HiSET › Properties of polygons and circles
A quadrilateral is shown, and the angle measures of 3 interior angles are given. Find x, the missing angle measure.
The sum of the measures of the interior angles of a quadrilateral is 360 degrees. The sum of the measures of the interior angles of any polygon can be determined using the following formula:
, where
is the number of sides.
For example, with a quadrilateral, which has 4 sides, you obtain the following calculation:
Solving for requires setting up an algebraic equation, adding all 4 angles to equal 360 degrees:
Solving for is straightforward: subtract the values of the 3 known angles from both sides:
The depicted circle has a radius of 3 cm. The arc length between the two points shown on the circle is cm. Find the area of the enclosed sector (highlighted in green).
First, find the area and circumference of the circle using the radius and the following formulae for circles:
Substituting in 3 for yields:
Next, find what fraction of the total circumference is between the two points on the circle (the arc length).
Finally, use this fraction to calculate the area of the enclosed sector. Note that this area is proportional to the above fraction. In other words:
So the Sector Area is one sixth of the total area.
Cross multiply:
Divide both sides by 6, then simplify to get the final answer:
The depicted circle has a radius of 3 cm. The arc length between the two points shown on the circle is cm. Find the area of the enclosed sector (highlighted in green).
First, find the area and circumference of the circle using the radius and the following formulae for circles:
Substituting in 3 for yields:
Next, find what fraction of the total circumference is between the two points on the circle (the arc length).
Finally, use this fraction to calculate the area of the enclosed sector. Note that this area is proportional to the above fraction. In other words:
So the Sector Area is one sixth of the total area.
Cross multiply:
Divide both sides by 6, then simplify to get the final answer:
A quadrilateral is shown, and the angle measures of 3 interior angles are given. Find x, the missing angle measure.
The sum of the measures of the interior angles of a quadrilateral is 360 degrees. The sum of the measures of the interior angles of any polygon can be determined using the following formula:
, where
is the number of sides.
For example, with a quadrilateral, which has 4 sides, you obtain the following calculation:
Solving for requires setting up an algebraic equation, adding all 4 angles to equal 360 degrees:
Solving for is straightforward: subtract the values of the 3 known angles from both sides:
Give the perimeter of a regular octagon in yards if the length of each side is feet.
yards
yards
yards
yards
yards
The perimeter of a regular octagon - the sum of the lengths of its (eight congruent) sides - is eight times the common sidelength, so the perimeter of the octagon is feet. One yard is equivalent to three feet, so divide this by conversion factor 3 to get
yards - the correct response.
Give the perimeter of a regular octagon in yards if the length of each side is feet.
yards
yards
yards
yards
yards
The perimeter of a regular octagon - the sum of the lengths of its (eight congruent) sides - is eight times the common sidelength, so the perimeter of the octagon is feet. One yard is equivalent to three feet, so divide this by conversion factor 3 to get
yards - the correct response.
Refer to the above figure. Give the ratio of the area of Sector 2 to that of Sector 1.
The ratio of the area of the larger Sector 2 to that of smaller Sector 1 is equal to the ratio of their respective arc measures - that is,
.
Therefore, it is sufficient to find these arc measures.
If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,
Letting , since the total arc measure of a circle is 360 degrees,
We are also given that
Making substitutions, and solving for :
Multiply both sides by 2:
Add 360 to both sides:
Divide both sides by 2:
,
the degree measure of .
It follows that
By the Arc Addition Principle,
Since , the central angle which intercepts
, is a right angle,
. By substitution,
,
and
The ratio is equal to
,
a 5 to 1 ratio.
and
are complementary angles.
and
are supplementary angles.
Evaluate .
and
are supplementary angles, so, by definition,
, so substitute and solve for
:
and
are complementary angles, so, by definition,
Substitute and solve for :
- that is, the angles have the same measure. Therefore,
.
is the center of the above circle. Calculate
.
is the central angle that intercepts
, so
.
Therefore, we need to find to obtain our answer.
If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,
Letting , since the total arc measure of a circle is 360 degrees,
We are also given that
Making substitutions, and solving for :
Multiply both sides by 2:
Subtract 360 from both sides:
Divide both sides by :
,
the measure of and, consequently, that of
.
is the center of the above circle, and
. Evaluate the length of
.
The radius of the circle is given to be . The total circumference
of the circle is
times this, or
.
The length of
is equal to
Thus, it is first necessary to find the degree measure of . If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,
Letting , since the total arc measure of a circle is 360 degrees,
We are also given that
Making substitutions, and solving for :
Multiply both sides by 2:
Subtract 360 from both sides:
Divide both sides by :
,
the degree measure of .
Thus, the length of
is