Angles of Polygons practice questions

Steps to Solve

1. Formula for the sum of interior angles

The formula for the sum of the interior angles of a polygon is given by:

$S = (n-2) \times 180^\circ$

where $n$ is the number of sides of the polygon.

2. Formula for each interior angle of a regular polygon

The formula to find the measure of each interior angle of a regular polygon is:

$A = \frac{(n-2) \times 180^\circ}{n}$

where $n$ is the number of sides of the regular polygon.

3. Sum of interior angles of a 35-gon

For a 35-gon, $n = 35$. Substitute this value into the formula for the sum S:

$S = (35-2) \times 180^\circ = 33 \times 180^\circ = 5940^\circ$

4. Finding the seventh angle of a heptagon

A heptagon has 7 sides. The sum of its interior angles is:

$S = (7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$

The sum of the six given angles is:

$107^\circ + 139^\circ + 131^\circ + 110^\circ + 145^\circ + 128^\circ = 760^\circ$

Let the seventh angle be $x$. Then:

$760^\circ + x = 900^\circ$

$x = 900^\circ - 760^\circ = 140^\circ$

5. Finding the number of sides given the sum of interior angles

Given that the sum of the interior angles is $3780^\circ$, we have:

$(n-2) \times 180^\circ = 3780^\circ$

$n-2 = \frac{3780^\circ}{180^\circ} = 21$

$n = 21 + 2 = 23$

6. Measure of each interior angle of a regular 18-gon

For a regular 18-gon, $n = 18$. Using the formula for each interior angle:

$A = \frac{(18-2) \times 180^\circ}{18} = \frac{16 \times 180^\circ}{18} = 16 \times 10^\circ = 160^\circ$

7. Sum of exterior angles of any polygon

The sum of the exterior angles of any polygon is always $360^\circ$.

8. Measure of each exterior angle of a regular 30-gon

For a regular 30-gon, the measure of each exterior angle is:

$\frac{360^\circ}{30} = 12^\circ$

9. Finding the number of sides given an exterior angle

If the exterior angle of a regular polygon is $24^\circ$, then the number of sides is:

$n = \frac{360^\circ}{24^\circ} = 15$

10. Finding the number of sides given an interior angle

If the interior angle of a regular polygon is $162^\circ$, then the exterior angle is $180^\circ - 162^\circ = 18^\circ$. The number of sides is:

$n = \frac{360^\circ}{18^\circ} = 20$

11. Solving for x in the heptagon

The sum of the interior angles of a heptagon ($n=7$) is $(7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$. Thus:

$139 + (10x - 11) + (8x + 15) + 121 + (9x - 17) + (9x - 14) + 153 + (7x - 6) = 900$

Combining like terms:

$(10x + 8x + 9x + 9x + 7x) + (139 - 11 + 15 + 121 - 17 - 14 + 153 - 6) = 900$

$43x + 380 = 900$

$43x = 900 - 380$

$43x = 520$

$x = \frac{520}{43} \approx 12.09$

12. Solving for x in the regular pentagon

The figure is a regular pentagon, so all its interior angles are equal. The sum of the interior angles of a pentagon is $(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$. Each interior angle of a regular pentagon is $\frac{540^\circ}{5} = 108^\circ$.

Therefore, $7x + 31 = 108$

$7x = 108 - 31$

$7x = 77$

$x = \frac{77}{7} = 11$

13. Solving for x in the hexagon

The figure is a hexagon ($n=6$), so the sum of its interior angles is $(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$.

$(11x - 5) + (4x + 29) + 67 + (9x - 15) + 61 + (6x + 13) = 720$

Combining like terms:

$(11x + 4x + 9x + 6x) + (-5 + 29 + 67 - 15 + 61 + 13) = 720$

$30x + 150 = 720$

$30x = 720 - 150$

$30x = 570$

$x = \frac{570}{30} = 19$