Pentagon Angle Calculation Using Ratios
The sum of the interior angles of a polygon with n sides is given by the formula:
$ \text{Sum} = (n-2) \times 180^\circ $
For a pentagon, n = 5. Therefore, the sum of its interior angles is:
$ (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ $
Ratio Analysis for Pentagon Angles
The interior angles are in the ratio 1 : 3 : 5 : 7 : 11. Let the angles be represented as $1x$, $3x$, $5x$, $7x$, and $11x$, where x is a common multiplier.
The sum of these angles is:
$ 1x + 3x + 5x + 7x + 11x = 27x $
Determining the Smallest Angle
Equating the sum of the ratio parts to the total sum of the interior angles of the pentagon:
$ 27x = 540^\circ $
Now, solve for x:
$ x = \frac{540^\circ}{27} $
$ x = 20^\circ $
The smallest interior angle corresponds to the smallest part of the ratio, which is $1x$. Therefore, the smallest angle is:
$ 1x = 1 \times 20^\circ = 20^\circ $
Similar Questions
- The adjacent sides of a parallelogram are $4a$ and $3a$. If the angle between them is $60^{\circ}$, then one of the diagonals of the parallelogram will be:
- Every interior angle of a regular octagon is $135^\circ$. Find the exterior angle of the octagon.
- PQ is a diameter of circle whose centre is O. If a point R lies on a circle and $\angle RPO$ is $42^\circ$, then find $\angle RQP$.
- ABC is an equilateral triangle and O is its circumcentre. If the side of triangle is 6 cm, then the $\angle BOC$ is:
- If every interior angle of a regular polygon is $144^\circ$, then the polygon has ______ sides.
- The ratio of the numbers of sides of two regular polygons is $1 : 2$. If each interior angle of the first polygon is $140^\circ$, then the measure of each interior angle of the second polygon is:
- If the sum of the interior angles of a regular polygon is equal to four times the sum of its exterior angles, then what is the number of diagonals in the polygon?
- Find the ratio of the measure of an angle of a regular pentagon to that of a regular octagon.
- What is the measure of each exterior angle of a regular octagon?
- The difference between the interior and exterior angles at a vertex of a regular polygon is $140^\circ$. The number of sides of the polygon is:
Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
