Polygons - ACT Math

A regular polygon has each interior angle measuring 108°. How many sides does it have?
Answer: 5 This question asks how many sides a regular polygon has when each interior angle is 108°. For a regular polygon, each interior angle equals 180(n-2)/n degrees. Setting 180(n-2)/n = 108 and solving: 180n - 360 = 108n, so 72n = 360, giving n = 5. A regular pentagon has interior angles of 108° each.
A regular polygon has interior angles of $140^\circ$ each. How many sides does it have?
Answer: 9 This problem asks how many sides a polygon has when each interior angle is 140°. For a regular polygon, each interior angle equals 180(n-2)/n degrees. Setting this equal to 140°: 180(n-2)/n = 140. Solving: 180(n-2) = 140n, which gives 180n - 360 = 140n, so 40n = 360, and n = 9. Choice A might represent confusion with the 150° case.
How many sides does a polygon have if the sum of its interior angles is 1260°?
Answer: 9 This question asks how many sides a polygon has when the sum of interior angles is 1260°. Using the formula 180(n-2) = 1260, we solve: n-2 = 1260/180 = 7, so n = 9. The polygon has 9 sides (nonagon). Choice B would be incorrect as it represents an 8-sided polygon with sum 1080°.
How many sides does a polygon have if each interior angle in a regular polygon measures $120^\circ$ ?
Answer: 6 Given that each interior angle of a regular polygon is 120°, we need to find the number of sides. Using 180(n-2)/n = 120: 180(n-2) = 120n, which gives 180n - 360 = 120n, so 60n = 360, therefore n = 6. The polygon is a hexagon.
What is the sum of the interior angles of a nonagon?
Answer: 1260° This question asks for the sum of interior angles of a nonagon (9 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 9: 180(9-2) = 180(7) = 1260°. Choice A would result from using an octagon (8 sides) with sum 180(6) = 1080°.
A convex polygon has 7 sides. What is the sum of its interior angles?
Answer: $900^\circ$ This question asks for the sum of the interior angles of a convex 7-sided polygon. The standard formula is 180(n-2) degrees for the interior angle sum of any convex polygon with n sides. For n=7, this is 180(7-2)=180(5)=900 degrees. Note that this holds for convex polygons, where each interior angle is less than 180 degrees. Choice C, 720 degrees, is the sum for a pentagon (n=5), perhaps from subtracting wrong as 7-3=4, 180(4)=720.
A regular polygon has interior angles of $135^\circ$ . How many sides does it have?
Answer: 8 A regular polygon has interior angles of 135°, and we need to find the number of sides. Using 180(n-2)/n = 135: 180(n-2) = 135n, which gives 180n - 360 = 135n, so 45n = 360, therefore n = 8. The polygon is an octagon.
A regular polygon has each interior angle measuring $108^\circ$ . How many sides does it have?
Answer: 5 We need to find the number of sides when each interior angle is 108°. For a regular n-gon, each angle = 180(n-2)/n. Setting equal to 108: 180(n-2)/n = 108, so 180n - 360 = 108n, giving 72n = 360, thus n = 5. A regular pentagon has interior angles of 108°.
A regular polygon has each interior angle measuring 120°. How many sides does it have?
Answer: 6 This question asks how many sides a regular polygon has when each interior angle is 120°. For a regular polygon, each interior angle equals 180(n-2)/n degrees. Setting 180(n-2)/n = 120 and solving: 180n - 360 = 120n, so 60n = 360, giving n = 6. A regular hexagon has interior angles of 120° each.
A regular polygon has 7 sides. What is the sum of the interior angles of this polygon?
Answer: $900^\circ$ The question seeks the sum of interior angles for a 7-sided polygon. Apply the formula 180(n-2) degrees. For n=7, 180(7-2)=180*5=900 degrees. Calculate: 7-2=5, 180*5=900. Choice B, 720 degrees, is the sum for a hexagon (n=6), a common off-by-one error.
How many sides does a polygon have if the sum of its interior angles is $1080^\circ$ ?
Answer: 8 This problem asks for the number of sides when the sum of interior angles is 1080°. Using the formula 180(n-2) = sum of interior angles, we have 180(n-2) = 1080. Dividing both sides by 180: n-2 = 6, so n = 8. Choice A represents a hexagon with sum 720°, while choice B represents a heptagon with sum 900°.
