Plane Geometry

These Plane Geometry questions are aligned with the ACT from ACT, Inc. in United States.

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Plane Geometry is the largest content area on the ACT Math, covering angles, triangles, circles, quadrilaterals, perimeter, area, and basic three-dimensional figures. ACT Plane Geometry questions often involve multiple steps—applying a formula is usually not enough; you also need to set up the relationship between geometric figures. Practice with these tests to develop the spatial reasoning and multi-step problem solving that Plane Geometry demands.

Plane Geometry makes up about 20-25% of the ACT Math section — the largest content area alongside Pre-Algebra. This topic covers properties of shapes, angles, area, perimeter, volume, and geometric reasoning in two dimensions.

Angles

  • Complementary angles sum to 90°
  • Supplementary angles sum to 180°
  • Vertical angles are equal
  • When a transversal crosses parallel lines: alternate interior angles are equal, corresponding angles are equal, co-interior (same-side interior) angles are supplementary
Example: Two angles are supplementary. One is 3x + 10 and the other is 2x + 20. Find x.
(3x + 10) + (2x + 20) = 180 → 5x + 30 = 180 → 5x = 150 → x = 30. The angles are 100° and 80°.

Example: A transversal crosses two parallel lines forming an angle of 65°. Find all eight angles.
The eight angles come in two sets: four angles equal 65° (corresponding and vertical) and four equal 180° − 65° = 115°.

Triangles

The angle sum of any triangle is 180°. Key triangle types:
  • Equilateral: all sides and angles equal (60° each)
  • Isosceles: two equal sides, two equal base angles
  • Right: one 90° angle, follows the Pythagorean theorem (a² + b² = c²)
Area of a triangle = ½ × base × height.

Example: In a triangle, two angles measure 45° and 70°. Find the third angle.
180° − 45° − 70° = 65°.

Example: An isosceles triangle has a vertex angle of 40°. What are the base angles?
(180° − 40°)/2 = 70° each.

Special Right Triangles

Special Right Triangles — Must Memorize! 45°-45°-90° x x x√2 45° 45° Ratio: 1 : 1 : √2 30°-60°-90° x√3 x 2x 30° 60° Ratio: 1 : √3 : 2 Pythagorean triples to memorize: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples like 6-8-10, 9-12-15).

Example 1: A ladder leans against a wall. Its base is 6 feet from the wall and the ladder is 10 feet long. How high up the wall does it reach?
6² + h² = 10² → 36 + h² = 100 → h² = 64 → h = 8 feet (a 6-8-10 = 2×(3-4-5) triple).

Example 2: An equilateral triangle has side length 8. What is its height?
Cut it in half to form a 30-60-90 triangle. Short side = 4, hypotenuse = 8, so height = 4√3 ≈ 6.93.
Area = ½ × 8 × 4√3 = 16√3 ≈ 27.7.

Example 3: A square has a diagonal of 10. What is its side length?
Diagonal of a square = side × √2. So side = 10/√2 = 10√2/2 = 5√2 ≈ 7.07.

Quadrilaterals

  • Rectangle: Area = lw, Perimeter = 2l + 2w
  • Square: Area = s², Perimeter = 4s, Diagonal = s√2
  • Parallelogram: Area = base × height
  • Trapezoid: Area = ½(b₁ + b₂) × h
  • Rhombus: Area = ½ × d₁ × d₂ (where d₁, d₂ are diagonals)
Example: A trapezoid has parallel sides of 12 and 8 and a height of 5. Find its area.
Area = ½(12 + 8) × 5 = ½(20)(5) = 50.

Circles

  • Circumference = 2πr = πd
  • Area = πr²
  • Arc length = (θ/360°) × 2πr
  • Sector area = (θ/360°) × πr²
  • Central angle = intercepted arc
  • Inscribed angle = ½ intercepted arc
Circle: Key Measurements r θ Formulas Circumference = 2πr Area = πr² Arc length = (θ/360) × 2πr Sector area = (θ/360) × πr² Central angle = arc Inscribed angle = ½ arc Example 1: A circle has radius 6. Find the area of a sector with a 60° central angle.
Sector area = (60/360) × π(6²) = (1/6) × 36π = 6π ≈ 18.85

Example 2: A pizza has a diameter of 16 inches. One slice corresponds to a 45° central angle. What is the area of one slice?
Radius = 8. Sector area = (45/360) × π(64) = (1/8)(64π) = 8π ≈ 25.1 square inches.

Example 3: An inscribed angle intercepts an arc of 100°. What is the inscribed angle?
Inscribed angle = ½ × 100° = 50°.

