Circumference of Earth

Clearly, the arc $AB$ subtends an angle of $4.92^{o}$ at the centre. Now, if $\theta$ is the angle (in radians) which an arc of length $S$ subtends at the centre of a circle of radius $r$ ,

$S=r\times \theta$

So for the Earth, as $r$ is constant (perfect sphere assumption),

$\frac{S_{1}}{S_{2}}=\frac{\theta_{1}}{\theta_{2}}$

Given that $\theta_{1}$ in degrees is $4.92^{o}$ and $S_{1}$ is $500$ km, circumference $S_{2}$ can be evaluated by putting $\theta_{2}$ as $360^{o}$ .

So, $S_{2} = \frac{\theta_{2}\times S_{1}}{\theta_{1}}$ $S_{2}=\frac{360^{o}\times 500}{4.92^{o}}$ km

Note: As the ratio of the angles is required, the unit (degree or radian) is immaterial since they follow a linear relationship.

The arc length is $s = r \theta$ where $\theta$ is the angle, and $r$ is the radius. So, If you have that $s = 500\text{km},$ was due to an angle of 4.92°, then you have that $r=\frac{500\text{ km}}{4.92\text{°}},$ but this is also true for $r= \frac{\text{circumference}}{360\text{°}}.$ Equating both expressions for $r$ , you get $\text{circumference}=\frac{360\text{°}\times 500\text{ km}}{4.92\text{°}}.$

Well, I just thought that if 4.92 degrees gave us 500km, then dividing the 500km by 4.92 would give us how many km in a degree. Multiply that by 360 to find the total for the whole circumference: 500/4.92 x 360/1 or 500x360/4.92

4.92 degree corresponds to 500 km. so 1 degree will correspond to 500/4.92 km, and hence 360 degree will correspond to (500*360)/4.92 km

4,92/360*X=500

X=500 /(4,92/360)

X = 500*360/4,92

Since clearly, the angle of inclination from the center forms the 500km arc, we can therefore derive it from the formula in getting the arc length or s of the circle in the formula: s=r(Angle in Radians) . That's 500 = r x (4.92degrees) then multiply it by 360 degrees to get the circumference of the circle :) . Since, C=2pi or 360 degrees x (r)^2 and r = 500/4.92 degrees .

C = 360 degrees x 500 km / 4.92 degrees

Answer analysis lets you solve this even if you are stuck.

It cannot be A, since the degrees do not cancel out. B and C both leave you with numbers too small to be relevant for the circumference of the Earth. Leaving D.

The exact answer is 36,585.36585 (repeating)

360÷4.92= target ratio

Target ratio×100=percentage of whole circumference

500×4.92=2460

360×500=180000

2460÷180000=target ratio

180000÷4.92= 36,585.36585 (repeating) This is the Circumference

2460÷4.92=500 (for checking purposes)

Let C = 2 $\pi$ r, where C is the circumference of Earth and r is the radius of Earth. Let L = $\theta$ $\frac{\pi}{180^{\circ}}$ r, where L is the arc length from points B to A, and $\theta$ is the angle, in degrees, which is subtended by the arc length. Since the angle subtended by the arc and the angle at point A are corresponding angles, $\theta$ = $4.92^{\circ}$ . Using the arc length to solve for the radius r, while plugging in $\theta$ = $4.92^{\circ}$ , and the arc length L = 500 km, we have r = $\frac{180^{\circ}\times500km}{4.92^{\circ}\pi}$ . Plugging in r into the circumference C, we have C = $\frac{2\pi\times180^{\circ}\times500km}{4.92^{\circ}\times\pi}$ = $\frac{500\times360^{\circ}}{4.92^{\circ}}$ km.

