RS Aggarwal Class 8 Mathematics Solutions Chapter 14 Polygons

Exercise 14A

Question 1. Find the exterior angle of a regular polygon with (i) 5 sides (ii) 6 sides (iii) 7 sides (iv) 10 sides (v) 15 sides.
Answer: The exterior angle formula for an n-sided regular polygon is \( \left(\frac{360}{n}\right)^{\circ} \).
(i) Pentagon: \( n = 5 \implies \left(\frac{360}{5}\right)^{\circ} = 72^{\circ} \)
(ii) Hexagon: \( n = 6 \implies \left(\frac{360}{6}\right)^{\circ} = 60^{\circ} \)
(iii) Heptagon: \( n = 7 \implies \left(\frac{360}{7}\right)^{\circ} = 51.43^{\circ} \)
(iv) Decagon: \( n = 10 \implies \left(\frac{360}{10}\right)^{\circ} = 36^{\circ} \)
(v) 15-sided polygon: \( n = 15 \implies \left(\frac{360}{15}\right)^{\circ} = 24^{\circ} \)
In simple words: Divide 360 by the number of sides to find each exterior angle.

Exam Tip: Remember that the sum of all exterior angles of any polygon is always 360°. Use this formula whenever you need to find one exterior angle of a regular polygon.

Question 2. Can you have a polygon with each exterior angle equal to 50°?
Answer: Each exterior angle of an n-sided polygon is \( \left(\frac{360}{n}\right)^{\circ} \). If the exterior angle is 50°, then \( \frac{360}{n} = 50 \implies n = 7.2 \). Since the number of sides must be a whole number and 7.2 is not an integer, it is not possible to have a polygon where each exterior angle equals 50°.
In simple words: When you divide 360 by 50, you get 7.2 sides, which is impossible. Polygons must have a whole number of sides.

Exam Tip: For a valid regular polygon, 360 divided by the exterior angle must always give a whole number.

Question 3. Find the interior angle of a regular polygon with (i) 10 sides (ii) 15 sides.
Answer: For a regular polygon with n sides, each interior angle is \( 180 - \left(\frac{360}{n}\right) \).
(i) For a 10-sided polygon: Each exterior angle is \( \frac{360}{10} = 36^{\circ} \), so each interior angle is \( 180 - 36 = 144^{\circ} \)
(ii) For a 15-sided polygon: Each exterior angle is \( \frac{360}{15} = 24^{\circ} \), so each interior angle is \( 180 - 24 = 156^{\circ} \)
In simple words: An interior angle and its matching exterior angle always add to 180°. First find the exterior angle, then subtract it from 180.

Exam Tip: The relationship "interior angle + exterior angle = 180°" is the key shortcut for all regular polygon problems.

Question 4. If each interior angle of a regular polygon is 100°, find the number of sides.
Answer: Each interior angle of a regular polygon with n sides is \( 180 - \left(\frac{360}{n}\right) = \frac{180n - 360}{n} \). If each interior angle is 100°:
\( 100 = \frac{180n - 360}{n} \)
\( 100n = 180n - 360 \)
\( 180n - 100n = 360 \)
\( 80n = 360 \)
\( n = \frac{360}{80} = 4.5 \)
Since n is not a whole number, it is not possible to have a regular polygon with each interior angle equal to 100°.
In simple words: Setting up the equation and solving gives us 4.5 sides, which is impossible. A polygon cannot have a fractional number of sides.

Exam Tip: Always verify that your answer for n is a whole number - if not, the polygon cannot exist.

Question 5. Find the sum of the interior angles of (i) a pentagon (ii) a hexagon (iii) a nonagon (iv) a 12-sided polygon.
Answer: The sum of the interior angles of an n-sided polygon is \( (n - 2) \times 180^{\circ} \).
(i) Pentagon: \( n = 5 \implies (5 - 2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ} \)
(ii) Hexagon: \( n = 6 \implies (6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ} \)
(iii) Nonagon: \( n = 9 \implies (9 - 2) \times 180^{\circ} = 7 \times 180^{\circ} = 1260^{\circ} \)
(iv) 12-sided polygon: \( n = 12 \implies (12 - 2) \times 180^{\circ} = 10 \times 180^{\circ} = 1800^{\circ} \)
In simple words: Subtract 2 from the number of sides and then multiply by 180°. This tells you the total of all the inside angles.

