Pentagons - ISEE Upper Level: Quantitative Reasoning
Card 1 of 52
The measures of the angles of a pentagon are 
. If 
, what is 
?
The measures of the angles of a pentagon are
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The angles of a pentagon measure a total of 
. From the information, we know that:
If 
, then
The angles of a pentagon measure a total of
If
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A convex pentagon has four angles that measure 
each. What is the measure of the fifth angle?
A convex pentagon has four angles that measure
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The angles of a pentagon measure a total of . If we let the unknown angle measure be 
, then from this information:
Since an angle measure cannot be negative, this pentagon cannot exist.
The angles of a pentagon measure a total of
Since an angle measure cannot be negative, this pentagon cannot exist.
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The measures of the angles of a pentagon are 
. If 
, what is 
?
The measures of the angles of a pentagon are
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The angles of a pentagon measure a total of . From the information, we know:
If 
, then the above becomes:
The angles of a pentagon measure a total of
If
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Solve for 
:
Solve for
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The sum of the interior angles of a pentagon can be determined by the following equation, where 
is the number of sides:
Therefore:
The sum of the interior angles of a pentagon can be determined by the following equation, where
Therefore:
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In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?
In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?
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In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that 
, the sum of 2 of the sides will be 
In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that
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If a pentagon has a perimeter of 85in, what is the length of one side of the pentagon?
If a pentagon has a perimeter of 85in, what is the length of one side of the pentagon?
Tap to reveal answer
A pentagon has 5 equal sides. We know the formula to find perimeter of a pentagon is
where a is the length of one side of the pentagon. Now, we know the total perimeter is 85in. Knowing this, we will substitute into the formula. We get
Therefore, the length of one side of the pentagon is 17in.
A pentagon has 5 equal sides. We know the formula to find perimeter of a pentagon is
where a is the length of one side of the pentagon. Now, we know the total perimeter is 85in. Knowing this, we will substitute into the formula. We get
Therefore, the length of one side of the pentagon is 17in.
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Find the perimeter of a pentagon with a side of length 12in.
Find the perimeter of a pentagon with a side of length 12in.
Tap to reveal answer
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know the length of one side of the pentagon is 12in.
Knowing this, we can substitute into the formula. We get
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know the length of one side of the pentagon is 12in.
Knowing this, we can substitute into the formula. We get
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Find the perimeter of a pentagon with a side having a length of 21in.
Find the perimeter of a pentagon with a side having a length of 21in.
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A pentagon has 5 sides. To find the perimeter, we will use the following formula:
where a is the length of a side of the pentagon.
Now, we know the length of a side of the pentagon is 21in.
Knowing this, we will substitute into the formula. We get
A pentagon has 5 sides. To find the perimeter, we will use the following formula:
where a is the length of a side of the pentagon.
Now, we know the length of a side of the pentagon is 21in.
Knowing this, we will substitute into the formula. We get
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Find the perimeter of a pentagon with a side having a length of 15cm.
Find the perimeter of a pentagon with a side having a length of 15cm.
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To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon. Because a pentagon has 5 equal sides, we can use any side in the formula.
So, we know the length of one side of the pentagon is 15cm. So, we can substitute. We get
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon. Because a pentagon has 5 equal sides, we can use any side in the formula.
So, we know the length of one side of the pentagon is 15cm. So, we can substitute. We get
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Find the perimeter of a pentagon with a side having a length of 9in.
Find the perimeter of a pentagon with a side having a length of 9in.
Tap to reveal answer
A pentagon has 5 equal sides. To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know a side of the pentagon has a length of 9in. So, we get
A pentagon has 5 equal sides. To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know a side of the pentagon has a length of 9in. So, we get
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A pentagon has a side of length 11in. Find the perimeter.
A pentagon has a side of length 11in. Find the perimeter.
Tap to reveal answer
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know the length of one side of the pentagon is 11in.
Knowing this, we can substitute into the formula. We get
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know the length of one side of the pentagon is 11in.
