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ACT · Domain 4

Geometry

12–15% of the ACT math section

Shapes, solids and their measures, and trigonometry through the unit circle. The ACT prints no formula sheet: area, volume, and the special triangles have to be carried in.

Geometry · Topic 1 of 4

Shapes, Solids & Their Measures

The SAT hands you a reference sheet; the ACT does not. Circle area and circumference, the volume of a box, a cylinder, a cone and a sphere, the angle sum of a polygon, and the Pythagorean triples — all of it is assumed, and a question that needs one of them is meant to be fast.

The problem
A right circular cylinder has a radius of 3 and a height of 10. What is its volume?
  1. A90π90\pi
  2. B30π30\pi
  3. C60π60\pi
  4. D300π300\pi
Working1 / 3
  1. A cylinder is a circle stacked to a heightV=πr2hV = \pi r^2 h
Geometry · Topic 2 of 4

Circles, Arcs & Sectors

An arc is a fraction of the circumference and a sector is the same fraction of the area, and the fraction is the central angle over 360. That one sentence is every arc and sector question on the ACT. Circumference 2πr2\pi r and area πr2\pi r^2 are not printed; carry them in.

The problem
A circle has a radius of 6. What is the area of a sector of the circle with a central angle of 60∘60^\circ?
  1. F6π6\pi
  2. G2π2\pi
  3. H36π36\pi
  4. J12π12\pi
Working1 / 3
  1. The fraction of the circle60360=16\frac{60}{360} = \frac{1}{6}
Geometry · Topic 3 of 4

Trigonometry

SOH CAH TOA covers the right-triangle questions. The ACT goes further: angles past 90∘90^\circ, radians, and the graphs of sine and cosine. For an angle past 90∘90^\circ, find the reference angle it makes with the xx-axis, take the sine or cosine of that, and give it the sign of the quadrant.

The problem
What is the value of sin⁡(210∘)\sin(210^\circ)?
  1. A−12-\dfrac{1}{2}
  2. B12\dfrac{1}{2}
  3. C−32-\dfrac{\sqrt{3}}{2}
  4. D32\dfrac{\sqrt{3}}{2}
Working1 / 3
  1. 210∘210^\circ is 30∘30^\circ past 180∘180^\circ: reference angle 30∘30^\circ210−180=30210 - 180 = 30
Geometry · Topic 4 of 4

Coordinate Geometry

Distance is Pythagoras with the legs read off the coordinates; the midpoint is the average of the two ends; a perpendicular slope is the negative reciprocal. Three formulas, and they are usually one step of a longer question — the distance is a radius, the midpoint is a centre.

The problem
A circle in the standard (x,y)(x, y) coordinate plane has a diameter with endpoints (−2,1)(-2, 1) and (4,9)(4, 9). Which of the following is an equation of the circle?
  1. F(x−1)2+(y−5)2=25(x - 1)^2 + (y - 5)^2 = 25
  2. G(x−1)2+(y−5)2=100(x - 1)^2 + (y - 5)^2 = 100
  3. H(x+1)2+(y+5)2=25(x + 1)^2 + (y + 5)^2 = 25
  4. J(x−2)2+(y−10)2=25(x - 2)^2 + (y - 10)^2 = 25
Working1 / 3
  1. The centre is the midpoint of the diameter(−2+42,  1+92)=(1,5)\left(\frac{-2 + 4}{2},\; \frac{1 + 9}{2}\right) = (1, 5)