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  1. #1
    Newbie Zyger's Avatar
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    Exclamation geometry help

    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
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  2. #2
    Eater of Worlds
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    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.
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  3. #3
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    Quote Originally Posted by Zyger View Post
    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
    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: $\displaystyle \frac{1}{2}\pi (3\sqrt{2})^2 - (3\sqrt{2})^2 = 9\pi - 18$.
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  4. #4
    Newbie Zyger's Avatar
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    may please have some more help on the others?
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Zyger View Post
    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
    A useful formula to know is that

    For a regular n-gon of sidelength b

    $\displaystyle A_{n-gon}=\frac{1}{4}nb^2\cot\bigg(\frac{\pi}{n}\bigg)$
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  6. #6
    Newbie Zyger's Avatar
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    Quote Originally Posted by Zyger View Post
    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


    All I need now is the one with the pentagon
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Zyger View Post
    All I need now is the one with the pentagon
    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
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  8. #8
    Newbie Zyger's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    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
    I am so sorry, but that math looks like a foreign language to me. Can you go a little slower with it please?
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  9. #9
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Zyger View Post
    I am so sorry, but that math looks like a foreign language to me. Can you go a little slower with it please?
    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

    $\displaystyle A=\frac{1}{2}\cdot{107.8}\cdot{15}=808.75$
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  10. #10
    Newbie Zyger's Avatar
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    SORRY POR MY POST, i HAVE TO EDIT NOW
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  11. #11
    Newbie Zyger's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    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

    $\displaystyle A=\frac{1}{2}\cdot{107.8}\cdot{15}=808.75$

    Thank you sooooooooooooooooooooooooooooooooo much
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