Area of rectangle 10 by 6 with a semicircle cut out Step by Step

Step by step

The question

Question: find the area of this shape: a 10 by 6 rectangle with a semicircle cut out of one end.

1 The whole and the hole

The whole shape would be a 10 by 6 rectangle.

A semicircle is cut out of one end.

That part is not there any more.

So we take it away.

2 Area of each part

The whole rectangle: length times width.

10 times 6 is 60.

whole = 10 × 6 = 60

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 Take away the hole

The area is the whole, take away the hole: 60 − 4.5π.

We keep π in the answer.

That is the exact answer.

A = 60 − 4.5π

4 About how much

To see about how much, put 3.14 for π.

A ≈ 60 − 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 60 − 14.13.

A ≈ 60 − 14.13

60 minus 14.13 is 45.87.

So the area is about 45.87 square units.

A ≈ 45.87

5 Check

Let's check: put the semicircle back.

That is about 14.13.

We should get the whole rectangle.

45.87 + 14.13 = 60

We get 60, the whole rectangle.

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.

Answer