Other Polygons - SAT Math
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If angle A and angle C are complementary angles and B and D are supplementary angles, which of the following must be true?
If angle A and angle C are complementary angles and B and D are supplementary angles, which of the following must be true?
This question is very misleading, because while each answer COULD be true, none of them MUST be true. Between angle A and C, onne of the angles could be very small (0.001 degrees) and the other one could be very large. For instance, if A = 89.9999 and C = 0.0001, AC = 0.009. On the other hand, the two angles could be very siimilar. If B = 90 and D = 90 then BD = 8100 and BD > AC. If we use these same values we disprove AD = BC as 8100 ≠ .009. Finally, if B is a very small value, then B/C will be very small and smaller than A/D.
This question is very misleading, because while each answer COULD be true, none of them MUST be true. Between angle A and C, onne of the angles could be very small (0.001 degrees) and the other one could be very large. For instance, if A = 89.9999 and C = 0.0001, AC = 0.009. On the other hand, the two angles could be very siimilar. If B = 90 and D = 90 then BD = 8100 and BD > AC. If we use these same values we disprove AD = BC as 8100 ≠ .009. Finally, if B is a very small value, then B/C will be very small and smaller than A/D.
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In isosceles triangle ABC, the measure of angle A is 50 degrees. Which is NOT a possible measure for angle B?
In isosceles triangle ABC, the measure of angle A is 50 degrees. Which is NOT a possible measure for angle B?
If angle A is one of the base angles, then the other base angle must measure 50 degrees. Since 50 + 50 + x = 180 means x = 80, the vertex angle must measure 80 degrees.
If angle A is the vertex angle, the two base angles must be equal. Since 50 + x + x = 180 means x = 65, the two base angles must measure 65 degrees.
The only number given that is not possible is 95 degrees.
If angle A is one of the base angles, then the other base angle must measure 50 degrees. Since 50 + 50 + x = 180 means x = 80, the vertex angle must measure 80 degrees.
If angle A is the vertex angle, the two base angles must be equal. Since 50 + x + x = 180 means x = 65, the two base angles must measure 65 degrees.
The only number given that is not possible is 95 degrees.
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In triangle ABC, the measure of angle A = 70 degrees, the measure of angle B = x degrees, and the measure of angle C = y degrees. What is the value of y in terms of x?
In triangle ABC, the measure of angle A = 70 degrees, the measure of angle B = x degrees, and the measure of angle C = y degrees. What is the value of y in terms of x?
Since the three angles of a triangle sum to 180, we know that 70 + x + y = 180. Subtract 70 from both sides and see that x + y = 110. Subtract x from both sides and see that y = 110 – x.
Since the three angles of a triangle sum to 180, we know that 70 + x + y = 180. Subtract 70 from both sides and see that x + y = 110. Subtract x from both sides and see that y = 110 – x.
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What is the measure, in degrees, of each interior angle of a regular convex polygon that has twelve sides?
What is the measure, in degrees, of each interior angle of a regular convex polygon that has twelve sides?
The sum of the interior angles, in degrees, of a regular polygon is given by the formula 180(n – 2), where n is the number of sides. The problem concerns a polygon with twelve sides, so we will let n = 12. The sum of the interior angles in this polygon would be 180(12 – 2) = 180(10) = 1800.
Because the polygon is regular (meaning its sides are all congruent), all of the angles have the same measure. Thus, if we divide the sum of the measures of the angles by the number of sides, we will have the measure of each interior angle. In short, we need to divide 1800 by 12, which gives us 150.
The answer is 150.
The sum of the interior angles, in degrees, of a regular polygon is given by the formula 180(n – 2), where n is the number of sides. The problem concerns a polygon with twelve sides, so we will let n = 12. The sum of the interior angles in this polygon would be 180(12 – 2) = 180(10) = 1800.
Because the polygon is regular (meaning its sides are all congruent), all of the angles have the same measure. Thus, if we divide the sum of the measures of the angles by the number of sides, we will have the measure of each interior angle. In short, we need to divide 1800 by 12, which gives us 150.
The answer is 150.
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A regular polygon has a measure of 
for each of its internal angles. How many sides does it have?
