1 · Angle facts
Angles on a straight line add to 180°, and angles around a point add to 360°. Vertical angles (across from each other where two lines cross) are equal.
When a line crosses two parallel lines, the matching (corresponding) angles are equal, and the two inside angles on the same side add to 180°.
2 · Polygon angles
The inside angles of a polygon with n sides add to (n − 2) × 180°. A hexagon: 4 × 180° = 720°. In a regular hexagon each angle is 720° ÷ 6 = 120°.
The outside (exterior) angles of any polygon add to 360°. Each exterior angle of a regular polygon is 360° ÷ n, and the interior angle is 180° minus that.
3 · Triangle facts
An exterior angle of a triangle equals the sum of the two angles that are not next to it. If two angles are 50° and 70°, the exterior angle at the third corner is 120°.
Any side must be shorter than the other two added together and longer than their difference. Sides 5 and 9 mean the third side is between 4 and 14.
4 · The Pythagorean theorem
In a right triangle, a² + b² = c², where c is the side across from the right angle. Legs 6 and 8: 36 + 64 = 100, so c = 10.
Learn the common triples and their multiples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. To find a leg, subtract: a² = c² − b².
5 · Special right triangles
A 45°-45°-90° triangle has legs s and hypotenuse s√2. That is also the diagonal of a square with side s.
A 30°-60°-90° triangle has short leg s, long leg s√3, and hypotenuse 2s. The short leg is across from the 30° angle.
6 · Similar figures
Similar figures have the same shape, so matching sides share one scale factor. If the factor is k, areas grow by k² and volumes by k³.
Shadows make similar triangles: a 6-foot person with a 4-foot shadow and a tree with a 30-foot shadow give 6/4 = h/30, so the tree is 45 feet tall.
7 · Circles, arcs, and sectors
Circumference = 2πr and area = πr². An arc or sector is a fraction of the whole: central angle ÷ 360°.
Radius 9, 60° arc: 60/360 = 1/6 of the circumference 18π, which is 3π. The 60° sector is 1/6 of the area 81π, which is 13.5π.
8 · Angles in circles
An inscribed angle (its corner on the circle) is half the central angle that cuts off the same arc. Central angle 110° → inscribed angle 55°.
An angle that cuts off a diameter is always a right angle. So a triangle drawn with one side as a diameter and its third corner on the circle has a 90° angle at that corner.
9 · Distance and midpoint
The distance between two points is the hypotenuse of a right triangle: from (1, 2) to (7, 10) the legs are 6 and 8, so the distance is 10.
The midpoint averages the coordinates: ((1 + 7)/2, (2 + 10)/2) = (4, 6). If you know one end and the midpoint, take the same step again to reach the other end.
10 · Slopes and lines
Parallel lines have the same slope. Perpendicular slopes are negative reciprocals: a line with slope 2/3 is perpendicular to one with slope −3/2.
For the line through (2, 1) and (6, 9): the slope is 8 ÷ 4 = 2, and 1 = 2(2) + b gives b = −3, so y = 2x − 3.
Ready to practice?
The quiz gives you new questions each time and explains every answer. You'll also find more quizzes and full practice tests.