. If a spherical Triangle on Earth has an excess of 30 degrees, find the area of this spherical triangle measured in square miles.
b. If a spherical triangle on the moon has an excess of 30 degrees, find the area of this spherical triangle in sqare miles.
c. Find the ratio of the above answers to a and b.
area=sphericalExcessinRadians*radius^2
a. area=PI/6*re^2
b. area=PI/6*rm^2
c. (re/rm)^2
Confused? What does that mean?
I gave you the formula for area of a spherical triangle given the excess angle, and the radius of the object. re is the radius of Earth. rm is the radius of the moon.
To find the area of a spherical triangle, you need to know the excess angle and the radius of the sphere (in this case, the Earth or the Moon). Additionally, you need to convert the measurements to the desired unit (square miles).
a. Find the area of the spherical triangle on Earth:
1. First, convert the excess angle of 30 degrees to radians. Since 1 degree = π/180 radians, the excess angle in radians is (30 * π/180) = π/6 radians.
2. The formula to calculate the area of a spherical triangle is A = R^2 * E, where A represents the area, R is the radius of the sphere, and E is the excess angle in steradians.
3. The Earth's radius is approximately 3,959 miles. So, substituting the values into the formula, the area of the spherical triangle on Earth is A = (3959)^2 * (π/6) square miles.
b. Find the area of the spherical triangle on the Moon:
1. Repeat steps 1 and 2 from part (a), but now use the radius of the Moon. The Moon's average radius is approximately 1,079 miles. So, the area of the spherical triangle on the Moon is A = (1079)^2 * (π/6) square miles.
c. Find the ratio of the answers from parts (a) and (b):
To find the ratio, divide the area of the spherical triangle on Earth by the area of the spherical triangle on the Moon:
Ratio = (3959)^2 * (π/6) square miles / (1079)^2 * (π/6) square miles.
Simplify the formula by canceling out common factors, such as (π/6) from both numerator and denominator. The final ratio will be the square of the ratio of the radii:
Ratio = (3959/1079)^2.
Now, you can calculate the ratio by evaluating the expression (3959/1079)^2 using a calculator or a computer program.