ACT Geometry & Trigonometry Reference

What You Need to Know

ACT Geometry & Trigonometry questions are mostly formula recognition + clean setup. You’ll be asked to find lengths, angles, areas, volumes, slopes, and trig values using a small set of high-frequency rules.

Your core toolkit:

• Right triangles: Pythagorean Theorem + basic trig ratios.
• Similarity: proportional sides and scale factors.
• Circles: radius/diameter, arc/sector, tangents.
• Coordinate geometry: slope, distance, midpoint.
• Area/volume: memorize the common shapes.

ACT reality check: you’re usually not proving theorems. You’re matching the diagram to the right formula, keeping track of scale factors, and avoiding traps (radius vs diameter, degrees vs radians, perimeter vs area scaling).

Step-by-Step Breakdown

A. Solving most geometry problems (universal process)

1. Mark the diagram: write given lengths/angles directly on it; label unknowns.
2. Identify the shape family: right triangle, circle, similar triangles, polygon, coordinate plane, 3D solid.
3. Choose the fastest relationship:
   • Right triangle: $a^2+b^2=c^2$ or $\sin(\theta),\cos(\theta),\tan(\theta)$.
   • Similar: side ratios and scale factors.
   • Circle: circumference/area, arc/sector, tangent facts.
   • Coordinate: slope/distance.
4. Set up one equation (or one proportion) before calculating.
5. Check units and reasonableness (bigger angle faces bigger side; hypotenuse is longest; area can’t exceed bounding box, etc.).

B. Similarity + scale factor (the most common “hidden” method)

1. Confirm similarity using AA (two angles equal) or parallel lines creating equal angles.
2. Write the scale factor $k=\frac{\text{new}}{\text{old}}$ using corresponding sides.
3. Scale:
   • Lengths multiply by $k$.
   • Areas multiply by $k^2$.
   • Volumes multiply by $k^3$.

Mini-example (scale factor):
If triangle $A$ is similar to triangle $B$ and $\frac{\text{side}_B}{\text{side}_A}=\frac{3}{2}$, then

• a corresponding perimeter scales by $\frac{3}{2}$,
• a corresponding area scales by $\left(\frac{3}{2}\right)^2=\frac{9}{4}$.

C. Right-triangle trig workflow

1. Identify the angle $\theta$ you’re using.
2. Label sides relative to $\theta$: opposite, adjacent, hypotenuse.
3. Choose ratio:
   • $\sin(\theta)=\frac{\text{opp}}{\text{hyp}}$
   • $\cos(\theta)=\frac{\text{adj}}{\text{hyp}}$
   • $\tan(\theta)=\frac{\text{opp}}{\text{adj}}$
4. Solve for the unknown; if needed use inverse trig (calculator) for angles.

Mini-example (trig ratio):
If $\tan(\theta)=\frac{5}{12}$ in a right triangle, you can treat opposite $=5$ and adjacent $=12$, so hypotenuse $=13$ by $5\text{-}12\text{-}13$.

Key Formulas, Rules & Facts

A. Triangles (including right triangles)

Special right triangles (memorize):

• $45^\circ\text{-}45^\circ\text{-}90^\circ$: sides $x,\,x,\,x\sqrt{2}$
• $30^\circ\text{-}60^\circ\text{-}90^\circ$: sides $x,\,x\sqrt{3},\,2x$ (short leg $x$ opposite $30^\circ$)

Common Pythagorean triples:

• $3\text{-}4\text{-}5$ and multiples
• $5\text{-}12\text{-}13$ and multiples
• $8\text{-}15\text{-}17$ and multiples

B. Trigonometry essentials

Degree-radian conversion:

• $180^\circ=\pi$ radians
• $\theta_{\text{rad}}=\theta_{\circ}\times\frac{\pi}{180}$
• $\theta_{\circ}=\theta_{\text{rad}}\times\frac{180}{\pi}$

C. Circles

D. Polygons & angle sums

Other polygon facts:

• Each exterior angle (regular) is $\frac{360^\circ}{n}$.
• Number of diagonals in an $n$-gon: $\frac{n(n-3)}{2}$.

E. Coordinate geometry

F. Area (2D) and Volume/Surface Area (3D)

Area and perimeter basics

3D solids

Reminder: For cones, slant height $\ell$ is not the same as height $h$. Often $\ell$ comes from a right triangle: $\ell^2=r^2+h^2$.

