Example Question - hexagon angles
Here are examples of questions we've helped users solve.
Calculating Interior Angles of a Hexagon
The diagram provided is of a hexagon, which has six sides. The sum of the interior angles of a hexagon can be calculated using the formula: Sum of interior angles = (n - 2) × 180° where n is the number of sides in the polygon. For a hexagon, n = 6. So the sum of the interior angles of our hexagon will be: (6 - 2) × 180° = 4 × 180° = 720° From the image, we can see three interior angles are shown. Two of them have the measures of 120° and 135°, and the third angle is labeled as x. We also know that the remaining three interior angles are equal, which means each one has the same measure as x. Since the sum of all the interior angles is 720°, we can set up the following equation: 120° + 135° + x + x + x + x = 720° Now, we combine the known angles and the x's: 255° + 4x = 720° Subtracting 255° from both sides gives us: 4x = 720° - 255° 4x = 465° Finally, we solve for x by dividing both sides by 4: x = 465° ÷ 4 x = 116.25° So, each of the remaining three interior angles, including the one labeled as x in the diagram, measures 116.25°.
Solving for x in a Hexagon with Known Angles
To solve for \( x \) in the hexagon shown in the image, we will first determine the sum of the interior angles of a hexagon. The sum of the interior angles of any polygon can be found using the formula: \[ S = (n - 2) \times 180^\circ \] where \( S \) is the sum of interior angles and \( n \) is the number of sides. For a hexagon (\( n = 6 \)), this formula gives: \[ S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \] The problem states that three of the angles we know (120°, x, and 135°), and the remaining three interior angles are equal. Let's call each of the three unknown equal angles \( y \). We can set up an equation because we know that the sum of all angles must equal 720°: \[ 120^\circ + x + 135^\circ + 3y = 720^\circ \] Combining the known angles gives us: \[ 255^\circ + x + 3y = 720^\circ \] Now subtract 255° from both sides of the equation to solve for \( x + 3y \): \[ x + 3y = 720^\circ - 255^\circ \] \[ x + 3y = 465^\circ \] Since we do not have the individual values for \( x \) and \( y \), let's find \( x \) in terms of \( y \). Now we express \( x \) as: \[ x = 465^\circ - 3y \] To find the value of one of the equal angles \( y \), we need additional information which the question seems to omit. However, in typical hexagon problems, if the angles are not provided, it may be assumed that the hexagon is a regular hexagon, where all angles would be equal. In this case, the problem states that three angles are equal and the others are not, making it impossible to calculate \( x \) without further information about \( y \). Please check the problem again to see if there’s any missing information that would allow us to solve for \( x \).
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