SME
Grade 10 Math · Geometry
Quadrilaterals and other polygons: angle sums and properties
Use this for “Find the sum of the interior angles of a 9-sided polygon, and each angle when it is regular.”
Diagram
A regular nonagon: 9 equal sides and 9 equal angles.
Diagonals from A: n − 2 = 7 triangles, 180° each.
Interior sum: S = 7 × 180° = 1,260°.
Each angle: e = 1,260° ÷ 9 = 140°.
Each exterior angle: x = 360° ÷ 9 = 40°, and the 9 of them add to 360°.
This is an example. Type your own values to start a new problem.
S
°
=
(
n
−
2)
×
180°
e
Each interior angle
calculated · °
x
Each exterior angle
calculated · °
Formulas
S
=
(
n
−
2)
×
180
1,260 = (9 − 2) × 180
e
=
S
n
140
=
1,260
9
x
=
360
n
40
=
360
9
e
+
x
=
180
140 + 40 = 180
Assumptions
•
The diagonals from one corner cut a convex polygon with n sides into n − 2 triangles.
•
Each triangle’s angles add to 180°, so the interior angles add to (n − 2) × 180°.
•
A regular polygon’s angles are equal: each is the sum ÷ n. Its exterior angles add to 360°, so each is 360° ÷ n.
Step-by-step
WE KNOW
n = 9
FIND
sum of the interior angles (S), each interior angle (e), each exterior angle (x)
Step 1 · Find sum of the interior angles (S)
Use S = (n − 2) × 180
The
n
−
2
triangles
from
one
corner
each
add
180°.
S
=
(9
−
2)
×
180
S
=
7
×
180
S
=
1,260°
Step 2 · Find each interior angle (e)
Use e = S ÷ n
The
n
equal
angles
share
the
sum.
e
=
1,260
9
e
=
140°
Step 3 · Find each exterior angle (x)
Use x = 360 ÷ n
The
n
equal
exterior
angles
add
to
360°.
x
=
360
9
x
=
40°
Check
Put the numbers back in. Both sides should match.
1,260 = (9 − 2) × 180 ✓
140
=
1,260
9
✓
40
=
360
9
✓
140 + 40 = 180 ✓
Topic check
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