How to find an angle in a pentagon - Math

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Question

The angles at 3 verticies of a pentagon are 60, 80 and 100. Which of the following could NOT be the measures of the other 2 angles?

Answer

The sum of the angles in a polygon is

For a pentagon, this equals 540. Since the first 3 angles add up to 240, the remaining 2 angles must add up to

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Question

What is the sum of three angles in a hexagon if the perimeter of the hexagon is ?

Answer

The perimeter in this question is irrelevant. Use the interior angle formula to determine the total sum of the angles in a hexagon.

$\Theta =(n-2)\cdot180$

There are six interior angles in a hexagon.

$\Theta =(6-2)\cdot180 = $720^{\circ}$$

Each angle will be a sixth of the total angle.

$\frac{720}{6}$= $120^{\circ}$

Therefore, the sum of three angles in a hexagon is:

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Question

Add four interior angles in a regular pentagon. What is the result?

Answer

Use the interior angle formula to find the total sum of angles in a pentagon.

$\sum \Theta=(n-2) \cdot 180$

for a pentagon, so substitute this value into the equation and solve:

$\sum \Theta=(5-2) \cdot 180 = $540^{\circ}$$

Divide this number by 5, since there are five interior angles.

The sum of four interior angles in a regular pentagon is:

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Question

What is the sum of two interior angles of a regular pentagon if the perimeter is 6?

Answer

The perimeter of a regular pentagon has no effect on the interior angles of the pentagon.

Use the following formula to solve for the sum of all interior angles in the pentagon.

$\sum\theta=(n-2)\cdot 180$

Since there are 5 sides in a pentagon, substitute the side length .

$\sum\theta=(5-2)\cdot 180=540$

Divide this by 5 to determine the value of each angle, and then multiply by 2 to determine the sum of 2 interior angles.

$$\frac{540}{5}$=108$

The sum of 2 interior angles of a pentagon is .

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Question

Let the area of a regular pentagon be . What is the value of an interior angle?

Answer

Area has no effect on the value of the interior angles of a pentagon. To find the sum of all angles of a pentagon, use the following formula, where is the number of sides:

$\sum\theta=(n-2)\cdot 180$

There are 5 sides in a pentagon.

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each angle.

$$\frac{540}{5}$=108^\circ$

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Question

Suppose an interior angle of a regular pentagon is . What is ?

Answer

The pentagon has 5 sides. To find the value of the interior angle of a pentagon, use the following formula to find the sum of all interior angles.

$\sum\theta=(n-2)\cdot 180$

Substitute .

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each interior angle.

$$\frac{540}{5}$=108$

Every interior angle is 108 degrees. The problem states that an interior angle is . Set these two values equal to each other and solve for .

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Question

True or false: Each of the five angles of a regular pentagon measures .

Answer

A regular polygon with sides has congruent angles, each of which measures

Setting , the common angle measure can be calculated to be

The statement is therefore false.

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Question

True or false: Each of the exterior angles of a regular pentagon measures $72 ^{\circ }$ .

Answer

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is . Each exterior angle of a regular pentagon has the same measure, so if we let be that common measure, then

$5t = 360 ^{\circ }$

Solve for :

$5t \div 5 = 360 ^{\circ } \div 5$

$t = 72 ^{\circ }$

The statement is true.

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Question

Given: Pentagon .

$m \angle P = 108 ^{\circ }$

True, false, or undetermined: Pentagon is regular.

Answer

Suppose Pentagon is regular. Each angle of a regular polygon of sides has measure

A pentagon has 5 sides, so set ; each angle of the regular hexagon has measure

Since one angle is given to be of measure $108 ^{\circ }$ , the pentagon might be regular - but without knowing more, it cannot be determined for certain. Therefore, the correct choice is "undetermined".

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Question

What is the interior angle measure of any regular pentagon?

Answer

To find the angle of any regular polygon you find the number of sides . For a pentagon, .

You then subtract 2 from the number of sides yielding 3.

