Model Textbook of Mathematics Grade 7 (FBISE / NBF)
Class 7 Mathematics Federal Board of Intermediate and Secondary Education (FBISE) (FBISE) 📚 Model Textbook of Mathematics Grade 7 (FBISE / NBF)

Class 7 Mathematics Ch 9 Mastery Guide: Angles of Polygons, Regular Polygons, Quadrilaterals & Circle Geometry (FBISE)

📖 Chapter 9: Geometry 📅 Updated: Sep 13, 2026

🗺️ Teacher & Parent Roadmap: Unit 9 (Geometry)

Geometry is the language of spatial reasoning, shapes, angles, and real-world design. In this unit, students advance from basic shape recognition to formal geometric deduction. They will uncover the underlying mathematical laws governing polygons, master interior and exterior angle sums, explore the rich hierarchy of quadrilaterals (parallelograms, rectangles, rhombuses, trapeziums, and kites), and dissect circular geometry (radii, diameters, chords, arcs, sectors, and segments).

🎯 Core Learning Objectives:
  • Calculate interior and exterior angles of regular & irregular polygons using (n-2) × 180° and 360° rules.
  • Compute diagonals from a single vertex (n-3) and total diagonals.
  • Identify, classify, and solve properties of special quadrilaterals using algebraic systems.
  • Distinguish between chords, diameters, secants, tangents, major/minor arcs, and segments in circles.
⚠️ Common Misconceptions & Pitfalls:
  • Misconception: Thinking exterior angle sum increases with more sides. (Fact: Exterior angle sum is always 360° for ANY polygon).
  • Error: Confusing a circular region (area inside) with the circle itself (curved boundary line).
  • Pitfall: Forgetting that opposite angles in a parallelogram are equal, while adjacent angles sum to 180°.

📐 Unit Overview: The World of Polygons, Quadrilaterals & Circles

Everything around us in architecture, art, engineering, and nature is composed of geometric figures. From the hexagonal structure of honeycombs and tile patterns to the circular gears in clockwork and planetary orbits, understanding geometric relationships empowers us to analyze and design the physical world.

1. Polygons: Convex vs. Concave & Diagonals

A polygon is a 2D closed figure bounded by three or more straight line segments joined end to end.

Convex All angles < 180°
Convex Polygon
> 180° (Reflex)
Concave Polygon

Diagonals from a Single Vertex (Formula: n - 3)

From any vertex of an n-sided polygon, you cannot draw diagonals to itself or its two immediate neighbors. Therefore, the number of diagonals from one vertex is always n - 3.

Vertex A Pentagon (n=5) Diagonals: 5 - 3 = 2
Pentagon: 2 Diagonals from Vertex
Hexagon (n=6) Diagonals: 6 - 3 = 3
Hexagon: 3 Diagonals from Vertex
Polygon Name Sides (n) Interior Sum (n-2)×180° Each Interior Angle Each Exterior Angle Diagonals (n-3)
Triangle 3 (3-2)×180° = 180° 180° / 3 = 60° 360° / 3 = 120° 3 - 3 = 0
Quadrilateral 4 (4-2)×180° = 360° 360° / 4 = 90° 360° / 4 = 90° 4 - 3 = 1
Pentagon 5 (5-2)×180° = 540° 540° / 5 = 108° 360° / 5 = 72° 5 - 3 = 2
Hexagon 6 (6-2)×180° = 720° 720° / 6 = 120° 360° / 6 = 60° 6 - 3 = 3
Heptagon 7 (7-2)×180° = 900° 900° / 7 ≈ 128.57° 360° / 7 ≈ 51.43° 7 - 3 = 4
Octagon 8 (8-2)×180° = 1080° 1080° / 8 = 135° 360° / 8 = 45° 8 - 3 = 5
Decagon 10 (10-2)×180° = 1440° 1440° / 10 = 144° 360° / 10 = 36° 10 - 3 = 7

2. Hierarchy & Visual Guide of Special Quadrilaterals

O (Bisects)
Parallelogram
Opposite sides parallel & equal; diagonals bisect
Rectangle
4 Right Angles (90°); Equal Diagonals
90°
Rhombus
4 Equal Sides; Perpendicular (90°) Diagonals
90° Bisect
Square
All 4 sides equal & 4 right angles (90°)
Trapezium
Exactly 1 pair of parallel opposite sides
Kite
2 pairs adjacent equal sides; Perpendicular diagonals

3. Anatomy of a Circle & Circular Regions

A circle is a set of all points in a 2D plane equidistant from a fixed interior point (the centre).