The angle of center of earth is 360 degree. now,as a similar angle the given angle focus on the area of center angle of earth=4.92* so, for 4.92 degree distance goes to =500 km 1 degree distance goes to =500/4.92 & 360 degree " " " =(500*360)/4.92 km :)

ACLARACIÓN: 1- Los puntos A y B en el diagrama se muestra sobre la tierra, entonces 500km es la longitud de un arco. Si asumimos la distancia AB como recta no podríamos presentar la respuesta dentro de las posibles dadas. 2- Las discusión sobre los rayos del sol no son parte del problema, el diagrama los muestra paralelos. 3- De las posibles respuestas, 360*500/4.92 es la única posible si pensamos en la magnitud y unidades (km) del número sin hacer cálculos, esta es otra solución, para un examen contra reloj ... pero no para hacer matemática

Using the equation for arc length of a circle,

$S = \pi d \frac{\theta}{360^\circ}$

The given values are:

$S = 500$ , $\theta = 4.92^\circ$

By substitution,

$500 = \pi d \frac{4.92^\circ}{360^\circ}$

Since the circumference of a circle is equal to pi times the circle's diameter,

$C = \pi d$

By substitution,

$500 = C \frac{4.92^\circ}{360^\circ}$

Thus, by solving for C,

$500 \frac{360^\circ}{4.92^\circ} = C \frac{4.92^\circ}{360^\circ} \times \frac{360^\circ}{4.92^\circ}$

$500 \frac{360^\circ}{4.92^\circ} = C$

or

$C = 500 \frac{360^\circ}{4.92^\circ}$

Therefore,

$C = \frac{500\times360^\circ}{4.92^\circ}$

or

$\boxed{C = \frac{360^\circ \times 500}{4.92^\circ}}$

To see the solution to the problem as the author intended it to be interpreted, see the top answer.

If you were a little puzzled and were getting something like "r = 500km * sin(87.54 deg)/sin(4.92 deg)", you aren't wrong, exactly, but the author didn't mean the problem the way that you were reading it.

There are two sticking points for me. Maybe you had an issue too.

• The diagram have A and B backward. If you think about it, from point A in the picture, the sun appears to be 4.92 degrees away from directly above.

• "Distance from A to B" should have been clarified in the wording of the problem - it is interpreted by the author as an arclength rather than as a straight line's length. Both are reasonable measurements to think about. If you interpret 500km as the arclength, or the distance you would swim/walk along the Earth, then the answer is given as the correct multiple choice canswer, (4.92/360)*500km.

• If you assume that the Euclidean straight line distance is 500km, it's a different problem, but very similar answer will result because the angle is small, and the arc is "almost" a line segment. But let's do it.

Assuming the arc in the picture is actually a straight line, the simplest way to solve the triangle is probably the Law of Sines. The triangle in the picture has two long legs of length r, forming angle A=4.92 degrees at the center of the earth, and opposite side a=500km. The other angles are (180-4.92)/2 = 87.54 degrees. Call one of these angles B, and its opposite side has length r (we would call this side length b, typically, in the Law of Sines).

Law of sines: sinA/a = sinB/b = sinB/r sin(4.92 deg)/500km = sin(87.54 deg)/r

r = 500km * sin(87.54 deg)/sin(4.92 deg) = 500km * sin(1.528)/sin(.086) (At the end, I converted to radians, but you can use a degree mode setting on a calculator if you don't do radians)

r = 500km * 11.64 = 5825km So circumference = 2pi r = 36,596km

The problem where you interpret "distance" as "earthling travel distance" a.k.a. "arclength", you get circumference = 36,585km. If you can sketch a 5 degree angle and see how small it is, you'll see that the arc straight line are almost the same, so no wonder the answers are so close.

Just by looking at the answers, the 3rd answer is the only one that comes close to a planet sized circumference. The rest are much to small

R 4.92 pi/180 = 500 Hence Circumference = 2 pi R = 360*500/4.92 Wording of problem should be like, " Sun rays are vertical at A. At point B 500 km north of A, sun rays make an angle of 4.92, what is the circumference of the earth?

its simple

4.92 deg makes arc of length= 500 km

1 deg will make arc of length= 500/4.92

360 deg will make arc of length= 500/4.92 *360