Similar Triangles

If two triangles are similar, their corresponding sides are proportional. If the ratio of sides is k, then the ratio of areas is k² and the ratio of volumes (for 3D) is k³.

Example 1: Triangle ABC is similar to triangle DEF with a side ratio of 3:5. If the area of ABC is 27, what is the area of DEF?
Area ratio = (3/5)² = 9/25. So 27/Area_DEF = 9/25 → Area_DEF = 75.

Example 2: A tree casts a 15-foot shadow. At the same time, a 6-foot person casts a 4-foot shadow. How tall is the tree?
Similar triangles: 6/4 = h/15 → h = (6 × 15)/4 = 22.5 feet.

Volume and Surface Area

  • Rectangular prism: V = lwh, SA = 2(lw + lh + wh)
  • Cylinder: V = πr²h, SA = 2πr² + 2πrh
  • Cone: V = ⅓πr²h
  • Sphere: V = (4/3)πr³, SA = 4πr²
Example: A cylinder has radius 3 and height 10. A cone has the same radius and height. How much more volume does the cylinder have?
Cylinder V = π(9)(10) = 90π. Cone V = ⅓π(9)(10) = 30π. Difference = 60π ≈ 188.5.

Shaded Region Problems

Shaded Region Strategy Square with inscribed circle How to Solve 1. Find the area of the WHOLE shape 2. Find the area of the UNSHADED part 3. Shaded = Whole − Unshaded Example: Square side 10, inscribed circle r=5 Shaded = 100 − 25π ≈ 100 − 78.5 = 21.5

Common Mistakes

Top 5 Plane Geometry Traps 1. Using diameter instead of radius (or vice versa) in circle formulas. Area uses r, not d. 2. Confusing perimeter with area. The ACT will include the other as a wrong answer choice. 3. Using the slant height instead of the perpendicular height for triangle area. 4. In similar figures, scaling the area by k instead of k². Sides scale by k, area by k². 5. Assuming a figure is drawn to scale when it says "not drawn to scale."

Quick Reference — Formula Sheet

Plane Geometry Quick Reference Triangles Angle sum = 180° Area = ½ × base × height Pythagorean: a² + b² = c² Special Right Triangles 45-45-90: x, x, x√2 30-60-90: x, x√3, 2x Quadrilaterals Rectangle: A=lw, P=2l+2w Parallelogram: A=bh Trapezoid: A=½(b₁+b₂)h Rhombus: A=½d₁d₂ Pythagorean Triples 3-4-5 | 5-12-13 | 8-15-17 7-24-25 (and multiples) Circles C = 2πr = πd A = πr² Arc = (θ/360)×2πr Sector = (θ/360)×πr² 3D Shapes Box: V=lwh Cylinder: V=πr²h Cone: V=⅓πr²h Sphere: V=⁴⁄₃πr³ Similar Figures Side ratio: k Area ratio: k² Volume ratio: k³ Shaded = Whole − Unshaded

ACT Strategies for Plane Geometry

  • Know your triples: Memorize 3-4-5, 5-12-13, 8-15-17 and their multiples to save significant time.
  • Special triangles everywhere: Watch for 30-60-90 and 45-45-90 triangles hidden inside other problems (equilateral triangles cut in half, squares with diagonals).
  • Circle area vs. circumference: Identify which one the question asks for before calculating.
  • Similar triangle proportions: Set up proportions carefully — match corresponding sides.
  • Not drawn to scale: Do not assume angles or lengths from the picture unless stated.
  • Break down complex shapes: Decompose into simpler shapes (rectangles, triangles) to find areas.
  • Shaded region = whole − unshaded: This classic pattern appears on nearly every ACT.
  • Interior angle sum: For any polygon with n sides, the sum of interior angles = (n − 2) × 180°.
  • Exterior angle theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Practice Walkthrough

ACT-Style Problem: A rectangular garden is 12 meters long and 8 meters wide. A circular fountain with radius 2 meters is placed in the center. What is the area of the garden NOT covered by the fountain?

A. 96 − 2π    B. 96 − 4π    C. 96 − 8π    D. 80 − 4π

Step 1 — Rectangle area: 12 × 8 = 96 square meters.
Step 2 — Circle area: π(2²) = 4π square meters.
Step 3 — Subtract: 96 − 4π.
Elimination: Choice A uses πr instead of πr². Choice C uses π(2r)² = 16π, then halves it — wrong. Choice D starts with 80 (perhaps subtracting the fountain twice from the perimeter somehow). Only B (96 − 4π) is correct.
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