Diagonals of Polygons

Show your work for each problem.

1.
What is a diagonal in a polygon?
1. a)A side of the polygon
2. b)A line connecting two non-adjacent vertices
3. c)The perimeter of the polygon
4. d)An angle inside the polygon
Answer: A line connecting two non-adjacent vertices
A diagonal is a line segment that connects two vertices that are not next to each other (non-adjacent). Sides connect adjacent vertices.
2.
How many diagonals does a triangle have?
1. a)0
2. b)3
3. c)1
4. d)2
Answer: 0
A triangle has 0 diagonals because every vertex is connected to every other vertex by a side. Using the formula:
3.
How many diagonals does a quadrilateral (4 sides) have?
Answer: 2
. A quadrilateral has 2 diagonals.
4.
How many diagonals does a pentagon (5 sides) have?
Answer: 5
. A pentagon has 5 diagonals.
5.
In a square ABCD, which segments are diagonals?
1. a)AB and CD
2. b)AB and BC
3. c)AC and BD
4. d)BC and DA
Answer: AC and BD
In square ABCD, vertices A and C are opposite (not adjacent), as are B and D. So AC and BD are the diagonals. AB, BC, CD, and DA are sides because they connect adjacent vertices.
6.
How many diagonals does a hexagon (6 sides) have?
Answer: 9
. A hexagon has 9 diagonals.
7.
How many diagonals does an octagon (8 sides) have?
Answer: 20
. An octagon has 20 diagonals.
8.
Calculate the number of diagonals in a heptagon (7 sides).
Answer: 14
1. What is for a heptagon? 7
2. What is ? 4
3. What is ? 28
4. What is ? 14
9.
From one vertex of a hexagon, how many diagonals can be drawn?
1. a)5
2. b)6
3. c)4
4. d)3
Answer: 3
From one vertex, you can draw diagonals to other vertices (not to yourself, not to your 2 adjacent neighbors). For a hexagon: diagonals from each vertex.
10.
How many diagonals does a decagon (10 sides) have?
Answer: 35
. A decagon has 35 diagonals.
11.
A decorative tile is shaped like a nonagon (9 sides). How many diagonals does it have?
Answer: 27
1. What is for a nonagon? 9
2. What is ? 6
3. What is ? 54
4. What is ? 27
12.
A polygon has 5 diagonals. How many sides does it have?
Answer: 5
1. Set up the equation: . Multiply both sides by 2. What is ? 10
2. Expand: . Rearrange to standard form: . Factor. What are the two factors? (n-5)(n+2)
3. From , we get or . Which is valid? 5
13.
A polygon has 35 diagonals. How many sides does it have?
Answer: 10
. Since , . It's a decagon.
14.
A polygon has 44 diagonals. How many sides does it have?
Answer: 11
1. Set up: . Multiply by 2. What is ? 88
2. Expand: . Write in standard form: . What are two numbers that multiply to -88 and add to -3? -11 and 8
3. Factor: . What is the positive solution? 11
15.
A dodecagon (12 sides) has how many more diagonals than an octagon (8 sides)?
1. a)34
2. b)24
3. c)44
4. d)54
Answer: 34
Dodecagon: . Octagon: . Difference: .
16.
How many diagonals does a 15-sided polygon (pentadecagon) have?
Answer: 90
. A pentadecagon has 90 diagonals.