3. AB, BC and CD are the three consecutive sides of a regular polygon. If ∠BAC = 15o; find,
(i) Each interior angle of the polygon.
(ii) Each exterior angle of the polygon.
(iii) Number of sides of the polygon.
Solution:
(i) Let each angle of measure x degree.
Therefore, measure of each angle will be:
x – 180o – 2 × 15o = 150o
(ii) Let each angle of measure x degree.
Therefore, measure of each exterior angle will be:
x = 180o – 150o
= 30o
(iii) Let the number of each sides is n.
Now we can write
n . 150o = (2n – 4) × 90o
180o n – 150o n = 360o
300 n = 360o
n = 12
Thus, the number of sides are 12.
2. In a pentagon ABCDE, AB is parallel to DC and ∠A : ∠E : ∠D = 3 : 4 : 5. Find angle E.
1. Two angles of an eight-sided polygon are 142o and 176o. If the remaining angles are equal to each other; find the magnitude of each of the equal angles.
Solution:
Let the measure of each equal sides of the polygon is x.
Then we can write:
142o + 176o + 6x = (2 × 8 – 4) 90o
6x = 1080o – 318o
6x = 762o
x = 127o
Thus, the measure of each equal angles is 127o
5. Three angles of a seven-sided polygon are 132o each and the remaining four angles are equal. Find the value of each equal angle.
Solution:
Let the measure of each equal angles are x.
Then we can write
3 × 132o + 4x = (2 × 7 – 4) 90o
4x = 900o – 396o
4x = 504o
x = 126o
4. In a polygon there are 5 right angles and the remaining angles are equal to 195o each. Find the number of sides in the polygon.
Solution:
Let the number of sides of the polygon is n and there are k angles with measure 195o.
Therefore, we can write:
5 × 90o + k × 195o = (2n – 4) 90o
180on – 195o k = 450o – 360o
180on – 195ok = 90o
12n – 13k = 90o
In this linear equation n and k must be integer.
Therefore, to satisfy this equation the minimum value of k must be 6 to get n as integer.
Hence the number of sides are: 5 + 6 = 11.
3. One angle of a six-sided polygon is 140o and the other angles are equal. Find the measure of each equal angle.
Solution:
Let the measure of each equal angles are x.
Then we can write
140o + 5x = (2 × 6 – 4) × 90o
140o + 5x = 720o
5x = 580o
x = 116o
Therefore, the measure of each equal angles are 116o
2. The angles of a pentagon are in the ratio 4 : 8 : 6 : 4 : 5. Find each angle of the pentagon.
Let the angles of the pentagon are 4x, 8x, 6x, 4x and 5x.
Thus, we can write
4x + 8x + 6x + 4x + 5x = 540o
27x = 540o
x = 20o
Hence the angles of the pentagon are:
4×20o = 80o, 8×20o = 160o, 6×20o= 120o, 4×20o= 80o, 5×20o= 100o
1. The sum of the interior angles of a polygon is four times the sum of its exterior angles. Find the number of sides in the polygon.
4. The length, breadth and height of cuboid are in the ratio 6 : 5 : 3. If its total surface area is 504 cm2, find its volume.
720 cubed cm
3. The external dimensions of an open wooden box are 65 cm, 34 cm and 25 cm. If the box is made up of wood 2 cm thick, find the capacity of the box and the volume of wood used to make it.
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