Area And Perimeter

Perimeter

• **Perimeter** is the total distance around the outside of a 2D shape; for a circle it is called **circumference**.
• Perimeter is a length measured in units such as mm, cm, m.
• Add together the lengths of all sides of the shape.
• For a regular polygon, perimeter = number of sides × side length (e.g., square side x → 4x).
• For compound shapes, split into rectangles, triangles, or parts of circles; use properties like equal sides or Pythagoras' theorem to find missing lengths.
• In an L-shape, the sum of two shorter horizontal lengths equals the longer opposite length.

Area

• **Area** is the amount of space within the perimeter of a 2D shape, measured in square units $(mm^{2}, cm^{2}, m^{2})$.
• Rectangle area = length × width $(A = lw)$.
• Triangle area = ½ × base × perpendicular height $(A =$ ½ bh).
• Trapezium area = ½ $\times (sum$ of parallel $sides) \times$ perpendicular height $(A =$ ½ (a+b)h).
• Parallelogram area = base × perpendicular height $(A = bh)$.
• The perpendicular height is the distance between the base and the opposite side, not necessarily a side length.

Adding & Subtracting Areas

• **Compound shapes** can be split into standard shapes (rectangles, triangles, etc.) to find area.
• Find areas of each standard shape and add them together.
• Alternatively, add an extra shape to form a larger standard shape, then subtract the area of the extra shape.
• Always check for missing lengths using given dimensions and shape properties.

Problem Solving with Areas

• Real-life problems often involve area and cost (e.g., carpet, paint, tiles).
• Read the context carefully: look for key words like area, cost, minimum, maximum.
• Use compound units (e.g., £/m²) to identify needed calculations.
• Annotate diagrams with known and missing lengths.
• Compare different options (e.g., companies) by calculating total cost using area.

Area and Perimeter with Bounds

• When measurements are given to a certain accuracy, find **upper and lower bounds** for perimeter or area.
• For a square side length correct to nearest cm, upper bound perimeter $= 4 \times (side + 0.5)$.
• For a rectangle with length and width correct to 1 d.p., upper bound perimeter $= 2 \times (length+0.05 + width+0.05)$.
• For area, upper bound $= (upper$ bound $length) \times (upper$ bound width).
• For a triangle with base and height given to nearest cm, upper bound area = ½ $\times (base+0.5) \times (height+0.5)$.

Common Shapes and Formulas

• **Square**: perimeter $= 4s$, area $= s^{2}$.
• **Rectangle**: perimeter $= 2(l+w)$, area $= lw$.
• **Triangle**: area = ½ bh; perimeter $= sum$ of three sides.
• **Trapezium**: area = ½ (a+b)h; perimeter $= sum$ of all four sides.
• **Parallelogram**: area = bh; perimeter $= 2(a+b)$ where a and b are adjacent sides.
• **Circle**: circumference $= 2\pi r$ or πd, area $= \pi r^{2}$.

Working with Compound Shapes

• Split the shape into rectangles, triangles, or parts of circles.
• Use subtraction method: area of large shape minus area of removed shape(s).
• Example: a path around a pond – area of outer rectangle minus area of inner rectangle.
• Example: a circle with two smaller circles removed – area of large circle minus $2\times$ area of small circle.

Problem Solving with Tiling

• For tiling a rectangular floor with square tiles, divide length by tile side to get number along length, and width by tile side to get number along width, then multiply (not add).
• Tiles are often sold in packs; calculate number of tiles needed, then number of packs (round up).
• Include fitting costs: area in $m^{2} \times$ rate per $m^{2}$, rounding up any fraction of a square metre.

Perimeter of a Compound L-Shape

Area of a Compound Shape (Pentagon)

Path Around a Pond (Subtracting Areas)

Circle with Two Smaller Circles Removed