Area & Volume

2D Area and Perimeter

For a rectangle, Area = $l \times w$. For a triangle, Area = $\frac{1}{2}bh$, where $h$ is the perpendicular height: the distance from the apex straight down to the base line at 90°. It is not the length of a sloping side. If you don't have the height, use the area of a triangle formula: Area = $\frac{1}{2}ab \sin C$, where $C$ is the angle between sides $a$ and $b$. For a trapezium, use Area = $\frac{a+b}{2} \times h$. For a triangle defined by coordinates, translate one vertex to the origin and use Area = $\frac{1}{2}|x_1y_2 - x_2y_1|$.

Circles, Sectors, and Arcs

For a circle, Area = $\pi r^2$ and Circumference = $2\pi r$. When dealing with a sector, the area is $\pi r^2 \times (\frac{\theta}{360^\circ})$ and arc length is $2\pi r \times (\frac{\theta}{360^\circ})$. If the angle $\theta$ is in radians, use the simpler versions: Area = $\frac{1}{2}r^2\theta$ and arc length = $r\theta$. Perimeter of a sector is the arc length plus two radii. Before you substitute, check if your angle is in degrees or radians.

3D Solids and Nets

A net is the flat shape formed by unfolding a solid. It shows which faces exist. A cylinder's net is two circles plus a rectangle ($2\pi r \times h$). A cone's net is a circle plus a sector ($l$ is the slant height). For any prism — a solid with the same cross-section all the way through — Volume = area of cross-section $\times$ length, and Surface area = $2 \times (\text{cross-section area}) + (\text{perimeter of cross-section}) \times \text{length}$. A rectangular block has Volume = $lbh$ and Surface area = $2(lb + lh + bh)$. A triangular prism's net is two identical triangles plus three rectangles, each as long as the prism; for right-angled, isosceles, or equilateral triangular bases, find the perpendicular height first to calculate the cross-section area. A cylinder is just a prism with a circular cross-section, which is why its volume is $\pi r^2 \times h$. Memorise these: Cylinder Volume = $\pi r^2h$, CSA = $2\pi rh$. Cone Volume = $\frac{1}{3}\pi r^2h$, CSA = $\pi rl$, with $l^2 = h^2 + r^2$. Sphere Volume = $\frac{4}{3}\pi r^3$, Area = $4\pi r^2$. A hemisphere is half: Volume = $\frac{2}{3}\pi r^3$, curved surface area = $2\pi r^2$, total surface area = $3\pi r^2$.

Composite Solids and Conservation of Volume

For composite solids, don't use a blanket subtraction rule. Work piece by piece and ask: is this face on the outside or is it glued to another piece? Add only the surfaces you could paint. When melting or reshaping solids, volume is conserved. Set the initial volume equal to the final volume and solve for the unknown dimension. If one solid sits inside another, draw the 2D cross-section and look for a right-angled triangle to connect the dimensions.

The Trapezoidal Rule

Use the trapezoidal rule to estimate the area of an irregular shape from $n$ equally spaced offsets $y_1, y_2, \dots, y_n$. These give $n - 1$ strips, each of the same width $h$, so if the offsets span a total distance $d$ then $h = \frac{d}{n - 1}$: Area $\approx \frac{h}{2}[y_1 + y_n + 2(y_2 + \dots + y_{n-1})]$. First and last heights appear once; middle heights are doubled. An end offset of zero is still an offset; keep it as 0 in the formula. If the boundary curves out, the rule underestimates; if it curves in, it overestimates.

Area Theorems

Three theorems from the formal geometry course are used to justify area steps. Theorem 16 states that for a triangle, base $\times$ height is constant regardless of base choice, explaining why $\frac{1}{2}bh$ works. Theorem 17 notes a diagonal bisects a parallelogram's area. Theorem 18 confirms a parallelogram's area is base $\times$ perpendicular height. Quote these by name if a geometry question asks you to justify an area step.

Key terms

Perpendicular Height

The distance from an apex straight down to the base line at a 90° angle.

Slant Height ($l$)

The distance along the sloped surface of a cone, linked to radius and height by $l^2 = h^2 + r^2$.

Curved Surface Area

The surface area of a solid excluding flat bases or tops.

Total Surface Area

The sum of all external surfaces of a 3D solid.

Net

The 2D flat pattern created by unfolding a 3D solid along its edges.

Sector

A wedge of a circle bounded by two radii and an arc.

Arc Length

The length of the curved boundary of a sector.

Trapezoidal Rule

An approximation method for irregular area using equally spaced vertical offsets.

Radian

A unit of angle measure where a full rotation is 2π radians.