I need help solving these math problems, I have no idea how to do them:
the circle one I need the are aof the shaded figure
the p-gram one, i need the area
the pentagon, I need the apothem, area, and side length
The area of the shaded regions in the circle.
The sides of the triangle with hypoteneuse 6 inches are the same, They are the radius of the circle:
$\displaystyle 6=\sqrt{a^{2}+a^{2}}$
$\displaystyle a=3\sqrt{2}$
The subtended angle is 90 degrees.
Area of circular segment:
$\displaystyle \frac{1}{2}(3\sqrt{2})^{2}(\frac{\pi}{2}-sin(\frac{\pi}{2}))$
That's one of them. Multiply by 2.
For the rhombus, find the area of one of the right triangles and multiply by 4.
For the circle problem, you need to compute the radius first, which is $\displaystyle 3\sqrt{2}$. Then, the area of the semicircle is $\displaystyle \frac{1}{2}\pi r^2$. We subtract out the area of the triangle to obtain the area of the shaded region: $\displaystyle \frac{1}{2}\pi (3\sqrt{2})^2 - (3\sqrt{2})^2 = 9\pi - 18$.
A general desire for the underyling is never hurtful
usign my formula $\displaystyle \frac{5}{4}\cdot{b^2}\cdot\cot\bigg(\frac{\pi}{5}\ bigg)=\frac{10(\sqrt{5}+1)\sqrt{2}x^2}{\sqrt{5-\sqrt{5}}}\approx{27.52b^2}$
So now apply it
plug in your b value
and use the fact that
$\displaystyle A=\frac{1}{2}p\cdot{a}$
to find the apothem
Of course this is a site to learn
Ok so basically we have that $\displaystyle A=\frac{1}{2}a\cdot{p}$
where a is apothem and p is permieter
Now since we know that the exterior angle of a n-gon is $\displaystyle \frac{360}{n}$ we know due to the linear pair postulate that the interior angle of an n-gon is
$\displaystyle 180-\frac{360}{n}$
so for a pentagon(5-gon)
the interior angle would be
$\displaystyle 180-\frac{360}{5}=108$
So now we know that the angle at a vertex is 108 we know that the apothem bisects it making
the angle that the apothem cuts off 44...so now we have a right triangle with angles ,90,44,46
So we need to calculate the apothem or in this case we are given it but we need to find sidelenght
so we use trig $\displaystyle \sin(46)=\frac{x}{15}\Rightarrow{x=\sin(46)\cdot{1 5}\approx{10.875}}$
and since that gives us half of our side we see that the sidelenghts are
21.75
now we go to our formula
$\displaystyle A=\frac{1}{2}a\cdot{p}$
Now since $\displaystyle P_{pentagon}=5n$ where n is the sidelength we see that $\displaystyle A=5\cdot{21.75}=107.8$
so now we see
$\displaystyle A=\frac{1}{2}\cdot{107.8}\cdot{15}=808.75$