ACT Math · Lesson 4 of 5
Geometry, circles, and trigonometry
Measure diagrams with the right relationship: angle facts, area and volume, circles, and trigonometric ratios.
Plane figures
Start by naming the figure and the requested measurement. A rectangle with length 8 and width 3 has area 24 and perimeter 22. Confusing those two answers is common because both calculations use the same two numbers.
A circle with radius r has area πr² and circumference 2πr. A sector is a fraction of the circle. A 90° sector is one quarter of both the area and the arc, but the numbers differ because one uses πr² and the other uses 2πr.
Trigonometry
In a right triangle, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. The adjacent side is next to θ but is not the hypotenuse.
Angle measure may be requested in degrees. A straight angle is 180°, and a full turn is 360°. If two lines are parallel, corresponding angles are equal, which often supplies the missing angle in a triangle.
Worked example
A right triangle has an angle of 30° and a hypotenuse of 10. What is the side opposite the 30° angle?
- Sine uses the opposite side: sin(30°) = opposite / 10.
- sin(30°) = 1/2, so opposite / 10 = 1/2.
- The opposite side is 5.
Why this works. Identify the sides relative to the stated angle, not relative to the picture’s orientation.
Check your understanding
- Separate area from perimeter before calculating.
- Use r or d consistently for a circle.
- Name the opposite and adjacent sides for the given angle.