Examples & Applications

1) Similar triangles + area scaling

Triangles are similar with side ratio $\frac{\text{big}}{\text{small}}=\frac{5}{2}$. If the small triangle’s area is $12$, find the big triangle’s area.

• Area scale factor: $\left(\frac{5}{2}\right)^2=\frac{25}{4}$
• Big area: $12\times\frac{25}{4}=75$

2) Coordinate geometry (distance + midpoint)

Find the distance and midpoint between $(-2,3)$ and $(4,-5)$.

• Distance: $d=\sqrt{(4-(-2))^2+(-5-3)^2}=\sqrt{6^2+(-8)^2}=\sqrt{100}=10$
• Midpoint: $\left(\frac{-2+4}{2},\,\frac{3+(-5)}{2}\right)=(1,-1)$

3) Trig in a right triangle

A ladder makes a $60^\circ$ angle with the ground and reaches $12\,\text{ft}$ up a wall (vertical height). How long is the ladder?

• Height is opposite the ground angle, ladder is hypotenuse.
• $\sin(60^\circ)=\frac{12}{L}$
• $L=\frac{12}{\sin(60^\circ)}=\frac{12}{\frac{\sqrt{3}}{2}}=\frac{24}{\sqrt{3}}=8\sqrt{3}$

4) Arc length vs sector area (degrees vs radians)

A circle has radius $6$ and central angle $120^\circ$. Find arc length.

• Degree method: $s=\frac{120^\circ}{360^\circ}\times 2\pi(6)=\frac{1}{3}\times 12\pi=4\pi$

(If using radians: $120^\circ=\frac{2\pi}{3}$, then $s=r\theta=6\times\frac{2\pi}{3}=4\pi$.)

Common Mistakes & Traps

1. Mixing up radius and diameter

   • Wrong: plugging diameter into $A=\pi r^2$ or $C=2\pi r$.
   • Fix: if you see diameter $d$, convert using $r=\frac{d}{2}$.

2. Forgetting area/volume scale differently than lengths

   • Wrong: if sides scale by $k$, assuming area scales by $k$.
   • Fix: memorize: lengths $k$, areas $k^2$, volumes $k^3$.

3. Using the wrong “height” in area formulas

   • Wrong: trapezoid/triangle height that is not perpendicular.
   • Fix: height is always the perpendicular distance to the base (often drawn with a right angle).

4. Trig ratio uses the wrong sides (relative to the angle)

   • Wrong: labeling opposite/adjacent without referencing $\theta$.
   • Fix: point to $\theta$ first, then label sides: opposite is across, adjacent touches, hypotenuse is across from $90^\circ$.

5. Arc/sector formulas with degrees plugged into radian formulas

   • Wrong: using $s=r\theta$ with $\theta$ in degrees.
   • Fix: use radians for $s=r\theta$ and $A=\frac{1}{2}r^2\theta$, or use the $\frac{\theta}{360^\circ}$ versions.

6. Assuming a figure is a square or a right angle exists without a marking

   • Wrong: using right-triangle rules just because it “looks” right.
   • Fix: rely on markings, given information, or provable facts (like tangent ⟂ radius).

7. Slope sign errors and flipped subtraction

   • Wrong: $\frac{y_2-y_1}{x_1-x_2}$ or inconsistent order.
   • Fix: keep the same order top and bottom: $\frac{y_2-y_1}{x_2-x_1}$.

8. Cone slant height confusion

   • Wrong: using $h$ instead of $\ell$ in lateral area $\pi r\ell$.
   • Fix: lateral surface area uses $\ell$; find $\ell$ with $\ell^2=r^2+h^2$ if needed.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can spot and use $a^2+b^2=c^2$ and common triples.
• You have $30\text{-}60\text{-}90$ and $45\text{-}45\text{-}90$ memorized.
• You can apply SOH-CAH-TOA with correct opposite/adjacent.
• You remember similarity scaling: lengths $k$, areas $k^2$, volumes $k^3$.
• You know circle basics: $C=2\pi r$, $A=\pi r^2$, and arc/sector formulas.
• You can do slope, distance, midpoint quickly and cleanly.
• You can compute polygon interior sum: $(n-2)\times 180^\circ$.
• You check for classic traps: radius vs diameter, degrees vs radians, slant height vs height.

You’ve got this: keep setups simple, label everything, and let the formulas do the work.