Question

What is the best approximation for the area of this figure?

Answer 1

This area can be seen as a semicircle plus a triangle. So, we will find the area of each figure and then add both to get the total area.

The area of a triangle is given by the following formula:

`A_(triangle)=(1)/(2)b.h` (1)

Where
`b` is the base and
`h` is the height.

`A_(triangle)=(1)/(2)(3)(7)`

`A_(triangle)=10.5 units` (2)

Now we are going to find the area of the cemicircle, which is the half of the area of a circle:

`A_(cemicircle)=(1)/(2)\pi r^(2)` (3)

Where
`r` is the radius, in order to find it we have to calculate the diameter of this semicircle first, and we will do it as follows:

We know the points of the ends of the diameter, which are:

`P_(1):(-5,-2)` and
`P_(2):(2,1)`

We have to use the Pithagorean theorem to calculate the distance between both points (taking into account the x-component and the y-component of each one)

`c^(2) =a^(2)+ b^(2)`

`c=\sqrt{a^(2)+b^(2)}`

`c=\sqrt{(-5-2)^(2)+(-2-1)^(2)}`

`c=√(58)`>>>>This is the diameter of the semicircle

Then, the radius is:

`r=(c)/(2)=(√(58))/(2)`

Now we can use the formula written in equation (3):

`A_(cemicircle)=(1)/(2)\pi ((√(58))/(2))^(2)`

`A_(cemicircle)=7.25\pi units` (4)

Adding (2) and (4):

`A_(triangle)+A_(cemicircle)=10.5 units+7.25\pi units`

`A_(triangle)+A_(cemicircle)=10.5+7.25\pi units^(2)`>>>>This is the answer