Every interior angle sum comes from cutting the shape into triangles.
Find the angle
What is the measure of each interior angle of a regular hexagon?
The interior angles of an n-sided polygon sum to (n − 2) × 180°, so a hexagon gives (6 − 2) × 180 = 720°. Regular means all six are equal, so each is 720 ÷ 6 = 120°. D is the total, not one angle. A is the exterior angle (360 ÷ 6), and B is the interior angle of a regular pentagon.
1
Naming by sides, and two words worth keeping straight.
Sides
3
4
5
6
7
8
10
Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
decagon
Convexno dent — every interior angle under 180°a stop sign
Concaveat least one angle over 180°an arrowhead, a star
Regularall sides and all angles equala rhombus is not regular
A rhombus has four equal sides but not four equal angles, so it is not regular. Regular needs both conditions.
2
Interior angles: cut it into triangles.
Sum = (n − 2) × 180°
Diagonals from one corner always make n − 2 triangles, and each triangle holds
180°. That is the whole formula — nothing to memorise beyond the picture.
3
Exterior angles always total 360°.
For any convex polygon, however many sides it has.
regular hexagon
six equal exterior angles
360 ÷ 6 = 60°
each exterior angle
180 − 60 = 120°
each interior angle — matching step 2
Think of walking once around the shape: by the time you are back where you started, facing the same way, you have turned a full 360°. The number of sides never changes that.
4
Triangles: the angle sum and the exterior angle theorem.
The three angles add to 180°, and an exterior angle equals the two opposite interior ones.
By sides
By angles
equilateral — 3 equal
acute — all under 90°
isosceles — 2 equal
right — one 90°
scalene — none equal
obtuse — one over 90°
An equilateral triangle has three 60° angles, so it is also acute. The two
classifications describe the same triangle from different angles.
5
Running the formulas backwards: from angles to the number of sides.
Exterior angle known: n = 360 ÷ exterior. Interior sum known: n − 2 = sum ÷ 180.
Each exterior angle of a regular polygon is 40°. How many sides does it have?
n × 40 = 360
the exterior angles always total 360°
n = 360 ÷ 40 = 9
a nonagon
Check: each interior angle is 180 − 40 = 140°, and 9 × 140 = 1,260 = (9 − 2) × 180 ✓
The interior angles of a polygon add to 1,800°. How many sides does it have?
(n − 2) × 180 = 1,800
the sum formula, with n unknown
n − 2 = 10
divide by 180
n = 12
add the 2 back
Stopping at the number of triangles
1,800 ÷ 180 = 10 counts the triangles, not the sides. A polygon always has 2 more sides than triangles, so the answer is 12; a decagon's angles add to only (10 − 2) × 180 = 1,440°.
6
Variations: angles in a ratio, a parallelogram's neighbours, and a polygon that cannot exist.
The same angle sums; these are the less direct ways they are asked.
The angles of a triangle are in the ratio 1 : 2 : 3. Find them.
1 + 2 + 3 = 6 parts = 180°
one part is 30°
30°, 60°, 90°
a right triangle
A ratio of angles is a sharing problem with 180° as the total.
One angle of a parallelogram is 3 times the angle next to it. Find both.
x + 3x = 180
neighbouring angles are supplementary
x = 45°, 3x = 135°
check: 45 + 135 + 45 + 135 = 360 ✓
Neighbours add to 180°; opposite angles are equal. That is why the four angles still total 360°.
Can a regular polygon have interior angles of 130°?
exterior = 180 − 130 = 50°
work with the exterior angle
360 ÷ 50 = 7.2
not a whole number, so no such polygon
The number of sides must be a whole number, so only exterior angles that divide 360 exactly are possible.
7
The quadrilateral family.
Shape
Opposite sides
Angles
Diagonals
Parallelogram
parallel and equal
opposite equal
bisect each other
Rectangle
parallel and equal
all 90°
equal
Rhombus
all four equal
opposite equal
perpendicular
Square
all four equal
all 90°
equal and perpendicular
Trapezoid
one pair parallel
—
—
The family tree runs one way
A square is a rectangle and a rhombus and a parallelogram. But a
rectangle is not a square. Statements like "every rhombus is a square" are false;
reverse them and they are true.
Remember the flow
Count the sides · Interior sum = (n − 2)180 · Regular? divide by n · Exterior angles always 360
A total and a single angle are different answers — check which the question wants.
On the exam
Polygon items ask for an interior-angle sum, one angle of a regular polygon, the number of sides from an angle, a missing angle in a triangle, or which quadrilateral statement is always true. The fastest route is the exterior angle, 360 ÷ n; the 10-second check is that interior plus exterior makes 180° at every corner.
Now you try
Practice problem
Find the number of sides
A regular polygon has interior angles of 144° each. How many sides does it have?
The quickest route is through the exterior angle: 180 − 144 = 36°, and the exterior angles total 360°, so there are 360 ÷ 36 = 10 sides. Checking with the interior formula: (10 − 2) × 180 = 1,440, and 1,440 ÷ 10 = 144 ✓ D is the exterior angle, mistaken for the count.
Hint: going via the exterior angle avoids solving an equation.
Find 180 − 144 first, then divide 360 by it.