O C D Diameter (2r) A Radius (r) X Y Chord (XY)
Circle Elements (Centre, Radius, Diameter, Chord)
Minor Segment Major Segment Segments of a Circle Minor Sector Major Sector Sectors of a Circle
Segments & Sector of a Circle

📝 Complete Step-by-Step Textbook Solutions

Exercise 9.1: Angles of Polygons & Quadrilaterals

Question 1: Calculate unknown angles in the following figures (not drawn to scale).

9.1-1(i).jpg

(i) Angle a:
a + 55° + 39° = 180° ⇒ a = 180° - 94° = 86°

9.1-1(ii).jpg


(ii) Angle b:
b + 60° + 60° = 180° ⇒ b = 60°

9.1-1(iii).jpg


(iii) Angle c:
c + 51° + 46° = 180° ⇒ c = 83°

9.1-1(iv).jpg


(iv) Exterior Angle d:
d = 87° + 63° = 150°

9.1-1(v).jpg


(v) Exterior Angle e:
Remote Interior angles: 21° + 31°
e = 21° + 31° = 52°

9.1-1(vi).jpg


(vi) Exterior Angle f:
Remote interior angles = 77° + 77°
f = 77° + 77° = 154°

9.1-1(vii).jpg


(vii) Angle g:
Remote interior angles = 39° + 26°
g = 39° + 26° = 65°

9.1-1(viii).jpg


(viii) Angle x:
Remote interior angles = 47° + 60°
x = 47° + 60° = 107°

9.1-1(ix).jpg


(ix) Angle y:
Remote angle x on right side:
x + 100° = 180° ⇒ x = 180° - 100° = 80°
Remote angle z on left side:
z + 120° = 180° ⇒ z = 180° - 120° = 60°
Finding exterior angle:
y + 40° = 180° 
⇒ y = 140°

Question 2: Calculate unknown angles in quadrilaterals.

(i) Parallelogram:

9.1-2(i).jpg


e = 125° (opposite angle)
f = 180° - 125° = 55° (adjacent angle)
(ii) Quadrilateral 85°, 71°, 87°, g:

9.1-2(ii).jpg

g + 85° + 71° + 87° = 360°
g = 360° - (85° + 71° + 87°) = 360° - 243° = 117°
(iii) Angles 90°, 80°, 120°, y:

9.1-2(iii).jpg

y + 90° + 80° + 120° = 360°
y = 360° - (90° + 80° + 120°) = 360° - 290° = 70°
(iv) Angles 84°, 100°, exterior 121°, x:

9.1-2(iv).jpg

Interior angle = 180° - 121° = 59°
x + 84° + 100° + 59° = 360°
x = 360° - (84° + 100° + 59°) = 117°
(v) Angles 90°, 68°, ext 50°, y:

9.1-2(v).jpg

Interior angle x = 180° - 50° = 80°
Interior angle z = 180° - 68° = 112°
y + 80° + 112° + 90° = 360°
y = 360° - (80° + 112° + 90°) = 28°
y = 28°
(vi) Parallelogram:

9.1-2(vi).jpg

Sum of interior angles
t + t + 125° + 125° = 360°
2t = 360° - 250°
t = 110°/2
t = 55°
Adjacent angles:
u = 180° - 125° = 55°

Exercise 9.2: Exterior Angles & Diagonals of Regular Polygons

Question 1: Sum of interior angles is 720°.
(i) (n-2)×180° = 720° ⇒ n-2 = 4 ⇒ n = 6 (Hexagon)
(ii) Interior angle = 720° / 6 = 120°
Question 2: Sum of exterior angles of regular octagon: 360° (Constant for all polygons).
Question 3: Exterior angle of regular pentagon: 360° / 5 = 72°.
Question 4: Exterior angle = 45°.
(i) n = 360° / 45° = 8. Sum = (8-2)×180° = 1080°
(ii) Name: Octagon.
Question 5: Interior angle = 144°.
(a) Exterior angle = 180° - 144° = 36°
(b) Sides = 360° / 36° = 10 sides
(c) Name: Decagon.
Question 6: Formula & Diagonals from a vertex (n - 3):
(a) Formula: d = n - 3
(b) (i) Quadrilateral: 4-3 = 1, (ii) Pentagon: 5-3 = 2, (iii) Hexagon: 6-3 = 3, (iv) Decagon: 10-3 = 7.
Question 7: Interior & Central angles of regular polygons:
  • n = 3 (Triangle): Interior = 60°, Central = 120°
  • n = 4 (Square): Interior = 90°, Central = 90°
  • n = 5 (Pentagon): Interior = 108°, Central = 72°
  • n = 6 (Hexagon): Interior = 120°, Central = 60°
  • n = 8 (Octagon): Interior = 135°, Central = 45°
  • n = 10 (Decagon): Interior = 144°, Central = 36°
Question 8: Unknown quantities in figures (i) to (ix):
(i) 

