SAT – Maths : Formula Sheet

Area of a Circle

\[
\begin{aligned}
& A=\pi r^2 \\
& C=2 \pi r
\end{aligned}
\]
\[
A=\pi r^2
\]
\(\pi\) is a constant that can, for the purposes of the SAT, be written as 3.14 (or 3.14159 )
 \(r\) is the radius of the circle (any line drawn from the center point straight to the edge of the circle)

Circumference of a Circle

\[
C=2 \pi r(\text { or } C=\pi d)
\]

 \(d\) is the diameter of the circle. It is a line that bisects the circle through the midpoint and touches two ends of the circle on opposite sides. It is twice the radius.

Area of a Rectangle

\(A=l w\)
 \(l\) is the length of the rectangle
 \(w\) is the width of the rectangle

Area of a Triangle

\[
\begin{aligned}
A & =\frac{1}{2} b h \\
\end{aligned}
\]

\(b\) is the length of the base of triangle (the edge of one side)
\(h\) is the height of the triangle
 In a right triangle, the height is the same as a side of the 90-degree angle. For non-right triangles, the height will drop down through the interior of the triangle, as shown above (unless otherwise given).

The Pythagorean Theorem

\[
a^2+b^2=c^2
\]

In a right triangle, the two smaller sides ( \(a\) and \(b\) ) are each squared. Their sum is the equal to the square of the hypotenuse (c, longest side of the triangle).

Properties of Special Right Triangle: Isosceles Triangle

An isosceles triangle has two sides that are equal in length and two equal angles opposite those sides.
An isosceles right triangle always has a 90-degree angle and two 45 degree angles.
The side lengths are determined by the formula: \(x, x, x \sqrt{2}\), with the hypotenuse (side opposite 90 degrees) having a length of one of the smaller sides * \(\sqrt{2}\).
E.g., An isosceles right triangle may have side lengths of 12,12 , and \(12 \sqrt{2}\).

Properties of Special Right Triangle: 30, 60, 90 Degree Triangle

body_306090_triangle.png

  •  A 30, 60, 90 triangle describes the degree measures of the triangle’s three angles.
  • The side lengths are determined by the formula: \(x, x \sqrt{3}\), and \(2 x\)
    • The side opposite 30 degrees is the smallest, with a measurement of \(x\).
    • The side opposite 60 degrees is the middle length, with a measurement of \(x \sqrt{3}\).
    • The side opposite 90 degrees is the hypotenuse (longest side), with a length of \(2 x\).
    • For example, a 30-60-90 triangle may have side lengths of \(5,5 \sqrt{3}\), and 10 .

Volume of a Rectangular Solid

\[
V=l w h
\]

\(l\) is the length of one of the sides.
\(h\) is the height of the figure.
\(w\) is the width of one of the sides.

Volume of a Cylinder

\[
V=\pi r^2 h
\]

\(r\) is the radius of the circular side of the cylinder.
\(h\) is the height of the cylinder.

Volume of a Sphere

\[
V=\left(\frac{4}{3}\right) \pi r^3
\]
 \(r\) is the radius of the sphere.

Volume of a Cone

       

\[
V=\left(\frac{1}{3}\right) \pi r^2 h
\]

 \(r\) is the radius of the circular side of the cone.
\(h\) is the height of the pointed part of the cone (as measured from the center of the circular part of the cone).

Volume of a Pyramid

\[
V=\left(\frac{1}{3}\right) l w h
\]
\(l\) is the length of one of the edges of the rectangular part of the pyramid.
\(h\) is the height of the figure at its peak (as measured from the center of the rectangular part of the pyramid).
\(w\) is the width of one of the edges of the rectangular part of the pyramid.

Law: the number of degrees in a circle is 360
Law: the number of radians in a circle is \(2 \pi\)
Law: the number of degrees in a triangle is 180

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