Exam Tip: The formula works for any polygon - triangle, quadrilateral, or any shape with straight sides.

Question 6. Find the number of diagonals in (i) a heptagon (ii) an octagon (iii) a 12-sided polygon.
Answer: The number of diagonals in an n-sided polygon is \( \frac{n(n-3)}{2} \).
(i) Heptagon: \( n = 7 \implies \frac{7(7-3)}{2} = \frac{7 \times 4}{2} = \frac{28}{2} = 14 \)
(ii) Octagon: \( n = 8 \implies \frac{8(8-3)}{2} = \frac{8 \times 5}{2} = \frac{40}{2} = 20 \)
(iii) 12-sided polygon: \( n = 12 \implies \frac{12(12-3)}{2} = \frac{12 \times 9}{2} = \frac{108}{2} = 54 \)
In simple words: Subtract 3 from the number of sides, multiply by the number of sides, then divide by 2. This counts all the line segments connecting non-adjacent vertices.

Exam Tip: Remember - (n-3) represents how many diagonals can be drawn from one vertex, and dividing by 2 avoids counting each diagonal twice.

Question 7. Find the number of sides of a regular polygon if each exterior angle is (i) 40° (ii) 36° (iii) 72° (iv) 30°.
Answer: The sum of all exterior angles of a regular polygon is 360°. The number of sides is found by dividing 360 by each exterior angle.
(i) Exterior angle = 40° \( \implies \) Number of sides = \( \frac{360}{40} = 9 \)
(ii) Exterior angle = 36° \( \implies \) Number of sides = \( \frac{360}{36} = 10 \)
(iii) Exterior angle = 72° \( \implies \) Number of sides = \( \frac{360}{72} = 5 \)
(iv) Exterior angle = 30° \( \implies \) Number of sides = \( \frac{360}{30} = 12 \)
In simple words: Divide 360 by the exterior angle to find the number of sides instantly.

Exam Tip: This is the quickest way to move from exterior angle to number of sides - memorize that exterior angles always sum to 360°.

Question 8. In a quadrilateral ABCD, the angles are \( m\angle ADC = 50^{\circ}, m\angle DAB = 115^{\circ}, m\angle BCD = 90^{\circ} \). Find x where \( m\angle ABC = x \).
Answer: The sum of all interior angles in a quadrilateral (n - 2) × 180° where n = 4 is \( (4 - 2) \times 180^{\circ} = 360^{\circ} \).
Given: \( m\angle ADC = 50^{\circ}, m\angle DAB = 115^{\circ}, m\angle BCD = 90^{\circ} \)
\( 130^{\circ} + 65^{\circ} + 90^{\circ} + m\angle ABC = 360^{\circ} \)
\( 285^{\circ} + m\angle ABC = 360^{\circ} \)
\( m\angle ABC = 75^{\circ} \)
\( m\angle CBF = 180 - 75 = 105^{\circ} \)
Therefore, \( x = 105 \)
In simple words: Add up three angles and subtract from 360° to get the fourth angle.

Exam Tip: Always check that all four angles sum to exactly 360° before finalizing your answer.

Question 9. The interior angle of a regular n-sided polygon is 108°. Find x where the polygon has 5 sides and \( x^{\circ} = 180 - \frac{360}{n} \).
Answer: For a regular n-sided polygon where each interior angle is given, we find n using the formula: Each interior angle = \( 180 - \left(\frac{360}{n}\right) \).
From the given information: \( n = 5 \)
\( x^{\circ} = 180 - \frac{360}{5} = 180 - 72 = 108^{\circ} \)
Therefore, \( x = 108 \)
In simple words: Plug in n = 5 into the interior angle formula to get 108°.

Exam Tip: Verify your work by checking that the exterior angle plus the interior angle equals 180°.

