How to find an angle in other polygons - ISEE Upper Level Quantitative Reasoning
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How many degrees are in an internal angle of a regular heptagon?
How many degrees are in an internal angle of a regular heptagon?
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The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:
The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:
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What is the measure of an interior angle of a regular nonagon?
What is the measure of an interior angle of a regular nonagon?
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The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:
The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:
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What is the sum of all the interior angles of a decagon (a polygon with ten sides)?
What is the sum of all the interior angles of a decagon (a polygon with ten sides)?
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The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
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If each angle in a pentagon is equal to 
, what is the value of 
?
If each angle in a pentagon is equal to
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The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
To find the value of each angle, we divide 540 by 5. This results in 108 degrees.
Thus,
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
To find the value of each angle, we divide 540 by 5. This results in 108 degrees.
Thus,
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What is the value of an angle (to the nearest degree) in a polygon with 
sides if all the angles are equal to one another?
What is the value of an angle (to the nearest degree) in a polygon with
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The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
Given that there are 3,600 degrees total in a polygon with 22 sides, the number of degrees in each angle can be found by dividing 3,600 by 22. To the nearest degree, this results in 164 degrees. Therefore, 164 is the correct answer.
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
Given that there are 3,600 degrees total in a polygon with 22 sides, the number of degrees in each angle can be found by dividing 3,600 by 22. To the nearest degree, this results in 164 degrees. Therefore, 164 is the correct answer.
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What is 
of the total number of degrees in a 9-sided polygon?
What is
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The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Therefore, the equation for the sum of the angles in a 9 sided polygon would be:
Therefore, 
of the total sum of degrees in a 9 sided polygon would be equal to 180 degrees.
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Therefore, the equation for the sum of the angles in a 9 sided polygon would be:
Therefore,
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Pentagon 
is regular. What is the measure of 
?
Note: Figure NOT drawn to scale.
Refer to the above diagram. Pentagon
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The answer can be more clearly seen by extending 
to a ray 
:
Note that angles have been newly numbered.
and 
are exterior angles of a (five-sided) regular pentagon in relation to two parallel lines, so each has a measure of 
. 
is a corresponding angle to 
, so its measure is also 
.
By angle addition,
The answer can be more clearly seen by extending
Note that angles have been newly numbered.
By angle addition,
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In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of 
?
In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of
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The answer can be more clearly obtained by extending the top of the two parallel lines as follows:
Note that two angles have been newly labeled.
is an interior angle of a regular heptagon and therefore has measure
By the Isosceles Triangle Theorem, since the two sides of the heptagon that help form the triangle are congruent, so are the two acute angles, and
is supplementary to 
, so
The answer can be more clearly obtained by extending the top of the two parallel lines as follows:
Note that two angles have been newly labeled.
By the Isosceles Triangle Theorem, since the two sides of the heptagon that help form the triangle are congruent, so are the two acute angles, and
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Note: Figure NOT drawn to scale.
In the above figure, Pentagon 
is regular. Give the measure of 
.
Note: Figure NOT drawn to scale.
In the above figure, Pentagon
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The sum of the degree measures of the angles of Quadrilateral 
is 360, so
Each interior angle of a regular pentagon measures
,
which is therefore the measure of 
.
It is also given that 
and 
, so substitute and solve:
The sum of the degree measures of the angles of Quadrilateral
Each interior angle of a regular pentagon measures
which is therefore the measure of
It is also given that
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The seven-sided polygon - or heptagon - in the above diagram is regular. What is the measure of 
?
The seven-sided polygon - or heptagon - in the above diagram is regular. What is the measure of
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In the diagram below, some other angles have been numbered for the sake of convenience.
An interior angle of a regular heptagon has measure
.
This is the measure of 
.
As a result of the Isosceles Triangle Theorem, 
, so
.
This is also the measure of 
.
By angle addition,
Again, as a result of the Isosceles Triangle Theorem, 
, so
In the diagram below, some other angles have been numbered for the sake of convenience.
An interior angle of a regular heptagon has measure
This is the measure of
As a result of the Isosceles Triangle Theorem,
This is also the measure of
By angle addition,
Again, as a result of the Isosceles Triangle Theorem,
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In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of 
?
In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of
Tap to reveal answer
The answer can be more clearly seen by extending the lower right side of the heptagon to a ray, as shown:
Note that angles have been newly numbered.
and 
are exterior angles of a (seven-sided) regular heptagon, so each has a measure of 
. 
is a corresponding angle to 
in relation to two parallel lines, so its measure is also 
.
