SME
Surveying
Distance and angle measurement
Use this for “A total station reads 312.45 m along the slope at a zenith angle of 86°15′20″. Find the horizontal distance and the rise.”
Diagram
V = S sin α = 20.4 m, up from the instrument to the prism.
Heights are drawn 5 times their scale; the tripod and rod are not to scale.
This is an example. Type your own values to start a new problem.
α
=
90°
−
z
H (m)
=
S (m)
×
cos(
α
)
V (m)
=
S (m)
×
sin(
α
)
Formulas
α
=
90°
−
z
3°44′40″ = 90° − 86°15′20″
H
=
S
×
cos(
α
)
311.783 = 312.45 × cos(3°44′40″)
V
=
S
×
sin(
α
)
20.405 = 312.45 × sin(3°44′40″)
Assumptions
•
The zenith angle z is read from straight up, the vertical angle α from level: α = 90° − z.
•
Instrument and target stand equally high, so V is the rise between the ground marks.
•
The sight is short: no curvature or refraction.
Step-by-step
WE KNOW
S = 312.45 m, z = 86°15′20″
FIND
vertical angle (α), horizontal distance (H), vertical difference (V)
Step 1 · Find vertical angle (α)
Use α = 90° − z
The zenith angle runs from straight up; the vertical angle from level is what is left of 90°.
α
=
90°
−
86°15′20″
α
=
3°44′40″
Step 2 · Find horizontal distance (H)
Use H = S × cos(α)
The
horizontal
side
is
next
to
α
,
with
the
slope
distance
as
hypotenuse.
H
=
312.45
×
cos(3°44′40″)
H
=
311.783
m
Step 3 · Find vertical difference (V)
Use V = S × sin(α)
The
vertical
side
is
opposite
α
.
V
=
312.45
×
sin(3°44′40″)
V
=
20.405
m
Check
Put the numbers back in. Both sides should match.
3°44′40″ = 90° − 86°15′20″ ✓
311.783 = 312.45 × cos(3°44′40″) ✓
20.405 = 312.45 × sin(3°44′40″) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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