ACT Math Geometry: angles and triangles, right-triangle trig, circles, area and volume, coordinate geometry and similarity

What the Geometry area covers

Geometry, about 12 to 15 percent of the ACT Math test, is one of its largest areas. It runs from angle chasing to trigonometry, circles, measurement and the coordinate plane. This guide ties together the dot points: angles, lines and triangles, right triangle trigonometry, circles and their equations, area, perimeter and volume, coordinate geometry, and similarity and congruence.

Angles, lines and triangles

Vertical angles are equal; a linear pair is supplementary. Across parallel lines, corresponding and alternate angles are equal and same-side interior angles are supplementary. A triangle's angles sum to $180°$, an exterior angle equals the sum of the two remote interior angles, and isosceles base angles are equal. The triangle inequality limits the third side.

Right triangle trigonometry

The Pythagorean theorem $a^{2} + b^{2} = c^{2}$ relates the legs and hypotenuse; common triples and the special triangles (45-45-90 as $x : x : x\sqrt{2}$, 30-60-90 as $x : x\sqrt{3} : 2x$) speed it up. SOH-CAH-TOA gives the ratios: $\sin = \frac{\text{opp}}{\text{hyp}}$, $\cos = \frac{\text{adj}}{\text{hyp}}$, $\tan = \frac{\text{opp}}{\text{adj}}$.

Circles and their equations

Circumference is $2\pi r$, area $\pi r^{2}$. An arc or sector is the fraction $\frac{\theta}{360}$ of the whole. The standard equation $(x - h)^{2} + (y - k)^{2} = r^{2}$ gives centre $(h, k)$ and radius $r$ (right side is $r^{2}$).

Area, perimeter and volume

Area: rectangle $lw$, triangle $\frac{1}{2}bh$, trapezoid $\frac{1}{2}(b_1 + b_2)h$, circle $\pi r^{2}$. Volume: prism or cylinder is (base area) times height, so a cylinder is $\pi r^{2}h$; cones and pyramids hold one third of that. Surface area is the total of all faces. Split composite figures into known pieces.

Coordinate geometry

Distance is $\sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$ (Pythagoras); the midpoint averages the coordinates; the slope is $\frac{y_2 - y_1}{x_2 - x_1}$. Parallel lines have equal slopes; perpendicular slopes are negative reciprocals.

Similarity and congruence

Congruent figures match in size and shape; similar figures have equal angles and proportional sides. Solve for a missing side with a proportion. With length scale factor $k$, areas scale by $k^{2}$ and volumes by $k^{3}$.

Check your knowledge

Try these, then read the solutions.

1. Two angles of a triangle are $45°$ and $65°$. Find the third. [1 point]
2. A right triangle has legs 8 and 6. Find the hypotenuse. [2 points]
3. A circle has radius 5. Find its area in terms of $\pi$. [1 point]
4. A cylinder has radius 2 and height 9. Find its volume in terms of $\pi$. [2 points]
5. Two similar figures have length scale factor 4. How many times larger is the area? [1 point]