Given an octagon, we need to find the ratio of longest diagonal to the shortest diagonal in the regular octagon.

1. Geometry Q11

2. Qn: Diagonals of octagon
(a) 3 : 1 (b) 2 : 1
(c) 2 : 3 (d) 2 : 1
What is the ratio of longest diagonal to the shortest diagonal in a
regular octagon?

3. Soln: Diagonals of octagon
Consider regular octagon ABCDEFGH
What is the ratio of longest diagonal to the shortest diagonal in a
regular octagon?
B
A
C
E
D
F
H
G
P Q
a
a
a
a a
a
a
a

4. Soln: Diagonals of octagon
Its longest diagonal would be AE or BF or CG or DH.
Let us try to find out AE.
Join AD and draw BP  AD and CQ  AD.
PQ = a
AP = QD
a2 = BP2 + AP2  a2 = 2 AP2 {since BP=AP}
 a = 2AP  AP =
𝑎
√2
AD =AP + PQ + QD =
𝑎
√2
+ a +
𝑎
√2
What is the ratio of longest diagonal to the shortest diagonal in a
regular octagon?

5. Soln: Diagonals of octagon
 a + a2
AE2 = AD2 + DE2
AE2 = (a + a2) 2 + a2
AE2 = (a2 + 2 x a x 22 + 2a2) + a2
AE2 = a2 (1 + 22 + 2) + a2
 a2 (4 + 22)
Shortest diagonal = AC or CE
AC2 = AB2 + BC2 – 2AB × BC cos135
What is the ratio of longest diagonal to the shortest diagonal in a
regular octagon?

6. Soln: Diagonals of octagon
(Alternatively, we can deduce this using AC2 = AQ2 + QC2. We use cosine rule
just to get some practice on a different method.)
= a2 + a2 – 2a2 × (
−1
√2
)
= 2a2 + 2a2
= a2 (2 + 2)
AE2 = a2 (4 + 22)
AE2
AC2 =
a2 (4 + 22)
a2 (2 + 2)
= 2
What is the ratio of longest diagonal to the shortest diagonal in a
regular octagon?

7. Soln: Diagonals of octagon
AE
AC
= 2
Remember, for a regular octagon.
Each internal angle = 135
Each external angle = 45 
So, we get a bunch of squares and isosceles right–angled s if we draw
diagonals.
A regular hexagon breaks into equilateral triangles. A regular octagon breaks
into isosceles right angled triangles.
Answer choice (d)
What is the ratio of longest diagonal to the shortest diagonal in a
regular octagon?