All lessons
Math 40 min

Geometry

Area, volume, angles, triangles, circles, coordinate geometry and trigonometry.

Board Buddy
Geometry on the ACT is all about knowing your formulas cold — there’s no formula sheet! Draw a quick sketch, label everything, and let the formulas do the work.

What it tests

Geometry questions cover shapes and space: angles, triangles, circles, area and volume, coordinate geometry, and right-triangle trigonometry. You need to know the formulas by heart and spot hidden shapes (like a right triangle inside a rectangle).

How it shows up

  • The Math test is 45 questions in 50 minutes — a little over a minute each. Every question has four answer choices, a calculator is allowed on the whole section, there is no penalty for guessing, and no formula sheet is given.
  • Geometry is one of the larger Preparing for Higher Math categories.
  • Figures are NOT necessarily drawn to scale — never measure or eyeball an answer from the picture.
  • Expect angle chases with parallel lines, right triangles, circle equations, arc length, volume of cylinders and cones, and SOH-CAH-TOA.

Key rules & concepts

The content you actually need to know.

Angles

  • Angles on a straight line add to 180°. Around a point: 360°. Vertical angles are equal.
  • Parallel lines cut by a transversal: corresponding and alternate interior angles are equal; same-side interior angles add to 180°.
  • Triangle angles add to 180°. Interior angles of an n-sided polygon add to (n − 2) · 180°.

Triangles

  • Area = ½ · base · height (height is perpendicular to the base).
  • Pythagorean theorem: a2 + b2 = c2. Common triples: 3-4-5, 5-12-13, 8-15-17 (and multiples like 6-8-10).
  • 45°-45°-90°: sides x, x, x√2. 30°-60°-90°: sides x (opposite 30°), x√3 (opposite 60°), 2x (hypotenuse).
  • Similar triangles have equal angles and proportional sides.
  • Triangle inequality: any two sides must add to more than the third side.

Circles

  • Area = πr2. Circumference = 2πr = πd.
  • Arc length = (θ/360°) · 2πr. Sector area = (θ/360°) · πr2.
  • Circle equation: (x − h)2 + (y − k)2 = r2, center (h, k), radius r.
  • A tangent line is perpendicular to the radius at the point of tangency.

Area and volume

  • Rectangle lw; parallelogram bh; trapezoid ½(b1 + b2)h.
  • Box lwh; cylinder πr2h; cone ⅓πr2h; sphere (4/3)πr3; pyramid ⅓ · (base area) · h.
  • Scaling: multiply all lengths by k → area multiplies by k2, volume by k3.

Coordinate geometry

  • Distance: √[(x2 − x1)2 + (y2 − y1)2] — it’s just the Pythagorean theorem.
  • Midpoint: ((x1 + x2)/2, (y1 + y2)/2).

Trigonometry

  • SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
  • sin2θ + cos2θ = 1 and tan θ = sin θ / cos θ.
  • Complementary angles: sin θ = cos(90° − θ).
  • π radians = 180°. Check your calculator is in the right mode (degrees vs. radians).

Strategy & traps

Step by step

  1. 1Sketch the figure (or redraw the given one) and label every given length and angle.
  2. 2Write the formula before plugging in numbers.
  3. 3Look for hidden right triangles, special triangles and similar triangles.
  4. 4Don’t trust how the picture looks — trust the numbers.

Diameter vs. radius

If the problem gives a diameter, halve it before using πr2.

Area vs. perimeter (or arc vs. sector)

Reread what the question wants: length or area?

Linear scaling of area

Doubling the sides of a square makes its area 4 times bigger, not 2.

Board Buddy
Board Buddy tip: No formula sheet on the ACT — if you can write it from memory, you own it. Label the drawing before you calculate.

Worked examples

Try each one first, then reveal the solution.

Example 1

In right triangle ABC, the right angle is at C, AC = 9 and BC = 12. What is sin A?

  • A3/5
  • B3/4
  • C4/5
  • D4/3

Example 2

What are the center and radius of the circle (x − 2)2 + (y + 5)2 = 49?

  • ACenter (2, −5), radius 7
  • BCenter (−2, 5), radius 7
  • CCenter (2, −5), radius 49
  • DCenter (−2, 5), radius 49

Example 3

A cylindrical can has a diameter of 6 inches and a height of 10 inches. What is its volume, in cubic inches?

  • A60π
  • B90π
  • C180π
  • D360π

Practice

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Question 1

Two parallel lines are cut by a transversal. Two same-side interior angles measure (3x + 10)° and (5x − 30)°. What is the value of x?

Question 2

In a 30°-60°-90° triangle, the side opposite the 30° angle is 7 units long. How long is the side opposite the 60° angle?

Question 3

A circle has a radius of 6. What is the length of an arc with a central angle of 120°?

Question 4

In the standard (x, y) coordinate plane, what is the midpoint of the segment with endpoints (−3, 8) and (5, −2)?

Question 5

For an acute angle θ, cos θ = 8/17. What is tan θ?