A regular polygon has a measure of 
To determine the measure of the angles of a regular polygon use:
Angle = (n – 2) x 180° / n
Thus, (n – 2) x 180° / n = 140°
180° n - 360° = 140° n
40° n = 360°
n = 360° / 40° = 9
To determine the measure of the angles of a regular polygon use:
Angle = (n – 2) x 180° / n
Thus, (n – 2) x 180° / n = 140°
180° n - 360° = 140° n
40° n = 360°
n = 360° / 40° = 9
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A regular seven sided polygon has a side length of 14”. What is the measurement of one of the interior angles of the polygon?
A regular seven sided polygon has a side length of 14”. What is the measurement of one of the interior angles of the polygon?
The formula for of interior angles based on a polygon with a number of side n is:
Each Interior Angle = (n-2)*180/n
= (7-2)*180/7 = 128.57 degrees
The formula for of interior angles based on a polygon with a number of side n is:
Each Interior Angle = (n-2)*180/n
= (7-2)*180/7 = 128.57 degrees
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In the figure above, polygon ABDFHGEC is a regular octagon. What is the measure, in degrees, of angle FHI?
In the figure above, polygon ABDFHGEC is a regular octagon. What is the measure, in degrees, of angle FHI?
Angle FHI is the supplement of angle FHG, which is an interior angle in the octagon. When two angles are supplementary, their sum is equal to 180 degrees. If we can find the measure of each interior angle in the octagon, then we can find the supplement of angle FHG, which will give us the measure of angle FHI.
The sum of the interior angles in a regular polygon is given by the formula 180(n – 2), where n is the number of sides in the polygon. An octagon has eight sides, so the sum of the angles of the octagon is 180(8 – 2) = 180(6) = 1080 degrees. Because the octagon is regular, all of its sides and angles are congruent. Thus, the measure of each angle is equal to the sum of its angles divided by 8. Therefore, each angle in the polygon has a measure of 1080/8 = 135 degrees. This means that angle FHG has a measure of 135 degrees.
Now that we know the measure of angle FHG, we can find the measure of FHI. The sum of the measures of FHG and FHI must be 180 degrees, because the two angles form a line and are supplementary. We can write the following equation:
Measure of FHG + measure of FHI = 180
135 + measure of FHI = 180
Subtract 135 from both sides.
Measure of FHI = 45 degrees.
The answer is 45.
Angle FHI is the supplement of angle FHG, which is an interior angle in the octagon. When two angles are supplementary, their sum is equal to 180 degrees. If we can find the measure of each interior angle in the octagon, then we can find the supplement of angle FHG, which will give us the measure of angle FHI.
The sum of the interior angles in a regular polygon is given by the formula 180(n – 2), where n is the number of sides in the polygon. An octagon has eight sides, so the sum of the angles of the octagon is 180(8 – 2) = 180(6) = 1080 degrees. Because the octagon is regular, all of its sides and angles are congruent. Thus, the measure of each angle is equal to the sum of its angles divided by 8. Therefore, each angle in the polygon has a measure of 1080/8 = 135 degrees. This means that angle FHG has a measure of 135 degrees.
Now that we know the measure of angle FHG, we can find the measure of FHI. The sum of the measures of FHG and FHI must be 180 degrees, because the two angles form a line and are supplementary. We can write the following equation:
Measure of FHG + measure of FHI = 180
135 + measure of FHI = 180
Subtract 135 from both sides.
Measure of FHI = 45 degrees.
The answer is 45.
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What is the measure of each angle in a regular octagon?
What is the measure of each angle in a regular octagon?
An octagon contains six triangles, or 1080 degrees. This means with 8 angles, each angle is 135 degrees.
An octagon contains six triangles, or 1080 degrees. This means with 8 angles, each angle is 135 degrees.
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What is the measure of each central angle of an octagon?
What is the measure of each central angle of an octagon?
There are 360 degrees and 8 angles, so dividing leaves 45 degrees per angle.
There are 360 degrees and 8 angles, so dividing leaves 45 degrees per angle.
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What is the average (arithmetic mean) of all 15 interior angles of a quadrilateral, pentagon, and hexagon?
What is the average (arithmetic mean) of all 15 interior angles of a quadrilateral, pentagon, and hexagon?
The 4 angles of a quadrilateral add to 360
The 5 angles of a pentagon add to 540
The 6 angles of a hexagon add to 720
The 4 angles of a quadrilateral add to 360
The 5 angles of a pentagon add to 540
The 6 angles of a hexagon add to 720
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Find the sum of the interior angles in a nonagon.
Find the sum of the interior angles in a nonagon.
To solve, simply use the formula for the total degrees in a polygon, where n is the number of vertices.
