Great Circle Distances

Topics

Try VividMath Premium to unlock full access

Start Free Trial

Lets see your results!

Points

Score

Time

Retry Quiz

Answered
Review

Question 1 of 8

1. Question
Find the position of $Y$ $Y$ relative to $X$ $X$ given the following coordinates:

$X (25 ° N, 150 ° E)$ $X(25°N,150°E)$

$Y (15 ° S, 150 ° W)$ $Y(15°S,150°W)$
- 1. $Y$ $Y$ is $60 °$ $60°$ West and $40 °$ $40°$ North of $X$ $X$
- 2. $Y$ $Y$ is $100 °$ $100°$ East and $60 °$ $60°$ North of $X$ $X$
- 3. $Y$ $Y$ is $40 °$ $40°$ West and $80 °$ $80°$ South of $X$ $X$
- 4. $Y$ $Y$ is $60 °$ $60°$ East and $40 °$ $40°$ South of $X$ $X$
Hint

Help Video
Correct

Great Work!
Incorrect
Need Text
Play
Current Time 0:00
/
Duration Time 0:00
Remaining Time -0:00
Stream TypeLIVE
Loaded: 0%
Progress: 0%
0:00
Fullscreen
Video Acceleration:
On
Off

00:00
Mute
Playback Rate
1x
2x
1.5x
1.25x
1x
0.75x
0.5x
Subtitles
subtitles off
Captions
captions off
English
Chapters
Chapters

The prime meridian is a line of longitude at $0 °$ $0°$ .

On the opposite side of the prime meridian lies the $180 °$ $180°$ meridian.

First, form a scale using the top view of the globe

Count starting from the longitude of $X$ $X$ to the longitude of $Y$ $Y$

$X (25 ° N,$ $X(25°N,$ $150 ° E$ $150°E$ $)$ $)$

$Y (15 ° S,$ $Y(15°S,$ $150 ° W$ $150°W$ $)$ $)$

From the scale above, we know that $Y$ $Y$ is $60 °$ $60°$ East of $X$ $X$

Lastly, add the latitude of $X$ $X$ and $Y$ $Y$

$X ($ $X($ $25 ° N$ $25°N$ $, 150 ° E)$ $,150°E)$

$Y ($ $Y($ $15 ° S$ $15°S$ $, 150 ° W)$ $,150°W)$

$25 ° + 15 °$ $25°+15°$ $=$ $=$ $40 °$ $40°$

Hence, $Y$ $Y$ is $40 °$ $40°$ South of $X$ $X$

$Y$ $Y$ is $60 °$ $60°$ East and $40 °$ $40°$ South of $X$ $X$

Question 2 of 8

Find the shortest distance between San Francisco and Seattle given the following coordinates:

San Fracisco

(38 ° N, 122 ° W)

$(38°N,122°W)$

Seattle

(47 ° N, 150 ° W)

$(47°N,150°W)$

Round your answer to the nearest whole number

(1005)km

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Seattle to San Francisco.

We also know that the radius of earth is $6400$ $6400$ km

Find $θ$ $theta$ by getting the difference between the two latitudes

San Fracisco

(38 ° N, 122 ° W)

$(38°N,122°W)$

Seattle

(47 ° N, 150 ° W)

$(47°N,150°W)$

$θ$ $theta$	$=$ $=$	$47 ° - 38 °$ $47°-38°$
$θ$ $theta$	$=$ $=$	$9 °$ $9°$

Lastly, substitute known values into the Arc Length Formula

θ = 9 °

$theta=9°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{9}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$\frac{1}{40} (40212.3)$ $1/40(40212.3)$

	$=$ $=$	$1005$ $1005$ km

Hence, the shortest distance between Seattle and San Francisco is

1005

$1005$ km

1005

$1005$ km

Question 3 of 8

Find the shortest distance between Melbourne and Cairns given the following coordinates:

Melbourne

(38 ° S, 145 ° E)

$(38°S,145°E)$

Cairns

(17 ° S, 145 ° E)

$(17°S,145°E)$

Round your answer to the nearest whole number

(2346)km

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Melbourne to Cairns.

We also know that the radius of earth is $6400$ $6400$ km

Find $θ$ $theta$ by getting the difference between the two latitudes

Melbourne

(38 ° S, 145 ° E)

$(38°S,145°E)$

Cairns

(17 ° S, 145 ° E)

$(17°S,145°E)$

$θ$ $theta$	$=$ $=$	$38 ° - 17 °$ $38°-17°$
$θ$ $theta$	$=$ $=$	$21 °$ $21°$

Lastly, substitute known values into the Arc Length Formula

θ = 21 °

$theta=21°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{21}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$2345.7$ $2345.7$

	$=$ $=$	$2346$ $2346$ km

Hence, the shortest distance between Melbourne and Cairns is

2346

$2346$ km

2346

$2346$ km

Question 4 of 8

Find the shortest distance between Adelaide and Tokyo given the following coordinates:

Adelaide

(35 ° S, 139 ° E)

$(35°S,139°E)$

Tokyo

(35 ° N, 139 ° E)

$(35°N,139°E)$

Round your answer to the nearest whole number

(7819)km

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Adelaide to Tokyo.

