Areas and Volumes - Yousef's Notes
Areas and Volumes

Areas and Volumes

#Areas

#Area of a Parallelogram

#Area of a Triangle

#Area of a Triangle of 3 points

#Volumes

#Exercises

Find the area of the parallelogram with edges

  • Find the area of the parallelogram with edges $\mathbf{v} = \begin{bmatrix} 3 \ 2 \end{bmatrix}$ and $\mathbf{w} = \begin{bmatrix} 1 \ 4 \end{bmatrix}$.
$$ \text{Area} = \begin{vmatrix} 3 & 2 \\ 1 & 4 \end{vmatrix} = 12 - 2 = 10 $$
  • A box has edges from $\begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}$ to $\begin{bmatrix} 3 \ 1 \ 1 \end{bmatrix}$ and $\begin{bmatrix} 1 \ 3 \ 1 \end{bmatrix}$ and $\begin{bmatrix} 1 \ 1 \ 3 \end{bmatrix}$. Find its volume.
$$ \text{Volume} = \begin{vmatrix} 3 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 3 \end{vmatrix} = 27 + 1 + 1 - 3 - 3 - 3 = 20 $$
  • The corners of a triangle are $\begin{bmatrix} 2 \ 1 \end{bmatrix}$, $\begin{bmatrix} 3 \ 4 \end{bmatrix}$, and $\begin{bmatrix} 0 \ 5 \end{bmatrix}$. What is the area?
$$ \text{Area} = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \frac{1}{2} \begin{vmatrix} 2 & 1 & 1 \\ 3 & 4 & 1 \\ 0 & 5 & 1 \end{vmatrix} = \frac{1}{2} (8 + 15 + 0 - 0 - 10 - 3) = \frac{1}{2} 10 = 5 $$