ACT Math Quiz: Polygons

What this quiz covers

This quiz focuses on Polygons, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 16
2. 32 (correct answer)
3. $16\pi$
4. 64

Explanation: This is an inscribed figures question testing the relationship between a circle's radius and an inscribed square's diagonal. Choice B (32) is correct — the diameter of the circle (8 inches) equals the diagonal of the inscribed square. Using the relationship diagonal = side × √2: 8 = s√2 → s = 8/√2 = 4√2. Area = s² = (4√2)² = 16 × 2 = 32 square inches. Choice A (16) uses the radius as the side length: 4² = 16 — confusing radius with the square's side. Choice C (16π) gives the area of the circle, not the inscribed square. Choice D (64) uses the diameter as the side length: 8² = 64 — the diameter is the diagonal, not the side. Pro tip: When a square is inscribed in a circle, the circle's diameter is the square's diagonal (it connects opposite corners through the center). From diagonal to side: s = d/√2 = d√2/2. Or use s² + s² = d² (Pythagorean theorem on the half-square): 2s² = 64 → s² = 32. The area is s², so no further calculation needed!

Question 2

1. 6
2. 7
3. 8 (correct answer)
4. 10

Explanation: Given that the sum of interior angles is 1080°, we need to find the number of sides. Using the formula 180(n-2) = 1080, we solve for n: 180(n-2) = 1080, so n-2 = 6, therefore n = 8. The polygon has 8 sides (octagon).

Question 3

1. 60° (correct answer)
2. 90°
3. 120°
4. 75°

Explanation: 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°.

Question 4

1. 9 (correct answer)
2. 8
3. 7
4. 10

Explanation: 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°.

Question 5

1. 96 (correct answer)
2. 144
3. 192
4. 384

Explanation: The correct answer is A (96). Use the perimeter formula to find the width: P = 2l + 2w → 40 = 2(12) + 2w → 40 = 24 + 2w → 2w = 16 → w = 8. Then compute area: A = l × w = 12 × 8 = 96. B (144) assumes the rectangle is a square with both dimensions equal to 12, giving 12 × 12 = 144. C (192) results from computing the width as 40 − 12 − 12 = 16 (correctly subtracting both lengths) but forgetting to divide by 2: 12 × 16 = 192. D (384) likely comes from multiplying the perimeter by the length. Always find the unknown dimension first before computing the area.

Question 6

1. 5 (correct answer)
2. 6
3. 8
4. 10

Explanation: 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.

Question 7

1. 450°
2. 540° (correct answer)
3. 360°
4. 720°

Explanation: This question asks for the sum of interior angles of a pentagon (5 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 5: 180(5-2) = 180(3) = 540°. Choice A would result from incorrectly using 180(2.5) for some fractional calculation.

Question 8

1. 7
2. 19
3. 38 (correct answer)
4. 96

Explanation: This is a perimeter and area question testing the relationship between area, dimensions, and perimeter. Choice C (38) is correct — find the width: Width = Area ÷ Length = 84 ÷ 12 = 7 inches. Perimeter = 2(length + width) = 2(12 + 7) = 2(19) = 38 inches. Choice A (7) stops after finding the width, reporting the intermediate step rather than the perimeter. Choice B (19) adds length + width = 12 + 7 = 19, but forgets to multiply by 2 — computing half the perimeter. Choice D (96) likely comes from multiplying area × length: 84 × 12 ÷ ... or adding area + length: 84 + 12 = 96. Pro tip: Finding a missing dimension from area is just the first step — remember to plug both dimensions into P = 2(l + w) to get the perimeter. The factor of 2 is easy to forget.

Question 9

1. $38°$
2. $86°$
3. $94°$ (correct answer)
4. $104°$

Explanation: This is a parallelogram angles question testing the supplementary consecutive angles property. Choice C (94°) is correct — consecutive (co-interior) angles in a parallelogram are supplementary (sum to 180°). Set up: (3x − 20) + (2x + 10) = 180 → 5x − 10 = 180 → 5x = 190 → x = 38. Compute both angles: 3(38) − 20 = 114 − 20 = 94° and 2(38) + 10 = 76 + 10 = 86°. The larger is 94°. Choice A (38°) reports x = 38, the variable value, not the angle measure. Choice B (86°) reports the smaller angle instead of the larger. Choice D (104°) comes from an arithmetic error: setting 5x − 10 = 180 → 5x = 190 → x = 38, but then computing 3(38) − 20 = 94 and mistakenly writing 104. Pro tip: Consecutive angles in a parallelogram are SUPPLEMENTARY (sum to 180°), while opposite angles are EQUAL. Don't confuse these. After finding x, always compute both angles and identify which is larger — the question specifies "the larger of these two angles.

