Areas and volumes
Areas of rhombi, trapezoids and polygons on grid paper (Pick's theorem), how area changes under similarity, volumes of prisms, cylinders, cones and a ball.
How to solve it
Area is how many unit squares fit into a figure, volume is how many unit cubes. Two rules follow: if every size grows $k$ times, the area grows $k^2$ times and the volume $k^3$ times. And a polygon on grid paper can be measured just by counting grid points.
Step by step
- A rhombus, or any quadrilateral with perpendicular diagonals: half the product of the diagonals. A trapezoid: the average of the bases times the height.
- A polygon with vertices at grid points: count the points inside ($I$) and on the boundary ($B$) and use Pick's theorem.
- Similarity: sides $k$ times larger make the area $k^2$ times larger and the volume $k^3$ times.
- The volume of a prism or a cylinder is the base area times the height; a pyramid or a cone is a third of that; a ball is $\frac{4}{3}\pi r^3$.
Common mistakes
- Thinking that sides $7$ times longer make the area $7$ times larger. It is $49$ times.
- Taking the product of a rhombus's diagonals without the half.
- Mixing up the inside and boundary points in Pick's theorem, or leaving the vertices out of the boundary.
- Computing a cone's volume like a cylinder's, without the $\frac{1}{3}$.