SAT Math · Lesson 4 of 4
Geometry and trigonometry
Connect diagrams, circles, and right-triangle ratios to exact relationships rather than to how a figure looks.
Lengths, areas, and circles
Similar triangles have corresponding angles equal and corresponding sides in proportion. Corresponding sides are across from equal angles, not merely the sides that look parallel in a sketch.
For a circle, radius and diameter are different: the diameter is twice the radius. The circumference is 2πr, and the area is πr². A central angle determines an arc length that is the same fraction of the circumference as the angle is of 360°, or of 2π radians.
Right-triangle trigonometry
Sine, cosine, and tangent are ratios for a chosen acute angle. Sine is opposite over hypotenuse. Changing the reference angle swaps which leg is opposite.
If a right triangle has legs 5 and 12, the hypotenuse is 13. The sine of the angle opposite the side of length 5 is 5/13. The sine of the other acute angle is 12/13.
Worked example
A circle has radius 6. What is its area, and what is the length of a 60° arc?
- Area = πr² = 36π.
- A 60° arc is 60/360 = 1/6 of the circle.
- The circumference is 12π, so the arc length is 12π / 6 = 2π.
Why this works. Area uses r². Arc length uses a fraction of the circumference, which depends on r once, not on r².
Check your understanding
- Mark the reference angle before choosing sine, cosine, or tangent.
- Use the radius, not the diameter, in πr².
- Match corresponding sides in similar figures by equal angles.