 60 + 14.13.
60 plus 14.13 is 74.13.
So the area is about 74.13 square units.
5 Check
Let's check it makes sense.
Draw a box around the whole shape.
The round end sticks out by its radius, 3.
So the box is 10 + 3 = 13 long and 6 high.
13 times 6 is 78.
The shape holds the rectangle and fits inside the box.
So the area must be more than the rectangle alone: 60.
And less than the box: 78.
74.13 is between them.
So the answer makes sense!
6 Pick the choice
Our answer is A ≈ 74.13.
That is choice B.
7 Why not the others
A is 60.
Wrong: forgot the semicircle.
C is 88.26.
Wrong: added a whole circle: a semicircle is half.
D is 116.52.
Wrong: used 6 as the radius: 6 is the diameter.
Let's look back.
First: split into parts.
Then: area of each part.
Then: add the parts.
Then: about how much.
Then: check.
Then: pick the choice.
Last: why not the others.