grade 7 Polygon & Circle

Polygon & Circle

A polygon is a shape enclosed by straight segments, and interior and exterior angles sit inside it. Draw diagonals from one vertex, much like slicing a pizza into n pieces, and the polygon splits into triangles, which is 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. Change the number of sides, the central angle, and the radius, and the angles and the sector update.

Intuition — slicing a pizza into n parts

Number of sides n5

🍕 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)°

⑤Number of diagonals n(n−3)/2; one exterior of a regular n-gon is 360/n

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

Central angle120°

Radius r5

🔍 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.

The interior-angle sum of an n-gon is 180° × (n − 2). Here n = 6.

180° × (6 − 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.)

Sector area = πr² × (central angle / 360°).

area = π × 6² × (60/360)

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°

A pentagon has interior-angle sum 180° × (5 − 2) = 540°.

180° × (5 − 2) = 540°

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

③Memorize regular polygon angles: hexagon 120°, octagon 135°

④(1/2)rl is a frequent constructed-response item

⑤Arc length and area are proportional to the central angle

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Construction & Congruence

Surface Area & Volume of Solids

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