In this particular case, a nonagon is a shape with nine sides and thus nine vertices.
Thus,
To solve, simply use the formula for the total degrees in a polygon, where n is the number of vertices.
In this particular case, a nonagon is a shape with nine sides and thus nine vertices.
Thus,
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Which of the following cannot be the measure of an exterior angle of a regular polygon?
Which of the following cannot be the measure of an exterior angle of a regular polygon?
If one exterior angle is taken at each vertex of any convex polygon, the sum of their measures is 
. In a regular polygon - one with congruent sides and congruent interior angles, each exterior angle is congruent to one another. If the polygon has 
sides, each exterior angle has measure 
.
Given the common measure 
,
Multiplying both sides by 
:
and
Since 
is equal to a number of sides, it is a whole number. Thus, we are looking for a value of 
which, when we divide 360 by it, yields a non-whole result. We see that 
is the correct choice, since'
A quick check confirms that 360 divided by 8, 10, 12, or 15 yields a whole result.
If one exterior angle is taken at each vertex of any convex polygon, the sum of their measures is
Given the common measure
Multiplying both sides by
and
Since
A quick check confirms that 360 divided by 8, 10, 12, or 15 yields a whole result.
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To the nearest whole degree, give the measure of each interior angle of a regular polygon with 17 sides.
To the nearest whole degree, give the measure of each interior angle of a regular polygon with 17 sides.
The measure of each interior angle of an 
-sided polygon can be calculated using the formula
Setting 
:
The correct choice is therefore 
.
The measure of each interior angle of an
Setting
The correct choice is therefore
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Each interior angle of a regular polygon has measure 
. How many sides does the polygon have?
Each interior angle of a regular polygon has measure
The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures 
, then each exterior angle, which is supplementary to an interior angle, measures
The measures of the exterior angles of a polygon, one per vertex, total 
; in a regular polygon, they are congruent, so if there are 
such angles, each measures 
. Since the number of vertices is equal to the number of sides, if we set this equal to 
and solve for 
, we will find the number of sides.
Multiply both sides by 
:
The polygon has 72 vertices and, thus, 72 sides.
The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures
The measures of the exterior angles of a polygon, one per vertex, total
Multiply both sides by
The polygon has 72 vertices and, thus, 72 sides.
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A square has an area of 36 cm2. A circle is inscribed and cut out. What is the area of the remaining shape? Use 3.14 to approximate π.
A square has an area of 36 cm2. A circle is inscribed and cut out. What is the area of the remaining shape? Use 3.14 to approximate π.
We need to find the area of both the square and the circle and then subtract the two. Inscribed means draw within a figure so as to touch in as many places as possible. So the circle is drawn inside the square. The opposite is circumscribed, meaning drawn outside.
Asquare = s2 = 36 cm2 so the side is 6 cm
6 cm is also the diameter of the circle and thus the radius is 3 cm
A circle = πr2 = 3.14 * 32 = 28.28 cm2
The resulting difference is 7.74 cm2
We need to find the area of both the square and the circle and then subtract the two. Inscribed means draw within a figure so as to touch in as many places as possible. So the circle is drawn inside the square. The opposite is circumscribed, meaning drawn outside.
Asquare = s2 = 36 cm2 so the side is 6 cm
6 cm is also the diameter of the circle and thus the radius is 3 cm
A circle = πr2 = 3.14 * 32 = 28.28 cm2
The resulting difference is 7.74 cm2
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In the square above, the radius of each half-circle is 6 inches. What is the area of the shaded region?
In the square above, the radius of each half-circle is 6 inches. What is the area of the shaded region?
We can find the area of the shaded region by subtracting the area of the semicircles, which is much easier to find. Two semi-circles are equivalent to one full circle. Thus we can just use the area formula, where r = 6:
π(62) → 36π
Now we must subtract the area of the semi-circles from the total area of the square. Since we know that the radius also covers half of a side, 6(2) = 12 is the full length of a side of the square. Squaring this, 122 = 144. Subtracting the area of the circles, we get our final terms,
= 144 – 36π
We can find the area of the shaded region by subtracting the area of the semicircles, which is much easier to find. Two semi-circles are equivalent to one full circle. Thus we can just use the area formula, where r = 6:
π(62) → 36π
Now we must subtract the area of the semi-circles from the total area of the square. Since we know that the radius also covers half of a side, 6(2) = 12 is the full length of a side of the square. Squaring this, 122 = 144. Subtracting the area of the circles, we get our final terms,
= 144 – 36π
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If square A has a side of length 5 inches, how many times bigger is the area of square B if it has a side of length 25 inches?
