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ACT Math Help: Polygons

Review real example questions for Polygons in ACT Math.

Question 1 / 10

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A square is inscribed in a circle with radius 4 inches. What is the area of the square, in square inches?

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Question 1

A square is inscribed in a circle with radius 4 inches. What is the area of the square, in square inches?

  1. 16
  2. 32 (correct answer)
  3. 16π16\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

How many sides does a polygon have if the sum of its interior angles is 1080∘1080^\circ?​

  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

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

  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

How many sides does a polygon have if the sum of its interior angles is 1260°?

  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

The perimeter of a certain rectangle is 40 inches. If the length of the rectangle is 12 inches, what is the area of the rectangle, in square inches?

  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

A regular polygon has each interior angle measuring 108°. How many sides does it have?

  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

What is the sum of the interior angles of a pentagon?

  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

The area of a rectangle is 84 square inches. If the length of the rectangle is 12 inches, what is the perimeter, in inches, of the rectangle?

  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

In a parallelogram, two consecutive angles measure (3x−20)°(3x - 20)° and (2x+10)°(2x + 10)°. What is the measure of the larger angle?

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

What is the measure of one interior angle of a regular octagon?

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