1. ## [SOLVED] latitude

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).

2. 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).

3. 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.

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

.

4. 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.

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

.

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

5. 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:
$\displaystyle S_{\text{sphere}} = 4 \, \pi \, r^2$

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

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.

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

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

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

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

Pi cancels.

.

6. Originally Posted by aidan
Surface Area of a sphere:
$\displaystyle S_{\text{sphere}} = 4 \, \pi \, r^2$

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

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.

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

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

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

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