[SOLVED] latitude

**nerdo** · Nov 28th 2009, 08:59 AM

How would i show the following:

Show that about 4% of the Earth's surface lies north of the Arctic circle (at latitude +66.5 degrees).

**nerdo** · Nov 28th 2009, 09:20 AM

How would i show the following:

Show that about 4% of the Earth's surface lies north of the Arctic circle (at latitude +66.5 degrees).

**aidan** · Nov 29th 2009, 01:59 AM

Originally Posted by nerdo

How would i show the following:

Show that about 4% of the Earth's surface lies north of the Arctic circle (at latitude +66.5 degrees).

Do an internet search for "SPHERICAL CAP"

or look here:Spherical Cap -- from Wolfram MathWorld

Since you are looking only for a per cent value, you do not require the radius of the earth in miles or kilometers, just use "R" for the radius.

Calculate the total surface area of a sphere,
then the area of the spherical cap.

$\dfrac{\text{AreaSphericalCap}}{\text{SurfaceAreaS phere} }100 = \text{percentage}$

.

**nerdo** · Nov 29th 2009, 08:56 AM

Originally Posted by aidan

Do an internet search for "SPHERICAL CAP"

or look here:Spherical Cap -- from Wolfram MathWorld

Since you are looking only for a per cent value, you do not require the radius of the earth in miles or kilometers, just use "R" for the radius.

Calculate the total surface area of a sphere,
then the area of the spherical cap.

$\dfrac{\text{AreaSphericalCap}}{\text{SurfaceAreaS phere} }100 = \text{percentage}$

.

Thanks for the help, but i would like to ask how would you calculate the radius.

**aidan** · Nov 29th 2009, 04:17 PM

Originally Posted by nerdo

Thanks for the help, but i would like to ask how would you calculate the radius.

Surface Area of a sphere:
$S_{\text{sphere}} = 4 \, \pi \, r^2$

Surface Area of a spherical cap
$S_{\text{cap}} = 2 \, \pi \, r \, h$

If you sliced the earth along the latitude 66.5 degress,
the distance from the plane ( the flat part )
to the north point is the distance h.

$h = R - R \sin (latitude)$
$h = R(1 - \sin (latitude))$

replacing h (in the spherical cap equation above)
$S_{\text{cap}} = 2 \, \pi \, r^2 (1-sin(latitude))$

the ratio:
$\dfrac{ \text{SphericalCapArea}}{\text{SurfaceAreaOfSphere }}$

$\dfrac{2\,\pi\,r^2(1-\sin(latitude))}{4\,\pi\,r^2}$

Radius cancels.
Pi cancels.

.

**nerdo** · Nov 29th 2009, 09:52 PM

Originally Posted by aidan

Surface Area of a sphere:
$S_{\text{sphere}} = 4 \, \pi \, r^2$

Surface Area of a spherical cap
$S_{\text{cap}} = 2 \, \pi \, r \, h$

If you sliced the earth along the latitude 66.5 degress,
the distance from the plane ( the flat part )
to the north point is the distance h.

$h = R - R \sin (latitude)$
$h = R(1 - \sin (latitude))$

replacing h (in the spherical cap equation above)
$S_{\text{cap}} = 2 \, \pi \, r^2 (1-sin(latitude))$

the ratio:
$\dfrac{ \text{SphericalCapArea}}{\text{SurfaceAreaOfSphere }}$

$\dfrac{2\,\pi\,r^2(1-\sin(latitude))}{4\,\pi\,r^2}$

Radius cancels.
Pi cancels.

.

Thank you for the help,

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