🧩 2D Combinations, Ratios & Scale

• Inscribed & compound shapes: split a compound figure into rectangles, triangles or semicircles and add or subtract their areas. When a circle is inscribed in a square the circle's diameter equals the square's side, so $r = \dfrac{a}{2}$. When a square is inscribed in a circle the square's diagonal equals the circle's diameter, so $a\sqrt{2} = 2r$ and the area is just $2r^2$.

• Ratio of areas: this is the single biggest time-saver. If two similar figures have lengths (sides, radii, diagonals, perimeters) in ratio $a:b$, then their areas are in ratio $a^2:b^2$, and for solids the volumes are in $a^3:b^3$. Square the linear ratio for area — never leave it linear.

• Perimeter & wire-bending: bending one wire into different shapes keeps the total length — the perimeter — constant. So equate perimeters: the square's $4a$ becomes the circle's $2\pi r$, find the new dimension, then answer whatever is asked.

• Scale & maps: a $1:n$ scale means real length $= n \times$ map length. Because area has two dimensions, real area $= n^2 \times$ map area. Square the scale before multiplying — that is the only trap.

The habit to build: read the figure, decide which of these four it is, then apply one formula. With $\pi = \dfrac{22}{7}$ a circumference of 44 always means radius 7 — keep that anchor handy.