ACT Geometry & Trigonometry: Complete Guide + Practice

ACT says Geometry accounts for roughly 17–20% of current ACT Math.

You should be comfortable with:

• lines and angles;
• triangles;
• circles;
• coordinate geometry;
• area and volume;
• right-triangle trigonometry.

Geometry is only one piece of ACT Math — the ACT statistics and probability guide covers another major chunk, and the Enhanced ACT complete guide ties every section together.

Angle facts

Straight line:

[ 180^\circ ]

Full turn:

[ 360^\circ ]

Triangle interior angles:

[ 180^\circ ]

Vertical angles are equal.

Pythagorean theorem

For a right triangle:

[ a^2+b^2=c^2 ]

where c is the hypotenuse.

Example:

legs 9 and 12:

[ c^2=81+144=225 ]

[ c=15 ]

Special triangles

45-45-90

[ 1:1:\sqrt2 ]

30-60-90

[ 1:\sqrt3:2 ]

These can save time.

SOHCAHTOA

[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} ]

[ \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} ]

[ \tan\theta=\frac{\text{opposite}}{\text{adjacent}} ]

Circle formulas

Circumference:

[ C=2\pi r ]

Area:

[ A=\pi r^2 ]

Arc length:

[ \frac{\theta}{360^\circ}(2\pi r) ]

Sector area:

[ \frac{\theta}{360^\circ}(\pi r^2) ]

Coordinate geometry

Slope:

[ m=\frac{y_2-y_1}{x_2-x_1} ]

Distance:

[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ]

Midpoint:

[ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) ]

Area and volume

Rectangle:

[ A=lw ]

Triangle:

[ A=\frac12 bh ]

Cylinder:

[ V=\pi r^2h ]

Rectangular prism:

[ V=lwh ]

Original practice

1

A right triangle has legs 8 and 15. Hypotenuse?

2

A circle has radius 6. Area?

3

What is the slope between (2,5) and (6,13)?

4

In a 30-60-90 triangle, the short leg is 7. Hypotenuse?

5

Relative to angle θ, opposite=9 and adjacent=12. What is (\tan\theta)?

Answers

1. 17
2. (36\pi)
3. 2
4. 14
5. 3/4

Common traps

• using diameter where radius is required;
• choosing the wrong side for sine/cosine/tangent;
• forgetting the square on area formulas;
• slope subtraction in inconsistent order;
• mixing degrees and radians.

Once these traps feel manageable, the how many ACT practice tests guide explains how many timed sets to run before test day, and the rest of our ACT guides cover every other section.

Official source

• ACT Math Test Description: https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/description-of-math-test.html

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