A regular polygon has each interior angle equal to $135^\circ$ . How many sides does this polygon have?
Answer: 8 The question asks how many sides a regular polygon has if each interior angle is 135 degrees. The formula is 180(n-2)/n =135. Solve: 180(n-2)=135n, 180n-360=135n, 45n=360, n=8. Step-by-step: subtract 135n from 180n to get 45n, then 360/45=8. Choice A, 6, might be from using the hexagon's 120 degrees or miscalculating.
What is the sum of the interior angles of a 7-sided polygon?
Answer: $900^\circ$ We need to find the sum of interior angles of a 7-sided polygon (heptagon). The formula for the sum of interior angles is 180(n-2)°, where n is the number of sides. Substituting n=7: 180(7-2) = 180(5) = 900°. Choice A (720°) incorrectly used n=6 instead of n=7.
What is the sum of the interior angles of a hexagon?
Answer: 720° This question asks for the sum of interior angles in a hexagon (6 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. For a hexagon: 180(6-2) = 180(4) = 720°. Choice C incorrectly used 180n instead of 180(n-2), giving 1080°.
What is the measure of each interior angle of a regular hexagon?
Answer: 120° This question asks for each interior angle in a regular hexagon (6 sides). For a regular polygon, each interior angle equals 180(n-2)/n degrees. Substituting n = 6: 180(6-2)/6 = 180(4)/6 = 720/6 = 120°. Choice B would be correct for a pentagon where each angle is 108°.
A regular hexagon (6-sided polygon) has all sides and angles equal. What is the measure of each interior angle of this regular hexagon?
Answer: $120^\circ$ The question asks for the measure of each interior angle in a regular hexagon, which has 6 sides. The formula for the sum of interior angles of any polygon with n sides is 180(n-2) degrees, and for a regular polygon, each interior angle is that sum divided by n. For a hexagon, n=6, so the sum is 180(6-2) = 180*4 = 720 degrees, and each angle is 720/6 = 120 degrees. This emphasizes how the interior angle formula derives from triangulating the polygon into n-2 triangles. Choice D, 60 degrees, might result from confusing it with the exterior angle, since 360/6=60 degrees.
What is the measure of each interior angle in a regular pentagon (5-gon)?
Answer: $108^\circ$ The interior angles of any $n$ -sided polygon sum to $180(n-2)$ degrees, and in a regular polygon all the angles are equal, so each one measures $\dfrac{180(n-2)}{n}$ . For a pentagon, $n=5$ gives $\dfrac{180(3)}{5}=\dfrac{540}{5}=108^\circ$ . The value $72^\circ$ is the exterior angle, found from $\dfrac{360}{5}$ , and it is tempting because interior and exterior angles are supplementary, with $108+72=180$ . The value $120^\circ$ is the interior angle of a regular hexagon, which comes from using $n=6$ , and $90^\circ$ belongs to a square. Compute the total $180(n-2)$ first and divide by $n$ for one interior angle, and use $\dfrac{360}{n}$ only when the question asks for an exterior angle.
What is the sum of the interior angles of a 12-sided polygon (a dodecagon)?
Answer: $1{,}800^\circ$ This question seeks the sum of the interior angles of a 12-sided polygon, known as a dodecagon. The formula for the sum of interior angles is 180(n-2) degrees for any convex polygon with n sides. For n=12, substitute to get 180(12-2)=180(10)=1800 degrees. This formula comes from the fact that a polygon can be divided into n-2 triangles, each contributing 180 degrees. Choice C, 1440 degrees, could result from using n-4 instead, as 180(8)=1440.
What is the measure of each interior angle of a regular triangle?
Answer: 60° This question asks for each interior angle in a regular triangle (3 sides). For a regular polygon, each interior angle equals 180(n-2)/n degrees. Substituting n = 3: 180(3-2)/3 = 180(1)/3 = 180/3 = 60°. An equilateral triangle has all angles equal to 60°.
What is the measure of each interior angle in a regular decagon (10-gon)?
Answer: $144^\circ$ This problem asks for each interior angle in a regular decagon (10-gon). For a regular n-sided polygon, each interior angle is 180(n-2)/n degrees. Substituting n = 10: 180(10-2)/10 = 180(8)/10 = 1440/10 = 144°. Choice B represents the angle for a 12-sided polygon, while choice C represents an octagon's angle.

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