Exercise 14B

Question 1. How many diagonals does a pentagon have?
Answer: For a pentagon: \( n = 5 \). Using the diagonal formula \( \frac{n(n-3)}{2} \):
\( \text{Number of diagonals} = \frac{5(5-3)}{2} = \frac{5 \times 2}{2} = 5 \)
In simple words: A pentagon has 5 diagonals. These are the line segments that connect vertices that are not next to each other.

Exam Tip: You can verify this by drawing a pentagon and counting the diagonals - there should be exactly 5.

Question 2. How many diagonals does a hexagon have?
Answer: For a hexagon: \( n = 6 \). Using the diagonal formula \( \frac{n(n-3)}{2} \):
\( \text{Number of diagonals} = \frac{6(6-3)}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9 \)
In simple words: A hexagon has 9 diagonals in total.

Exam Tip: For every vertex in a hexagon, you can draw 3 diagonals (n - 3 = 6 - 3 = 3), and since there are 6 vertices, dividing by 2 avoids counting each diagonal twice.

Question 3. How many diagonals does an octagon have?
Answer: For an octagon: \( n = 8 \). Using the diagonal formula \( \frac{n(n-3)}{2} \):
\( \frac{8(8-3)}{2} = \frac{8 \times 5}{2} = \frac{40}{2} = 20 \)
In simple words: An octagon has 20 diagonals.

Exam Tip: As the number of sides increases, the number of diagonals grows rapidly - notice how an 8-sided polygon has way more diagonals than a 5-sided polygon.

Question 4. How many diagonals does a 12-sided polygon have?
Answer: For a 12-sided polygon: \( n = 12 \). Using the diagonal formula \( \frac{n(n-3)}{2} \):
\( \frac{12(12-3)}{2} = \frac{12 \times 9}{2} = \frac{108}{2} = 54 \)
In simple words: A 12-sided polygon has 54 diagonals.

Exam Tip: Always subtract 3 first, then multiply by n, then divide by 2 - this three-step order prevents errors.

Question 5. If a polygon has 27 diagonals, how many sides does it have?
Answer: Using the diagonal formula \( \frac{n(n-3)}{2} = 27 \):
\( n(n-3) = 54 \)
\( n^2 - 3n - 54 = 0 \)
\( n^2 - 9n + 6n - 54 = 0 \)
\( n(n - 9) + 6(n - 9) = 0 \)
\( (n - 9)(n + 6) = 0 \)
\( n = - 6 \text{ or } n = 9 \)
Since the number of sides cannot be negative, \( n = 9 \). Therefore, the polygon has 9 sides.
In simple words: Set up the diagonal formula equal to 27 and solve the resulting quadratic equation. The answer must be positive, so only n = 9 works.

Exam Tip: When solving for n, always reject any negative answer - polygons cannot have negative sides.

Question 6. The sum of interior angles of a polygon is 540°. Find each interior angle if it is regular.
Answer: Using the formula for sum of interior angles: \( (n - 2) \times 180^{\circ} = 540^{\circ} \)
\( (n - 2) = \frac{540}{180} = 3 \)
\( n = 5 \)
For a regular polygon with 5 sides, each interior angle is:
\( 180 - \frac{360}{5} = 180 - 72 = 108^{\circ} \)
Alternatively, each angle = \( \frac{540}{5} = 108^{\circ} \)
\( x = 68^{\circ} \)
In simple words: First find how many sides the polygon has using the sum. Then divide the total by the number of sides to get each interior angle.

Exam Tip: Always verify: a 5-sided polygon should have interior angle sum of (5-2) × 180° = 540°, which matches the given information.

Question 7. Each exterior angle of a regular polygon is 40°. How many sides does it have?
Answer: For a regular n-sided polygon, each exterior angle is \( \frac{360}{n} = 40 \)
\( n = \frac{360}{40} = 9 \)
In simple words: Divide 360 by the exterior angle to find the number of sides.

Exam Tip: This is the most direct approach - you don't need to find interior angles first.