Knowing this, we can substitute into the formula. We get
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A pentagon with a perimeter of one mile has three congruent sides. The second-longest side is 250 feet longer than any one of those three congruent sides, and the longest side is 500 feet longer than the second-longest side.
Which is the greater quantity?
(a) The length of the longest side of the pentagon
(b) Twice the length of one of the three shortest sides of the pentagon
A pentagon with a perimeter of one mile has three congruent sides. The second-longest side is 250 feet longer than any one of those three congruent sides, and the longest side is 500 feet longer than the second-longest side.
Which is the greater quantity?
(a) The length of the longest side of the pentagon
(b) Twice the length of one of the three shortest sides of the pentagon
Tap to reveal answer
If each of the five congruent sides has measure 
, then the other two sides have measures 
and 
. Add the sides to get the perimeter, which is equal to 
feet, the solve for 
:
feet
Now we can compare (a) and (b).
(a) The longest side has measure 
feet.
(b) The three shortest sides each have length 856 feet; twice this is 
feet.
(b) is greater.
If each of the five congruent sides has measure
Now we can compare (a) and (b).
(a) The longest side has measure
(b) The three shortest sides each have length 856 feet; twice this is
(b) is greater.
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A regular pentagon has perimeter one yard. Which is the greater quantity?
(A) The length of one side
(B) 7 inches
A regular pentagon has perimeter one yard. Which is the greater quantity?
(A) The length of one side
(B) 7 inches
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One yard is equal to 36 inches. A regular pentagon has five sides of equal length, so one side of the pentagon has length
inches.
Since 
, (A) is greater.
One yard is equal to 36 inches. A regular pentagon has five sides of equal length, so one side of the pentagon has length
Since
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The measures of the angles of a pentagon are . If , what is ?
The measures of the angles of a pentagon are
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The angles of a pentagon measure a total of . From the information, we know that:
If , then
The angles of a pentagon measure a total of
If
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A convex pentagon has four angles that measure each. What is the measure of the fifth angle?
A convex pentagon has four angles that measure
Tap to reveal answer
The angles of a pentagon measure a total of . If we let the unknown angle measure be , then from this information:
Since an angle measure cannot be negative, this pentagon cannot exist.
The angles of a pentagon measure a total of
Since an angle measure cannot be negative, this pentagon cannot exist.
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The measures of the angles of a pentagon are . If , what is ?
The measures of the angles of a pentagon are
Tap to reveal answer
The angles of a pentagon measure a total of . From the information, we know:
If , then the above becomes:
The angles of a pentagon measure a total of
If
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Solve for :
Solve for
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The sum of the interior angles of a pentagon can be determined by the following equation, where is the number of sides:
Therefore:
The sum of the interior angles of a pentagon can be determined by the following equation, where
Therefore:
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In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?
In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?
Tap to reveal answer
In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that , the sum of 2 of the sides will be
In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that
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If a pentagon has a perimeter of 85in, what is the length of one side of the pentagon?
If a pentagon has a perimeter of 85in, what is the length of one side of the pentagon?
Tap to reveal answer
A pentagon has 5 equal sides. We know the formula to find perimeter of a pentagon is
where a is the length of one side of the pentagon. Now, we know the total perimeter is 85in. Knowing this, we will substitute into the formula. We get
Therefore, the length of one side of the pentagon is 17in.
A pentagon has 5 equal sides. We know the formula to find perimeter of a pentagon is
where a is the length of one side of the pentagon. Now, we know the total perimeter is 85in. Knowing this, we will substitute into the formula. We get
Therefore, the length of one side of the pentagon is 17in.
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Find the perimeter of a pentagon with a side of length 12in.
Find the perimeter of a pentagon with a side of length 12in.
Tap to reveal answer
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know the length of one side of the pentagon is 12in.
Knowing this, we can substitute into the formula. We get
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know the length of one side of the pentagon is 12in.
Knowing this, we can substitute into the formula. We get
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