Formulae Flashcards
Formulae and theorems important for AMC 10
Area of a triangle with sides a, b and included angle θ
1/2 · a · b · sin θ
a, b, c are the sides of the triangle with circumradius R.
abc/4R
Proof
Area of a kite
2.2 cm wide and 8 cm long
- d1</sub>d2
8. 8 cm2
Area of cyclic quadrilateral
- s = (a + b + c)/2
- sqrt( (s-a) * (s-b) * (s-c))
where s is the semiperimeter and a, b and c are the side lengths.
Area of triangle with sides a, b and c
- Let s= (a + b + c)/2
- Area = sqrt (s * (s-a) * (s-b) * (s-c))
Heron’s formula:
Area of regular polygon with perimeter p and apothem a.
- pa/2*
- p·cot 180/n* (where n is the number of sides)
- Inscribe the polygon a circle
- Draw a line from two adjacent vertices to the circumcenter.
- This creates a triangle that is 1/n of the total area.
Apothem: length of
The apothem of a regular polygon is the line segment drawn from the center of the polygon perpendicular to one of its edges. It is also the radius of the inscribed circle of the polygon.
Given the number of sides n and side length s the length of the apothem is s /( 2·tan (π/n) ).
Area of a regular hexagon with side s
A = (3/2)* sqrt(3) * s2
area of a hexagon with perimeter 24 cm
24*sqrt(3) square cm
~ 41.569 cm sq
Volume/Surface Area of a Cone
- V =πr2h/3*
- SA=πr2+πrl*
Where V is the volume, SA is the surface area, r is the radius of the circular base, h is the height, and l is the slant height.
Volume/Surface Area of a Sphere*
- V = (4/3 ) · πr3*
- SA = 4πr2*
Where V is the volume, SA is the surface area, and r is the radius of the sphere (which is radius of the central cross section/the base of the semisphere).
Volume/Surface Area of a Pyramid
- V = (1/2)bh*
- SA = 2sl + b*
Where V is the volume, SA is the surface area, b is the area of the base, h is the height, l is the slant height, and s is the length of a side of the base. Note that a pyramid can have a base of any polygon, but if none is specified, assume a square base. A pyramid with a triangular base is known as a tetrahedron.
Volume/Surface Area of a Prism*
- V =lwh*
- SA=2(lw+lh+wh)*
Where V is the volume, SA is the surface area, l is the length, w is the width, and h is the height.
Common Pythagorean triples
3 4 5
5 12 13
7 24 25
8 15 17
Important:or these side lengths multiplied by some factor, it is a right triangle)
distance between the line ax+by+c=0 and point (x1,y1) is
- |ax1 + by1+c| /sqrt(a2+b2)*