SME
Surveying · Problem type
Interior angle misclosure of a closed traverse
Use this for “The six interior angles of a closed traverse add to 719°59′42″. Find the misclosure and the correction to each angle.”
Diagram
Interior angles of 6 sides add to (n − 2) × 180° = 720°.
Misclosure −18″, shared equally: 3″ to each angle.
This is an example. Type your own values to start a new problem.
Sᵣ
=
(
n
−
2)
×
180°
E (″)
=
(
Sₘ
−
Sᵣ
)
×
3600
c (″)
=
−
(
E (″)
)
÷
n
Formulas
Sᵣ
=
(
n
−
2)
×
180°
720°00′00″ = (6 − 2) × 180°
E
=
(Sₘ
−
Sᵣ)
×
3,600
−18 = (719°59′42″ − 720°00′00″) × 3,600
c
=
−
E
n
+3
=
−
−18
6
Assumptions
•
The interior angles of a closed polygon of n sides add to (n − 2) × 180°.
•
Every angle was measured equally well, so each takes the same correction.
•
The misclosure is within what is allowed, often 10″√n to 30″√n.
Step-by-step
WE KNOW
n = 6, Sₘ = 719°59′42″
FIND
sum they should make (Sᵣ), misclosure (E), correction per angle (c)
Step 1 · Find sum they should make (Sᵣ)
Use Sᵣ = (n − 2) × 180°
Diagonals
from
one
corner
split
the
polygon
into
n
−
2
triangles
of
180°
each.
Sᵣ = (6 − 2) × 180°
Sᵣ = 4 × 180°
Sᵣ = 720°00′00″
Step 2 · Find misclosure (E)
Use E = (Sₘ − Sᵣ) × 3,600
What the angles miss by, in degrees, times 3600 to write it in seconds (3600″ to the degree).
E
=
(719°59′42″
−
720°00′00″)
×
3,600
E
=
−18″
Step 3 · Find correction per angle (c)
Use c = −E ÷ n
Each angle gives back an equal share, with the opposite sign.
c
=
−
−18
6
c
=
+3″
Check
Put the numbers back in. Both sides should match.
720°00′00″ = (6 − 2) × 180° ✓
−18 = (719°59′42″ − 720°00′00″) × 3,600 ✓
+3
=
−
−18
6
✓
Topic check
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