Solving Scale Problems
The solution will gradually “build”
for each problem
Some unit conversions
•
•
•
•
1 m = 100 cm = 1000 mm
1 km = 1000 m
1 foot = 12 inches (1’ = 12”)
1 mile = 63,360 inches
• 1 hectare (ha) = 10,000m2
The real world distance between 2 features is
1600m and the map distance is 8 cm.
What is the map scale ?
•
Scale = 1 / xxxxx = map dist / real world
distance
•
If we know 3 of the figures, then we solve
for the unknown by cross-multiply & then
divide
•
What are we solving for?
–
–
•
=
xxxx
1
Map dist.
Real world
dist.
=
xxxx
Map dist.
Real world
dist.
What do we know?:
–
–
•
Map scale is unknown
1 / xxx = map dist. / real world (RW) dist
1
Map dist = 8 cm
RW distance is 1,600 m = 160,000 cm
1 / xxx = 8 cm / 160,000 cm
–
–
–
xxx = (1*160,000 cm)/8 cm
xxx = 20,000
Scale is 1 : 20,000
1
xxxx
=
8cm
160,000cm
The distance between 2 features on a 1:10,000
map is 5 cm. What is the real world distance?
•
Scale = 1 / xxxx = map dist / real world
distance
•
If we know 3 of the figures, then we
solve for the unknown by crossmultiply & then divide
•
What are we solving for?
–
–
•
=
xxxx
1
Map dist.
Real world
dist.
=
xxxx
Map dist.
Real world
dist.
What do we know?:
–
–
•
Real world (RW) distance
1 /xxxx = map dist / RW dist
1
Scale of map (1/xxxx) is 1 / 10,000
Map dist = 5 cm
1 / 10,000 = 5 cm / RW dist. cm
–
–
–
RW dist (cm) = (5*10,000 cm)/1
RW dist cm = 50,000 cm
In metres, RW dist = 500 m
1
10,000
=
5 cm
Real world
dist. cm
What is the length in cm of one km
on a 1:25,000 map?
•
Scale = 1 / xxxxx = map dist / real
world distance
•
If we know 3 of the figures, then we
solve for the unknown by crossmultiply & then divide
•
What are we solving for?
–
–
•
=
xxxx
1
Map dist.
Real world
dist.
=
xxxx
Map dist.
Real world
dist.
What do we know?:
–
–
•
Length on map (in cm)
1 / x = map dist. / real world (RW) dist
1
Scale of map (1/x) is 1 / 25,000
1 km RW dist (1 km = 1,000 m =
100,000 cm)
1 / 25,000 = map dist cm / 100,000 cm
–
–
Map dist (cm) = (1*100,000 cm)/25,000
Map dist = 4 cm
1
25,000
=
Map dist. cm
100,000cm
You have a photo with a scale expressed as
1” = ¼ mile. What is the scale as a representative fraction?
•
•
•
•
•
•
•
•
This is really a unit conversion
question
Scale = 1 / xxx = map / RW
Scale of photo is 1” = ¼ mile
If you convert top and bottom
values to the same units, then you
have a representative fraction
(since the units will cancel out)
So the question really is … how
many inches in ¼ mile?
1 mile = 63,360 inches
¼ mile = 63,360 inches / 4 =
15,840 inches
Scale is 1 : 15,840 (NO need to
show units anymore)
1
=
xxxx
Real world dist.
1
=
xxxx
1
xxxx
1 inch
¼ mile
=
xxxx
1
Map dist.
1 in
15,840 in
=
1
15,840
A map scale is 1 cm = 750 m and a photo covers a portion of the map
area. The distance between two road intersections measures 9.8 cm on the
photo and 5.0 cm on the map. Determine the scale of the photo.
•
•
Photo scale = 1/xxx = photo/RW
We know
1
=
xxxx
Photo
RW
– photo measure of 9.8 cm
•
•
If we had the RW distance we
could determine the photo scale.
Can we find RW distance?
– Yes, as we have map scale and a
map measure
•
•
So first we find RW distance from
map, then use it to solve for photo
scale
Map scale is
– 1 cm = 750 m, or
– 1 cm = 75,000 cm, or
– 1 : 75,000
•
RW distance is
– RW = 5.0 cm * 75,000
– RW = 375,000 cm
1
=
xxxx
1
75,000
9.8 cm
RW
=
5.0 cm
RW
A map scale is 1 cm = 750 m and a photo covers a portion of the map
area. The distance between two road intersections measures 9.8 cm
on the photo and 5.0 cm on the map. Determine the scale of the photo.
•
•
From last slide
Photo scale is …
•
RW distance (from map measures) is
–
–
RW = 5.0 cm * 75,000
RW = 375,000 cm
1
xxxx
1
xxxx
•
Realize that RW distance is the same
no matter if we calculated from a map
or a photo – the RW distance is the
same
•
Now we can solve for photo scale
(xxxx)
–
–
–
xxxx = (1 * 375,000cm) / 9.8 cm
xxxx = 38,265
Scale is 1 : 38,265
=
9.8 cm
RW cm
=
9.8 cm
375000 cm
Solve for Area
• The unit we will use for an area measure
is the hectare (ha)
• 1 ha = 10,000 m2
• You can visualize this as a large square
that measures 100m on each side.
How many hectares are covered by a
standard air photo at a scale of 1:20,000
•
•
•
First remember that an air photo
measures 9” by 9”
We want to know what ground area is
covered in the photo.
First, since we will be solving for a
metric area measure (ha), it is best to
first convert 9” to cm
–
–
–
•
Second, we find the RW distance in
metres
–
–
•
Photo edge = 9 in
= 9 in * 2.54 cm/in
= 22.86 cm
RW = (20,000*22.86cm) / 1
RW = 457200 cm = 4572 m
Third, we find area (L * W)
–
–
–
Area = 4572m * 4572m
Area = 20903184m2
Area = 2090 ha
1
20000
=
22.86 cm
RW cm
How many hectares are covered by a standard air
photo at a scale of 1” = ¼ mile
•
The photo scale 1” = ¼ mi. needs to be
converted to a representative fraction
–
–
•
Next, convert the photo edge (9”) to metric
(cm)
–
–
–
–
•
Since we will be solving for a metric measure
(ha), it is best to first convert 9” to cm
Photo edge = 9 in
= 9 in * 2.54 cm/in
= 22.86 cm
Now we find the RW distance of the photo
edge
–
–
•
Solve by converting ¼ mi to inches
63,360 in/mi / 4 = 15,840”
RW cm = (22.86 cm*15840)/1
RW = 362,102 cm = 3621 m
Last we calculate area, in m2, then convert
to ha
–
–
Area = 3621 m * 3621 m = 13,111,641 m2
Area = 13,111,641 m2 / 10,000 = 1,311 ha
1
=
xxxx
1
¼ mi
=
15840
1”
15,840”
xxxx
1
1”
=
22.86 cm
RW cm