A polygon is a figure enclosed by line segments, and at each vertex there is an interior angle inside and an exterior angle outside.
Drawing diagonals from one vertex splits an n-sided polygon into (n − 2) triangles, so its interior angles add up to 180° × (n − 2).
The exterior angles always add up to 360°, however many sides there are, and for a sector, a slice of a circle, the arc length and area depend on what fraction of the whole circle its central angle is.
On this page you change the number of sides and count the triangles, then change the central angle and radius to check the arc and area of a sector.
Polygons split into triangles
seegongsik.com
A regular polygon split into triangles by diagonals from one vertex
5
🍕 Splitting into triangles
①From one vertex of an n-sided polygon you can draw (n − 3) diagonals: none to itself or to its two neighbours
②These diagonals split the polygon into (n − 2) triangles. Count the different-coloured pieces in the picture and there are n − 2
③Each triangle's angles add up to 180°, so the interior angles of an n-sided polygon add up to 180° × (n − 2)
④An n-sided polygon has n(n − 3)/2 diagonals in all
Interior and exterior angles
Sum of interior angles
180° × (n − 2)
Found by splitting the polygon into (n − 2) triangles
One interior angle of a regular polygon
180° × (n-2)n
All interior angles of a regular polygon are equal, so divide the sum by n
Sum of exterior angles
sum of exterior angles = 360°
It is 360° for every polygon, whatever the number of sides
💡 Why the exterior angles always add up to 360°
①At each vertex, the interior and exterior angles add up to 180°. An n-sided polygon has n vertices, so (interior sum) + (exterior sum) = 180° × n
②Subtract the interior sum 180° × (n − 2), and the exterior sum is 180° × 2 = 360°
③Or picture walking once around the edge: at each vertex you turn through the exterior angle, and you end up facing the way you started, so the turns add up to one full turn, 360°
④Each exterior angle of a regular n-sided polygon is 360° ÷ n
Circles and sectors
seegongsik.com
Arc length and area of a sector as the central angle and radius change
120°
5
🔍 A sector is a fraction of a circle
①A sector with central angle x° is x/360 of the whole circle (360°)
②So its arc length is x/360 of the circumference 2πr, and its area is x/360 of the circle's area πr²
③In one circle, arc length and area are directly proportional to the central angle: double the angle in the picture and both double
④The picture shows both the exact form with π and a decimal approximation. Answers are usually left with π
The formulas
Circumference
l = 2πr
Diameter × π. With diameter d, l = πd
Area of a circle
S = πr²
Square the radius and multiply by π
Arc length of a sector
l = 2πr × x360
Multiply the circumference by x/360
Area of a sector
S = πr² × x360 = 12rl
Multiply the circle's area by x/360. If you know the radius and arc length, (1/2) × r × l also works
💡 Why S = (1/2)rl
①Imagine cutting the sector into very thin slices from the centre
②Each slice is nearly a triangle with a tiny piece of arc as its base and the radius r as its height
③Adding all the slices gives (1/2) × r × (sum of the bases) = (1/2) × r × l
④So the radius and arc length alone give the area, without the central angle
Check with examples
Example 1
Find the sum of the interior angles of a hexagon.
1
The interior angles of an n-sided polygon add up to 180° × (n − 2), and a hexagon has n = 6.
180° × (6 − 2)
2
Calculate.
= 180° × 4 = 720°
▸ 720°
Set n to 6 in the picture and you get 4 triangles and an interior angle sum of 720°.
Example 2
Find the area of a sector with radius 6 and central angle 60°. (Leave π in the answer.)
1
The area of a sector is the circle's area multiplied by (central angle)/360°.
area = π × 6² × (60/360)
2
Calculate.
= 36π × 1/6 = 6π
▸ 6π
Set the radius to 6 and the central angle to 60° in the picture and the area shows 6π ≈ 18.85.
Exam checklist
Key formulas for polygons and sectors
180(n-2), πr², 12rl
interior angle sum, area of a circle, area of a sector
Grade 7 school test style
What is the size of one interior angle of a regular pentagon?
①100°
②108°
③110°
④120°
⑤135°
▸ ② 108°
1
The interior angles of a pentagon add up to 180° × (5 − 2) = 540°.
180° × (5 − 2) = 540°
2
A regular pentagon has five equal angles, so divide by 5.
540° ÷ 5 = 108°
🎯 Exam checklist
①The interior angles of an n-sided polygon add up to 180° × (n − 2)
②The exterior angles add up to 360° whatever n is; it is easy to think they grow with n
③One interior angle is 120° for a regular hexagon and 135° for a regular octagon
④Arc length and area are proportional to the central angle, so proportions solve them