Step by step
The question
Question: find the area of this shape: a 12 cm by 6 cm rectangle with a triangle on top, 5 cm 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 12 by 6.
The triangle sits on top.
Its base is 12 cm, its height 5 cm.
The round end is a semicircle: half of a circle.
2 Area of each part
The rectangle: length times width.
12 times 6 is 72.
The triangle: half of base times height.
12 times 5 is 60.
Half of 60 means 60 divided by 2.
That is 30.
So the triangle is 30 cm².
The semicircle sits on the end of the rectangle.
So its diameter is 6 cm.
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 three parts added: 72 + 30 + 4.5π.
72 plus 30 is 102.
So the area is 102 + 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.
Together, the numbers have 1 + 2 = 3 digits after the point.
So the answer has 3 too: 14.130, which is 14.13.
So we have 102 + 14.13.
102 plus 14.13 is 116.13.
So the area is about 116.13 cm².
5 Check
Let's check it makes sense.
The whole shape fits in a box 15 by 11: 165.
Let's check it makes sense.
The area must be more than the rectangle and the triangle: 102.
And less than the box: 165.
116.13 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.