Question 10

1. $45°$
2. $120°$
3. $135°$ (correct answer)
4. $144°$

Explanation: The correct answer is C (135°). The formula for one interior angle of a regular n-gon is (n − 2) × 180 ÷ n. For an octagon: (8 − 2) × 180 ÷ 8 = 6 × 180 ÷ 8 = 1080 ÷ 8 = 135°. A (45°) is the exterior angle of a regular octagon (360 ÷ 8 = 45°) — the student confuses interior and exterior angles. B (120°) is the interior angle of a regular hexagon (6 sides) — the student recalls the wrong polygon. D (144°) is the interior angle of a regular decagon (10 sides) — uses n = 10 instead of n = 8. Pro tip: memorize the formula (n−2)×180/n and double-check which polygon is referenced.

Question 11

1. 5
2. 6 (correct answer)
3. 8
4. 10

Explanation: The question asks how many sides a regular polygon has if each exterior angle is 60 degrees. The exterior angle sum is 360 degrees, so each is 360/n for regular polygons. Set 360/n=60, so n=360/60=6. This is a hexagon, with interior angles of 120 degrees each. Choice A, 5, would give 72-degree exterior angles, a common distractor for pentagons.

Question 12

1. 720° (correct answer)
2. 540°
3. 1080°
4. 900°

Explanation: This question asks for the sum of interior angles of a 6-sided polygon (hexagon). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 6: 180(6-2) = 180(4) = 720°. Choice B would result from using a pentagon (5 sides) with sum 180(3) = 540°.

Question 13

1. $135^\circ$ (correct answer)
2. $120^\circ$
3. $140^\circ$
4. $108^\circ$

Explanation: The question asks for the measure of each interior angle in a regular octagon, which has 8 sides. The formula for the sum of the interior angles of any polygon is 180(n-2) degrees, where n is the number of sides, and for a regular polygon, each interior angle is that sum divided by n. For n=8, the sum is 180*(8-2)=180*6=1080 degrees, so each angle is 1080/8=135 degrees. Remember that the sum of exterior angles is always 360 degrees, but here we focus on interior angles. Choice D, 108 degrees, likely comes from confusing it with a pentagon's interior angle.

Question 14

1. 540°
2. 180°
3. 360° (correct answer)
4. 720°

Explanation: This question asks for the sum of interior angles of a quadrilateral (4 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 4: 180(4-2) = 180(2) = 360°. Choice A would result from using a pentagon (5 sides) with sum 180(3) = 540°.

Question 15

1. 108° (correct answer)
2. 120°
3. 135°
4. 144°

Explanation: This question asks for each interior angle in a regular pentagon (5 sides). For a regular polygon, each interior angle equals 180(n-2)/n degrees. Substituting n = 5: 180(5-2)/5 = 180(3)/5 = 540/5 = 108°. Choice B would be correct for a hexagon where each angle is 120°.

Question 16

1. 1080°
2. 1260° (correct answer)
3. 1440°
4. 1620°

Explanation: 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°.

Question 17

1. 5
2. 6 (correct answer)
3. 8
4. 7

Explanation: 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.

Question 18

1. $60^\circ$
2. $90^\circ$
3. $120^\circ$ (correct answer)
4. $150^\circ$

Explanation: 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 a polygon with n sides is 180(n-2)°, and for a regular polygon, each interior angle is 180(n-2)/n °. For n=6, the sum is 180(6-2) = 180*4 = 720°, so each angle is 720/6 = 120°. The sum of exterior angles is always 360°, but here we focus on interior angles. Choice A of 60° might confuse with equilateral triangle angles, while the correct is 120°.

Question 19

1. 6
2. 8 (correct answer)
3. 10
4. 12

Explanation: The question asks for the number of sides in a regular polygon where each exterior angle is 45 degrees. Remember that the sum of exterior angles for any polygon is always 360 degrees, and for a regular polygon, each exterior angle is 360/n degrees. Set 360/n = 45, then solve for n: n=360/45=8. This shows the straightforward division: 360 divided by 45 equals 8 sides. Choice A, 6, could be from confusing with interior angles or using 180/ something.

Question 20

1. $120^\circ$ (correct answer)
2. $108^\circ$
3. $60^\circ$
4. $140^\circ$

Explanation: This question asks for the measure of each interior angle in a regular hexagon, which has 6 equal sides and angles. 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. Note that the sum of exterior angles for any polygon is always 360 degrees, but here we focus on interior angles. A common distractor might be confusing it with a pentagon's angle of 108 degrees, as in choice B.