9.2-8(i).jpgExterior angles:

180° - 108° = 72°
180° - 102° = 78°
180° - 120° = 60°
180° - 100° = 80°
x + 72° + 78° + 60° + 80° = 360°
x = 360° -  (72° + 78° +  60° +  80°)

x = 70° 
(ii) 

9.2-8(ii).jpg

b + 70° + 65° + 79° + 55° = 360°
b = 360° - (70° + 65° + 79° + 55°)
b = 91° 

(iii) 

9.2-8(iii).jpg

Unknown exterior angles
1st exterior angle 180° - 90° = 90°
2nd interior angle 90°
x + 90° + 60° + 90° + 70° = 360°
x = 360° - (90° + 60° + 90° + 70°)
x = 50° 
(iv) 

9.2-8(iv).jpg

a + 113° + 80° + 82° = 360°
a = 360° - (113° + 80° + 82°)
a = 85° 

(v) 

9.2-8(v).jpg

5y + 70° + 60° + 65° + 40° = 360°
5y = 360° - (70° + 60° + 65° + 40°)
5y = 125°
y = 25°
(vi) 

9.2-8(vi).jpg

4x + 62° + 3x + 46° + 93° + 47° = 360°
7x  = 360° - ( 62° + 46° + 93° +  47°)
x = 112°/7
x = 16° 

(vii) 

9.2-8(vii).jpg

x = 180°-60°=120°, y = 180°-100°=80°, z = 180°-130=50°
w + 60° + 80° + 50° + 85° = 360°
w  = 360° - (60° + 80° + 50° + 85°)
w = 85° 
(viii) 

9.2-8(viii).jpg

a = 180°-60°=120°, b =180°-110°= 70°, c = 180°-135°=45°
Sum of interior angles of a Pentagon = 540°
120° + 135° + (220°-120°) + 110° + d = 540°
120° + 135° + 100° + 110° + d = 540°
d = 540° - (120° + 135° + 100° + 110°)
d = 75° 

(ix) 

9.2-8(ix).jpg

Sum of interior angles of a Hexagon = 720°
8x + (5x+14) + (7x - 6) + 120° + 6x + (4x - 8) = 720°
30x + 120 = 720°
30x = 600°
x = 20°.

Exercise 9.3: Characteristics of Quadrilaterals

Question 1: In parallelogram ABCD, find x and y.

9.3-1(i).jpg


 3x = 18 ⇒ x = 6 cm
3y - 1 = 26 ⇒ 3y = 27 ⇒ y = 9 cm
Question 2: ABCD is a rhombus with ∠DAB = 56°. Determine ∠DBC.
∠ABC = 180° - 56° = 124°
∠DBC = 124° / 2 = 62°
Question 3: In rectangle ABCD, diagonals meet at O, OB = 2x + 4, OC = 3x + 1.

9.3-3.jpg


 3x + 1 = 2x + 4 ⇒ x = 3
OB = 2(3) + 4 = 10 cm ⇒ BD = 2 × 10 = 20 cm
Question 4: Adjacent angles (3x - 5)° and (3x + 11)°.
(3x - 5) + (3x + 11) = 180 ⇒ 6x + 6 = 180 ⇒ x = 29°
Four angles: 82°, 98°, 82°, 98°.
Question 5: Side 4.8 cm, other side 1 1/2 times.
Other side = 1.5 × 4.8 = 7.2 cm. Perimeter = 2(4.8 + 7.2) = 24 cm.
Question 6: Parallelogram with exterior angle 105° at B.

9.3-6.jpg


 x = 105°, y = 105°, z = 75°, p = 75°.
Question 7: Trapezium with AB = DC, ∠B = ∠C = 80°.
Other angles: 100°, 100°; 5x + 10 = 45 ⇒ x = 7°.
Question 8: Rhombus ∠P = (3x - 5)°, ∠Q = (2x + 15)°.
5x + 10 = 180 ⇒ x = 34°. Angles: 97°, 83°, 97°, 83°.
Question 9: Larger angle 24° more than smaller: 78° and 102° (Book key: 87°, 109°).
Question 10: Tangram puzzle shape decomposition.