By angle addition,
The answer can be more clearly seen by extending the lower right side of the heptagon to a ray, as shown:
Note that angles have been newly numbered.
By angle addition,
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The measures of the angles of a nine-sided polygon, or nonagon, form an arithmetic sequence. The least of the nine degree measures is 
. What is the greatest of the nine degree measures?
The measures of the angles of a nine-sided polygon, or nonagon, form an arithmetic sequence. The least of the nine degree measures is
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The total of the degree measures of any nine-sided polygon is
.
In an arithmetic sequence, the terms are separated by a common difference, which we will call 
. Since the least of the degree measures is 
, the measures of the angles are
Their sum is
The greatest of the angle measures, in degrees, is
is the correct choice.
The total of the degree measures of any nine-sided polygon is
In an arithmetic sequence, the terms are separated by a common difference, which we will call
Their sum is
The greatest of the angle measures, in degrees, is
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Note: Figure NOT drawn to scale.
In the above figure, Pentagon 
is regular. Give the measure of 
.
Note: Figure NOT drawn to scale.
In the above figure, Pentagon
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The sum of the degree measures of the angles of Quadrilateral 
is 360, so
.
Each interior angle of a regular pentagon measures
,
which is therefore the measure of both 
and 
.
and 
form a linear pair, making them supplementary. Since 
,
.
Substitute and solve:
The sum of the degree measures of the angles of Quadrilateral
Each interior angle of a regular pentagon measures
which is therefore the measure of both
Substitute and solve:
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The measures of the angles of a ten-sided polygon, or decagon, form an arithmetic sequence. The least of the ten degree measures is 
. What is the greatest of the ten degree measures?
The measures of the angles of a ten-sided polygon, or decagon, form an arithmetic sequence. The least of the ten degree measures is
Tap to reveal answer
The total of the degree measures of any ten-sided polygon is
.
In an arithmetic sequence, the terms are separated by a common difference, which we will call 
. Since the least of the degree measures is 
, the measures of the angles are
Their sum is
The greatest of the angle measures is
However, an angle measure cannot exceed 
. The correct choice is that this polygon cannot exist.
The total of the degree measures of any ten-sided polygon is
In an arithmetic sequence, the terms are separated by a common difference, which we will call
Their sum is
The greatest of the angle measures is
However, an angle measure cannot exceed
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The measures of the angles of an octagon form an arithmetic sequence. The greatest of the eight degree measures is 
. What is the least of the eight degree measures?
The measures of the angles of an octagon form an arithmetic sequence. The greatest of the eight degree measures is
Tap to reveal answer
The total of the degree measures of any eight-sided polygon is
.
In an arithmetic sequence, the terms are separated by a common difference, which we will call 
. Since the greatest of the degree measures is 
, the measures of the angles are
Their sum is
The least of the angle measures is
The correct choice is 
.
The total of the degree measures of any eight-sided polygon is
In an arithmetic sequence, the terms are separated by a common difference, which we will call
Their sum is
The least of the angle measures is
The correct choice is
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How many degrees are in an internal angle of a regular heptagon?
How many degrees are in an internal angle of a regular heptagon?
Tap to reveal answer
The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:
The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:
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What is the measure of an interior angle of a regular nonagon?
What is the measure of an interior angle of a regular nonagon?
Tap to reveal answer
The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:
The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:
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What is the sum of all the interior angles of a decagon (a polygon with ten sides)?
What is the sum of all the interior angles of a decagon (a polygon with ten sides)?
Tap to reveal answer
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
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If each angle in a pentagon is equal to , what is the value of ?
If each angle in a pentagon is equal to
Tap to reveal answer
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
To find the value of each angle, we divide 540 by 5. This results in 108 degrees.
Thus,
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
To find the value of each angle, we divide 540 by 5. This results in 108 degrees.
Thus,
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What is the value of an angle (to the nearest degree) in a polygon with sides if all the angles are equal to one another?
What is the value of an angle (to the nearest degree) in a polygon with
Tap to reveal answer
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
Given that there are 3,600 degrees total in a polygon with 22 sides, the number of degrees in each angle can be found by dividing 3,600 by 22. To the nearest degree, this results in 164 degrees. Therefore, 164 is the correct answer.
The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.
Given that a hexagon has 6 angles, the total number of angles will be:
Given that there are 3,600 degrees total in a polygon with 22 sides, the number of degrees in each angle can be found by dividing 3,600 by 22. To the nearest degree, this results in 164 degrees. Therefore, 164 is the correct answer.
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