We also know that the radius of earth is $6400$ $6400$ km

Find $θ$ $theta$ by adding the two angle measures

$θ$ $theta$	$=$ $=$	$35 ° + 35 °$ $35°+35°$
$θ$ $theta$	$=$ $=$	$70 °$ $70°$

Lastly, substitute known values into the Arc Length Formula

θ = 70 °

$theta=70°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{70}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$7819.08$ $7819.08$

	$=$ $=$	$7819$ $7819$ km

Hence, the shortest distance between Adelaide and Tokyo is

7819

$7819$ km

7819

$7819$ km

Question 5 of 8

Find the shortest distance between Stockholm and Cape Town given the following coordinates:

Stockholm

(59 ° N, 18 ° E)

$(59°N,18°E)$

Cape Town

(34 ° S, 18 ° E)

$(34°S,18°E)$

Round your answer to the nearest whole number

(10388)km

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Stockholm to Cape Town.

We also know that the radius of earth is $6400$ $6400$ km

Find $θ$ $theta$ by adding the two angle measures

$θ$ $theta$	$=$ $=$	$59 ° + 34 °$ $59°+34°$
$θ$ $theta$	$=$ $=$	$93 °$ $93°$

Lastly, substitute known values into the Arc Length Formula

θ = 93 °

$theta=93°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{93}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$10388.2$ $10388.2$

	$=$ $=$	$10388$ $10388$ km

Hence, the shortest distance between Stockholm and Cape Town is

10388

$10388$ km

10388

$10388$ km

Question 6 of 8

Find the shortest distance between these two points on the equator:

A (0 °, 18 ° W)

$A (0°,18°W)$

B (0 °, 21 ° E)

$B (0°,21°E)$

Round your answer to the nearest whole number

(4356)km

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are on the equator, meaning both of them lie on the same circle.

Draw the sector that covers the distance from

A

$A$ to

B

$B$ .

We also know that the radius of earth is $6400$ $6400$ km

Find $θ$ $theta$ by adding the two angle measures

$θ$ $theta$	$=$ $=$	$18 ° + 21 °$ $18°+21°$
$θ$ $theta$	$=$ $=$	$39 °$ $39°$

Lastly, substitute known values into the Arc Length Formula

θ = 39 °

$theta=39°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{39}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$4356.34$ $4356.34$

	$=$ $=$	$4356$ $4356$ km

Hence, the shortest distance between

A

$A$ and

B

$B$ is

4356

$4356$ km

4356

$4356$ km

Question 7 of 8

Find the shortest distance between these two points on the equator:

Q (0 °, 95 ° W)

$Q (0°,95°W)$

P (0 °, 120 ° E)

$P (0°,120°E)$

Round your answer to the nearest whole number

(16197)km

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are on the equator, meaning both of them lie on the same circle.

Draw the sector that covers the distance from

Q

$Q$ to

P

$P$ .

Remember that we are looking for the shortest distance, so

θ

$theta$ should be the angle subtended by the shorter arc

Find $θ$ $theta$ by subtracting the two angle measures from

360 °

$360°$

$θ$ $theta$	$=$ $=$	$360 ° - 95 ° - 120 °$ $360°-95°-120°$
$θ$ $theta$	$=$ $=$	$145 °$ $145°$

Lastly, substitute known values into the Arc Length Formula

θ = 145 °

$theta=145°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{145}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$16196.7$ $16196.7$

	$=$ $=$	$16197$ $16197$ km

Hence, the shortest distance between

Q

$Q$ and

P

$P$ is

16197

$16197$ km

16197

$16197$ km

Question 8 of 8

Find the shortest distance between Nairobi and Singapore given the following coordinates:

Nairobi

(0 °, 95 ° W)

$(0°,95°W)$

Singapore

(0 °, 120 ° E)

$(0°,120°E)$

Round your answer to the nearest whole number

(7596)km

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are on the equator, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Nairobi to Singapore.

Remember that we are looking for the shortest distance, so

θ

$theta$ should be the angle between Nairobi and Singapore

Find $θ$ $theta$ by getting the difference of the two angle measures

$θ$ $theta$	$=$ $=$	$104 ° - 36 °$ $104°-36°$
$θ$ $theta$	$=$ $=$	$68 °$ $68°$

Lastly, substitute known values into the Arc Length Formula

θ = 68 °

$theta=68°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{68}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$7595.67$ $7595.67$

	$=$ $=$	$7596$ $7596$ km

Hence, the shortest distance between Nairobi and Singapore is

7596

$7596$ km

7596

$7596$ km

Great Circle Distances

Try VividMath Premium to unlock full access

Quiz summary

Information

1. Question

Hint

Great Work!

2. Question

Hint

Correct!

Arc Length Formula

3. Question

Hint

Keep Going!

Arc Length Formula

4. Question

Hint

Fantastic!

Arc Length Formula

5. Question

Hint

Excellent!

Arc Length Formula

6. Question

Hint

Nice Job!

Arc Length Formula

7. Question

Hint

Well Done!

Arc Length Formula

8. Question

Hint

Exceptional!

Arc Length Formula

Quizzes