Take 3 and multiply it by 180 degrees to yield the total number of degrees in the regular pentagon.

Then to find one individual angle we divide 540 by the total number of angles 5, $$\frac{540}{5}$=108$

The answer is .

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Question

What is the measure of each interior angle of a regular pentagon?

Answer

The following equation can be used to determine the measure of an interior angle of a regular polygon, where $\small n$ equals the number of sides.

$\angle=\frac{180(n-2)}{n}$$

In a pentagon, $\small n=5$ .

$\angle=\frac{180(5-2)}{5}$$

Now we can solve for the angle.

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Question

Find the interior angle of the following regular pentagon:

Answer

The formula for the sum of the interior angles of a polygon is

,

where is the number of sides in the polygon

Plugging in our values, we get:

Dividing the sum of the interior angles by the number of angles in the polygon, we get the value for each interior angle:

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Question

What is the measure of an interior angle of a regular pentagon?

Answer

The measure of an interior angle of a regular polygon can be determined using the following equation, where equals the number of sides:

$\angle=\frac{180(n-2)}{n}$$

$\angle=\frac{180(5-2)}{5}$=\frac{180(3)}{5}$=108$

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Question

The angles at 3 verticies of a pentagon are 60, 80 and 100. Which of the following could NOT be the measures of the other 2 angles?

Answer

The sum of the angles in a polygon is

For a pentagon, this equals 540. Since the first 3 angles add up to 240, the remaining 2 angles must add up to

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Question

What is the sum of three angles in a hexagon if the perimeter of the hexagon is ?

Answer

The perimeter in this question is irrelevant. Use the interior angle formula to determine the total sum of the angles in a hexagon.

$\Theta =(n-2)\cdot180$

There are six interior angles in a hexagon.

$\Theta =(6-2)\cdot180 = $720^{\circ}$$

Each angle will be a sixth of the total angle.

$\frac{720}{6}$= $120^{\circ}$

Therefore, the sum of three angles in a hexagon is:

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Question

Add four interior angles in a regular pentagon. What is the result?

Answer

Use the interior angle formula to find the total sum of angles in a pentagon.

$\sum \Theta=(n-2) \cdot 180$

for a pentagon, so substitute this value into the equation and solve:

$\sum \Theta=(5-2) \cdot 180 = $540^{\circ}$$

Divide this number by 5, since there are five interior angles.

The sum of four interior angles in a regular pentagon is:

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Question

What is the sum of two interior angles of a regular pentagon if the perimeter is 6?

Answer

The perimeter of a regular pentagon has no effect on the interior angles of the pentagon.

Use the following formula to solve for the sum of all interior angles in the pentagon.

$\sum\theta=(n-2)\cdot 180$

Since there are 5 sides in a pentagon, substitute the side length .

$\sum\theta=(5-2)\cdot 180=540$

Divide this by 5 to determine the value of each angle, and then multiply by 2 to determine the sum of 2 interior angles.

$$\frac{540}{5}$=108$

The sum of 2 interior angles of a pentagon is .

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Question

Let the area of a regular pentagon be . What is the value of an interior angle?

Answer

Area has no effect on the value of the interior angles of a pentagon. To find the sum of all angles of a pentagon, use the following formula, where is the number of sides:

$\sum\theta=(n-2)\cdot 180$

There are 5 sides in a pentagon.

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each angle.

$$\frac{540}{5}$=108^\circ$

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Question

Suppose an interior angle of a regular pentagon is . What is ?

Answer

The pentagon has 5 sides. To find the value of the interior angle of a pentagon, use the following formula to find the sum of all interior angles.

$\sum\theta=(n-2)\cdot 180$

Substitute .

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each interior angle.

$$\frac{540}{5}$=108$

Every interior angle is 108 degrees. The problem states that an interior angle is . Set these two values equal to each other and solve for .

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Question

True or false: Each of the five angles of a regular pentagon measures .

Answer

A regular polygon with sides has congruent angles, each of which measures

Setting , the common angle measure can be calculated to be

The statement is therefore false.

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