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 + a2
AE2 = AD2 + DE2
AE2 = (a + a2) 2 + a2
AE2 = (a2 + 2 x a x 22 + 2a2) + a2
AE2 = a2 (1 + 22 + 2) + a2
a2 (4 + 22)
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 + 22)
AE2
AC2 =
a2 (4 + 22)
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?