面積と体積

The core area formulae are as follows. Rectangle: $A=\text{base}\times \text{height}$. Triangle: $A=\frac{1}{2}\times \text{base}\times \text{height}$. Parallelogram: $A=\text{base}\times \text{height}$, where the height is the perpendicular distance to the base. Trapezium: $A=\frac{1}{2}(a+b)h$, where $a$ and $b$ are the parallel sides and $h$ is the distance between them. For triangles and parallelograms, remember the height is the perpendicular distance, not the slanted side. Area is always measured in square units (cm${}^2$, etc.).

With radius $r$ and diameter $d=2r$, the distance around a circle (the circumference) is $C=2\pi r=\pi d$, and the area of the circle is $A=\pi r^2$. Here $\pi$ (pi) is a constant of about $3.142$; use the value the question specifies, or the calculator's $\pi$ otherwise. Take care not to confuse radius and diameter, and remember that for the area you square the radius first, then multiply by $\pi$. In this topic we use $\pi =3.142$ for numerical work.

A prism is a solid with the same cross-section all the way along its length. Its volume is always 'area of cross-section $\times$ length (height)'. A cuboid is a prism with a rectangular cross-section, so $V=\text{length}\times \text{width}\times \text{height}$. A cylinder is a prism with a circular cross-section, so we multiply the cross-sectional area $\pi r^2$ by the height $h$ to get $V=\pi r^{2}h$. Volume is always measured in cubic units (cm${}^3$, etc.).

The surface of a closed cylinder (like a solid can) is made of two circular ends and a curved side which unrolls into a rectangle. The two circles together have area $2\pi r^2$. The curved (lateral) surface is a rectangle whose width is the circumference $2\pi r$ and whose height is $h$, giving area $2\pi rh$. So the total surface area is $A=2\pi r^2+2\pi rh$. For an open tube or a container with no lid, be careful to add only the faces that are actually present.