Plane Geometry: Triangles, Circles, and Polygons — ACT Math

Introduction

ACT gives you NO formula sheet — so the student who has memorized the triangle area formula, circle formulas, and polygon angle rules has a massive edge.

Plane geometry accounts for roughly 14-18 questions per ACT (the largest single category). These questions span all difficulty levels, making geometry a must-master topic.

By the end of this lesson you will be able to:

Apply area, perimeter, and angle formulas for triangles, circles, and polygons
Use the Pythagorean theorem and special triangle ratios (30-60-90 and 45-45-90)
Solve problems involving arc length, sector area, and inscribed angles

You'll find the area of a shaded region between a square and an inscribed circle — combining circle and square formulas in a single problem.

The Concept

The Core Rule

Formulas to memorize — Triangle: Area = (1/2)bh; Pythagorean theorem: a² + b² = c². Circle: Area = πr², Circumference = 2πr, Arc length = (θ/360)·2πr, Sector area = (θ/360)·πr². Polygon interior angle sum = (n−2)·180°. Regular polygon interior angle = (n−2)·180°/n.

How the ACT tests this

Embedding a Pythagorean theorem calculation inside a word problem about distances
Asking for the area of an irregular shape by decomposing it into triangles and rectangles
Using arc length or sector area with a given central angle

Triangle Rules

Area = (1/2)bh. Triangle inequality: any side < sum of the other two sides. Similar triangles: corresponding sides are proportional. Special right triangles: 45-45-90 (legs a, a; hypotenuse a√2) and 30-60-90 (short leg a, long leg a√3, hypotenuse 2a).

Equilateral triangle area: (√3/4)s²
Exterior angle = sum of the two non-adjacent interior angles
Pythagorean triples to memorize: (3,4,5), (5,12,13), (8,15,17) and their multiples

Circle Rules

Area = πr². Circumference = 2πr = πd. A central angle equals its intercepted arc. An inscribed angle equals half its intercepted arc. A diameter subtends a 90° angle at any point on the circle.

Arc length = (central angle / 360°) × 2πr
Sector area = (central angle / 360°) × πr²
Tangent to circle: perpendicular to radius at the point of tangency

Polygons

Rectangle: Area = lw, Perimeter = 2l + 2w. Parallelogram: Area = bh. Trapezoid: Area = (1/2)(b₁ + b₂)h. Regular hexagon splits into 6 equilateral triangles. Sum of exterior angles of any convex polygon = 360°.

Interior angle sum of n-gon: (n−2)×180°
Diagonal count for an n-gon: n(n−3)/2
Square with side s: diagonal = s√2

Your strategy

Draw and label every geometry problem — a clear diagram prevents most errors
For composite shapes, add or subtract standard shape areas
Look for Pythagorean triple side lengths to avoid full theorem calculation
For circle problems, always identify whether you need radius, diameter, or arc before computing

Worked Examples

Easy Example 1 Using The Diameter (10) Instead Of The Radius (5) In The Area Formula

A circle has a radius of 5. What is its area?

A. 10π
B. 20π
C. 25π (Correct answer)
D. 50π
E. 100π

Step 1

Area formula: A = πr²

Step 2

Substitute r = 5: A = π(5)² = 25π

Step 3

Answer: 25π

Correct answer: C

Why C is correct

Correct: π × 25 = 25π

Why other options are wrong

A: 2r = 10; used circumference formula without the π factor correctly

B: Arithmetic error involving doubling

D: Used 2r² = 2(25) = 50; doubled needlessly

E: Used diameter (10) as radius: π(10)² = 100π

⚠ Trap: Using the diameter (10) instead of the radius (5) in the area formula

Medium Example 2 Confusing The 30-60-90 Ratio (1:√3:2) With The 45-45-90 Ratio (1:1:√2)

In a 30-60-90 triangle, the side opposite the 30° angle has length 7. What is the length of the hypotenuse?

A. 7
B. 7√2
C. 7√3
D. 14 (Correct answer)
E. 21

Step 1

In a 30-60-90 triangle, sides are in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse)

Step 2

Short leg (opposite 30°) = 7 = 1 × 7

Step 3

Hypotenuse = 2 × 7 = 14

Correct answer: D

Why D is correct

Correct: hypotenuse = 2 × short leg = 14

Why other options are wrong

A: Equal to the short leg; would be an equilateral triangle

B: Uses 45-45-90 ratio (hypotenuse = leg × √2) — wrong triangle type

C: Long leg = 7√3; this is the side opposite 60°, not the hypotenuse

E: 3 × 7 = 21; confuses the ratio multiplier

⚠ Trap: Confusing the 30-60-90 ratio (1:√3:2) with the 45-45-90 ratio (1:1:√2)

Hard Example 3 Using The Side Length (6) As The Radius Instead Of Half The Side Length (3)

A square with side length 6 has a circle inscribed in it (tangent to all four sides). What is the area of the region inside the square but outside the circle?

A. 36 − 9π (Correct answer)
B. 36 − 6π
C. 36 − 36π
D. 36 + 9π
E. 6 − 3π

Step 1

Square area: 6² = 36

Step 2

Inscribed circle: radius = half the side length = 6/2 = 3

Step 3

Circle area: π(3)² = 9π

Step 4

Shaded area = 36 − 9π

Correct answer: A

Why A is correct

Correct: 36 − 9π

Why other options are wrong

B: Used r = √6 somehow; incorrect radius derivation

C: Used radius = 6 instead of 3: π(6)² = 36π

D: Added instead of subtracted the circle area

E: Used side length instead of side² for the square area

⚠ Trap: Using the side length (6) as the radius instead of half the side length (3)

Strategy Tips

Memorize all formulas: Area = πr², C = 2πr, 30-60-90 ratio (1:√3:2), 45-45-90 ratio (1:1:√2)
Always identify 'radius vs diameter' at the start of every circle problem
For shaded regions: shaded = total shape area − unshaded shape area
Use Pythagorean triples (3-4-5, 5-12-13) to save time on right triangle problems
Draw altitude lines to decompose complex polygons into triangles and rectangles

Common pitfalls

Using diameter instead of radius in the area formula (off by factor of 4)

Confusing 30-60-90 and 45-45-90 side ratios

Forgetting to square the radius (using πr instead of πr²)

Simple area/perimeter: 30-45 seconds. Composite figures: 90 seconds. Circle arc/sector: 60 seconds. If you can't recall a formula in 10 seconds, derive it from the basic shape.

Summary

No formula sheet on the ACT — memorize Area = πr², C = 2πr, and special right triangle ratios
Shaded area = large shape area − small shape area
Sum of interior angles of any polygon = (n−2) × 180°

A circle of radius 4 is inscribed in a rectangle whose width equals the diameter. Find the area of the four corner regions. Express your answer in terms of π.

Next: Coordinate Geometry and Trigonometry All ACT Math lessons