9.3-10.jpg
Large Δ 1 Large Δ 2 Parall. Sm Δ 1 Square Sm Δ 2 Med Δ
7-Piece Tangram Decomposition
2 Large △, 1 Medium △, 2 Small △, 1 Square, 1 Parallelogram
Question 11: Angles of trapezium (x + 15)°, (2x + 15)°, (3x - 20)°, x°.

9.3-11.jpg


 7x + 10 = 360 ⇒ x = 50°. Angles: 50°, 65°, 115°, 130°.

Exercise 9.4: Circles, Chords, Arcs & Segments

Question 1: Circle Elements Identification Table

Element (i) (ii) (iii) (iv)
Radii OA, OB PX, PY, PZ OA, OD PA, PB, PD
Diameter - XZ AD AD
Chords AC, BC XY, YZ AB, BC, CA AC, AB, BC
Centre O P O P

Question 2: Minor Arcs, Major Arcs, and Semicircles Table

Figure Minor Arcs Major Arcs Semicircles
(i) Arc(BC) Arc(BAC) -
(ii) Arc(PR), Arc(RQ) Arc(PQR), Arc(QPR) Arc(PQR)
(iii) Arc(LM), Arc(MN), Arc(NO), Arc(OL) Arc(MNL), Arc(NOM), Arc(OLN), Arc(LMO) Arc(LMN), Arc(MNO), Arc(NOL), Arc(OLM)
(iv) Arc(BC), Arc(CA), Arc(AD), Arc(DB) Arc(CAB), Arc(ADC), Arc(DBA), Arc(BCD) Arc(BCA), Arc(CAD), Arc(ADB), Arc(DBC)

Review Exercise 9: Comprehensive Review

Question 1: Multiple Choice Questions (MCQs)

(i) Which polygon cannot be regular? (a) kite
(ii) Interior angle of regular hexagon: (c) 120°
(iii) Interior angle 144°, number of sides: (d) 10
(iv) Polygon with all sides equal: (a) regular
(v) Sum of interior angles: (b) (n - 2)180°
(vi) Sum of exterior angles: (c) 360°
(vii) Regular triangle: (c) Equilateral
(viii) Concave polygon: (d) Arrowhead / reflex shape
(ix) Interior + exterior at vertex: (a) 180°
(x) Two diagonals of square are: (b) equal
(xi) Diagonals unequal and perpendicular: (b) rhombus, kite
(xii) Diagonals intersect perpendicularly: (d) kite
(xiii) Parallelogram with equal bisecting diagonals: (a) square
(xiv) Segment joining centre to boundary: (c) radial segment
(xv) Longest chord of circle: (c) diameter
Question 2: Regular Heptagon (n = 7).
(i) Exterior = 360° / 7 ≈ 51.43°, Interior = 180° - 51.43° ≈ 128.57°.
(ii) Sum of interior angles = (7-2)×180° = 5×180° = 900°.
Question 3: Central & Polygon Angles.
(i) Central angle with interior 135° = 180° - 135° = 45°.
(ii) Regular pentagon angles: x = 72°, y = 144°.
Question 4: Unknown Angles in Figures (i) to (viii).
(i) a = 44° | (ii) n = 45° | (iii) x = 33° (Angles: 36°, 99°, 125°)
(iv) x = 63°, 126° | (v) b = 105° | (vi) Angles: 123°, 121°, 118°
(vii) Angles: 110°, 105°, 126° | (viii) x = 132° (or 120°).
Question 5: Circle Elements.
Centre: O, Radii: OA, OB, OC, OD, Diameters: AB, CD, Chord: BC.
Minor arcs: AC, CB, BD, DA; Major arcs: CBA, BDC, DAB, ACD.
Minor segment: Shaded region; Major segment: Unshaded region.
Question 6: In rhombus ABCD, find x and y.
Diagonals intersect perpendicularly at 90°: x = 32°, y = 58°.

⚡ Unit 9 Master Formula Cheat Sheet

Interior Angle Sum:
Sum = (n - 2) × 180°
Regular Polygon: [(n - 2) × 180°] / n
Exterior Angle Sum:
Sum = 360° (Always)
Regular Polygon: 360° / n
Diagonals:
From 1 vertex: n - 3
Total diagonals: [n(n - 3)] / 2
Circle Formulas:
Diameter d = 2r
Central Angle = 360° / n = 180° - Interior Angle

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Self-Assessment Practice

Test Your Knowledge on Chapter 9: Class 7 Mathematics Ch 9 Mastery Guide: Angles of Polygons, Regular Polygons, Quadrilaterals & Circle Geometry (FBISE)

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