If square A has a side of length 5 inches, how many times bigger is the area of square B if it has a side of length 25 inches?
First find the area of both squares using the formula 
.
For square A, s = 5.
For square B, s = 25.
The question is asking for the ratio of these two areas, which will tell us how many times bigger square B is. Divide the area of square B by the area of square A to find the answer.
First find the area of both squares using the formula
For square A, s = 5.
For square B, s = 25.
The question is asking for the ratio of these two areas, which will tell us how many times bigger square B is. Divide the area of square B by the area of square A to find the answer.
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If Bailey paints the wall shaped like above and uses one bucket per 5 square units, how many buckets does Bailey need?
If Bailey paints the wall shaped like above and uses one bucket per 5 square units, how many buckets does Bailey need?
To solve, we will need to find the area of the wall. We can do this by finding the areas of each section and adding them together. Break the area into a rectange and two triangles.
The area of the rectangle will be equal to the base times the height. The area of each triangle will be one half its given base times its height.
For the rectangle, the base is 12 and the height is 4 (both given in the figure).
The triangle to the right has a given base of 6, but we need to solve for its height. The height will be equal to the difference between the total height (6) and the height of the rectangle (4).
We now have the base and height of the triangle to the right, allowing us to calculate its area.
Now we need to solve the triangle to the left. We solved for its height (2), but we still need to solve for its base. The total base of the rectangle is 12. Subtract the base of the right-side triangle (6) and the small segment at the top of the rectangle (3) from this total length to solve for the base of the left triangle.
The left-side triangle has a base of 3 and a height of 2, allowing us to calculate its area.
Add together the two triangles and the rectangle to find the total area.
We know that each bucket of paint will cover 5 square units, and we have 57 square units total. Divide to find how many buckets are required.
We will need 11 full buckets and part of a twelfth bucket to cover the wall, meaning that we will need 12 buckets total.
To solve, we will need to find the area of the wall. We can do this by finding the areas of each section and adding them together. Break the area into a rectange and two triangles.
The area of the rectangle will be equal to the base times the height. The area of each triangle will be one half its given base times its height.
For the rectangle, the base is 12 and the height is 4 (both given in the figure).
The triangle to the right has a given base of 6, but we need to solve for its height. The height will be equal to the difference between the total height (6) and the height of the rectangle (4).
We now have the base and height of the triangle to the right, allowing us to calculate its area.
Now we need to solve the triangle to the left. We solved for its height (2), but we still need to solve for its base. The total base of the rectangle is 12. Subtract the base of the right-side triangle (6) and the small segment at the top of the rectangle (3) from this total length to solve for the base of the left triangle.
The left-side triangle has a base of 3 and a height of 2, allowing us to calculate its area.
Add together the two triangles and the rectangle to find the total area.
We know that each bucket of paint will cover 5 square units, and we have 57 square units total. Divide to find how many buckets are required.
We will need 11 full buckets and part of a twelfth bucket to cover the wall, meaning that we will need 12 buckets total.
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A square is inscribed within a circle with a radius 
. Find the area of the circle that is not covered by the square.
A square is inscribed within a circle with a radius
First, find the area of the circle.
Next, find the length of 1 side of the square using the Pythagorean Theorem. Two radii from the center of the circle to adjacent corners of the square will create a right angle at the center of the circle. The radii will be the legs of the triangle and the side of the square will be the hypotenuse.
Find the area of the square.
Subtract the area of the square from the area of the circle.
First, find the area of the circle.
Next, find the length of 1 side of the square using the Pythagorean Theorem. Two radii from the center of the circle to adjacent corners of the square will create a right angle at the center of the circle. The radii will be the legs of the triangle and the side of the square will be the hypotenuse.
Find the area of the square.
Subtract the area of the square from the area of the circle.
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If the following shape was going to be drawn in a circle, what is the minimum radius of the circle?
If the following shape was going to be drawn in a circle, what is the minimum radius of the circle?
IF you draw the longest diagonal across the shape, the length of it is 13.4. This means the radius must be at least 6.7. The answer is 7.
IF you draw the longest diagonal across the shape, the length of it is 13.4. This means the radius must be at least 6.7. The answer is 7.
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