SME
Surveying · Problem type
Trigonometric leveling
Use this for “A total station 1.55 m high reads 245.30 m at 4°30′ up to a prism 1.80 m high. Find the rise.”
Diagram
V = S sin α = 19.25 m, up from the instrument to the prism. ΔE = V + hᵢ − hᵣ = 19 m: the sight starts hᵢ above one ground and ends hᵣ above the other.
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.
V (m)
=
S (m)
×
sin(
α
)
ΔE (m)
=
V (m)
+
hᵢ (m)
−
hᵣ (m)
Formulas
V
=
S
×
sin(
α
)
19.246 = 245.3 × sin(4°30′00″)
ΔE
=
V
+
hᵢ
−
hᵣ
18.996 = 19.246 + 1.55 − 1.8
Assumptions
•
The vertical angle α is measured up from horizontal at the instrument.
•
The sight is short enough to ignore curvature and refraction.
•
ΔE is the ground elevation at the target less the ground elevation at the instrument.
Step-by-step
WE KNOW
S = 245.3 m, α = 4°30′00″, hᵢ = 1.55 m, hᵣ = 1.8 m
FIND
vertical component (V), rise in elevation (ΔE)
Step 1 · Find vertical component (V)
Use V = S × sin(α)
The
vertical
side
is
opposite
α
,
with
the
slope
distance
as
hypotenuse.
V
=
245.3
×
sin(4°30′00″)
V
=
19.246
m
Step 2 · Find rise in elevation (ΔE)
Use ΔE = V + hᵢ − hᵣ
The sight starts hᵢ above the ground and ends hᵣ above the target’s ground.
ΔE = 19.246 + 1.55 − 1.8
ΔE = 18.996 m
Check
Put the numbers back in. Both sides should match.
19.246 = 245.3 × sin(4°30′00″) ✓
18.996 = 19.246 + 1.55 − 1.8 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.