A polygon is a shape enclosed by straight segments, holding interior and exterior angles inside.
Draw diagonals from one vertex, much like slicing a pizza into n pieces, and the polygon splits into triangles, showing why the interior angles sum to 180°×(n−2).
The exterior angles always total 360° no matter the number of sides, and a sector takes a share of a circle's arc and area set by its central angle.
Drag the sliders for the number of sides, the central angle, and the radius to watch the angles and the sector change.
Intuition — slicing a pizza into n parts
seegongsik.com
Drawing diagonals from one vertex to triangulate the polygon
5
🍕 Think Pizza
①Cutting a pizza into n equal parts gives each slice a central angle of 360/n°
②A polygon is similar — diagonals from one vertex create triangles
③One triangle's interior angles sum to 180°
④There are (n-2) triangles, so the full sum is 180×(n-2)°
Interior and Exterior Angles
Sum of Interior Angles
Sum = 180° × (n - 2)
An n-gon decomposes into (n-2) triangles
Interior Angle of a Regular Polygon
180° × (n-2)n
All interior angles equal, so divide the sum by n
Sum of Exterior Angles
Sum = 360° (every polygon)
Interior + exterior = 180°, so exterior sum is always 360°
💡 Why Exterior Angles Always Sum to 360°
①Imagine walking around the polygon once
②The turn at each vertex is the exterior angle
③You turn 360° in total — so the sum is always 360°
Circles and Sectors
seegongsik.com
Sector arc length and area — adjust the central angle and radius
120°
5
🔍 A Sector is Part of a Circle
①A full circle = sector with 360° central angle
②A sector of x° equals x/360 of the circle
③Both arc length and area share the same x/360 ratio
Deriving Formulas
Circumference
l = 2πr
Diameter × π gives the circumference. Using d: l = πd
Area of a Circle
S = πr²
Square the radius and multiply by π
Arc Length of a Sector
l = 2πr × x360
x/360 of the circumference
Area of a Sector
S = πr² × x360 = 12rl
x/360 of the circle's area; also (1/2) × r × arc length
💡 Why S = (1/2)rl Works
①Imagine slicing the sector into many thin triangles
②Each base is a tiny arc piece, and the height is r
③Triangle area = (1/2) × base × height
④Summing them gives S = (1/2) × r × l
Work It Out
Example 1
Find the sum of the interior angles of a hexagon.
1
The interior-angle sum of an n-gon is 180° × (n − 2). Here n = 6.
180° × (6 − 2)
2
Compute.
= 180° × 4 = 720°
▸ 720°
Split a polygon into (n − 2) triangles to get the interior-angle sum.
Example 2
Find the area of a sector with radius 6 and central angle 60°. (Leave π as is.)
1
Sector area = πr² × (central angle / 360°).
area = π × 6² × (60/360)
2
Compute.
= 36π × 1/6 = 6π
▸ 6π
A sector is part of a circle, so multiply by the angle fraction (central/360°).
Exam Key Points
Polygon · Circle · Sector Summary
180(n-2), πr², 12rl
Three formulas — memorize them
Grade-7 school exam type
What is each interior angle of a regular pentagon?
①100°
②108°
③110°
④120°
⑤135°
▸ ② 108°
1
A pentagon has interior-angle sum 180° × (5 − 2) = 540°.
180° × (5 − 2) = 540°
2
A regular pentagon has five equal angles, so divide by 5.
540° ÷ 5 = 108°
🎯 Exam Key Points
①Substitute n into 180(n-2) for instant answers
②Exterior sum 360° is independent of n — watch for trick questions