10.9.3: Formula Review
- Page ID
- 129652
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)10.2 Angles
To translate an angle measured in degrees to radians, multiply by \(\frac{\pi}{180}\).
To translate an angle measured in radians to degrees, multiply by \(\frac{180}{\pi}\).
10.4 Polygons, Perimeter, and Circumference
The formula for the perimeter \(P\) of a rectangle is \(P=2 L+2 W\), twice the length \(L\) plus twice the width \(W\).
The sum of the interior angles of a polygon with \(n\) sides is given by
\[S=(n-2) 180^{\circ} . \nonumber \]
The measure of each interior angle of a regular polygon with \(n\) sides is given by
\[a=\frac{(n-2) 180^{\circ}}{n} . \nonumber \]
To find the measure of an exterior angle of a regular polygon with \(n\) sides we use the formula
\[b=\frac{360^{\circ}}{n} . \nonumber \]
The circumference of a circle is found using the formula \(C=\pi d\), where \(d\) is the diameter of the circle, or \(C=2 \pi r\), where \(r\) is the radius.
10.6 Area
The area of a triangle is given as \(A=\frac{1}{2} b h\), where \(b\) represents the base and \(h\) represents the height.
The formula for the area of a square is \(A=s \cdot s\) or \(A=s^2\).
The area of a rectangle is given as \(A=l w\).
The area of a parallelogram is \(A=b h\).
The formula for the area of a trapezoid is given as \(A=\frac{1}{2} h(a+b)\).
The area of a rhombus is found using one of these formulas:
- \(A=\frac{d_1 d_2}{2}\), where \(d_1\) and \(d_2\) are the diagonals.
- \(A=\frac{1}{2} b h\), where \(b\) is the base and \(h\) is the height.
The area of a regular polygon is found with the formula \(A=\frac{1}{2} a p\), where \(a\) is the apothem and \(p\) is the perimeter.
The area of a circle is given as \(A=\pi r^2\), where \(r\) is the radius.
10.7 Volume and Surface Area
The formula for the surface area of a right prism is equal to twice the area of the base plus the perimeter of the base times the height, \(S A=2 B+p h\), where \(B\) is equal to the area of the base and top, \(p\) is the perimeter of the base, and \(h\) is the height.
The formula for the volume of a rectangular prism, given in cubic units, is equal to the area of the base times the height, \(V=B \cdot h\), where \(B\) is the area of the base and \(h\) is the height.
The surface area of a right cylinder is given as \(S A=2 \pi r^2+2 \pi r h\).
The volume of a right cylinder is given as \(V=\pi r^2 h\).
10.8 Right Triangle Trigonometry
The Pythagorean Theorem states
\[a^2+b^2=c^2 \nonumber \]
where \(a\) and \(b\) are two sides (legs) of a right triangle and \(c\) is the hypotenuse.