Question 8. If each interior angle of a regular polygon is 108°, how many sides does it have?
Answer: Each interior angle for a regular n-sided polygon is \( 180 - \left(\frac{360}{n}\right) \).
If the interior angle is 108°:
\( 180 - \left(\frac{360}{n}\right) = 108 \)
\( \left(\frac{360}{n}\right) = 72 \)
\( n = \frac{360}{72} = 5 \)
In simple words: Rearrange the formula to find the exterior angle first (which is 72°), then divide 360 by this to get the number of sides.

Exam Tip: Notice that interior angle + exterior angle = 180°, so 180° - 108° = 72° gives you the exterior angle immediately.

Question 9. If each interior angle of a regular polygon is 135°, how many sides does it have?
Answer: Each interior angle for a regular n-sided polygon is \( 180 - \left(\frac{360}{n}\right) \).
If the interior angle is 135°:
\( 180 - \left(\frac{360}{n}\right) = 135 \)
\( \left(\frac{360}{n}\right) = 45 \)
\( n = \frac{360}{45} = 8 \)
In simple words: The exterior angle is 180° - 135° = 45°. Divide 360 by 45 to get 8 sides.

Exam Tip: An 8-sided polygon (octagon) is a common shape - knowing that its interior angles are 135° can help you recognize it quickly.

Question 10. A regular polygon has each exterior angle of 30°. How many sides does it have?
Answer: For a regular polygon where each exterior angle is 30°:
\( \frac{360}{n} = 30 \)
\( n = \frac{360}{30} = 12 \)
For a regular polygon with 12 sides:
Each exterior angle = \( \frac{360}{n} = \frac{360}{12} \)
Each interior angle = \( 180 - \frac{360}{n} = 180 - 30 = 150 \)
In simple words: Divide 360 by 30 to get 12 sides. A 12-sided regular polygon is called a dodecagon.

Exam Tip: Notice the pattern - smaller exterior angles mean more sides (the polygon is "rounder"), and larger exterior angles mean fewer sides.

Question 11. A regular polygon has each interior angle of 150°. How many sides does it have?
Answer: For a regular polygon with n sides:
Each exterior angle = \( \frac{360}{n} \)
Each interior angle = \( 180 - \frac{360}{n} \)
\( 180 - \frac{360}{n} = 3\left(\frac{360}{n}\right) \)
\( 180 = 4\left(\frac{360}{n}\right) \)
\( n = \frac{4 \times 360}{180} = 8 \)
In simple words: Set up the interior angle formula equal to 150°, then solve for n.

Exam Tip: Cross-check: if n = 8, then exterior angle = 360 ÷ 8 = 45°, and interior angle = 180 - 45 = 135°. Wait - let me recalculate. If interior angle is 150°, then exterior angle is 180 - 150 = 30°, so n = 360 ÷ 30 = 12 sides.

Question 12. The sum of interior angles of a hexagon is (2n - 4) right angles. What is the value of n?
Answer: The sum of interior angles of a hexagon where \( n = 6 \) is \( (n - 2) \times 180^{\circ} = (6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ} \). This equals \( (2n - 4) \) right angles where one right angle = 90°.
\( (2n - 4) \times 90^{\circ} = 8 \text{ right angles} \)
In simple words: A hexagon's interior angles sum to 720°, which is equivalent to 8 right angles (since 720 ÷ 90 = 8).

Exam Tip: Convert between degree measures and right angles by dividing or multiplying by 90.

Question 13. The sum of interior angles of a polygon is 1080°. How many sides does it have? If it is regular, find each interior angle.
Answer: Using the sum formula: \( (n - 2) \times 90^{\circ} = 1080^{\circ} \)
\( (2n - 4) = 12 \)
\( 2n = 16 \)
\( \text{or } n = 8 \)
Each interior angle = \( 180 - \frac{360}{n} = 180 - \frac{360}{8} = 180 - 45 = 135^{\circ} \)
In simple words: Divide the total interior angle sum by 180 and add 2 to find the number of sides. Then use the interior angle formula to find each angle in a regular polygon.

Exam Tip: Always verify: (8 - 2) × 180 = 6 × 180 = 1080°, which matches perfectly.