One-page revision sheet

• Perimeter is the distance round the edge. It is a length, measured in mm, cm, m or km.
• Area is the surface covered. It is measured in square units, because it is a length times a length.
• Rectangle: perimeter = 2(l + w) and area = l × w.
• Square of side s: perimeter = 4s and area = s².
• To work backwards from an area, divide: width = area ÷ length.
• To work backwards from a perimeter, take off the two sides you know first, then halve what is left.
• Two shapes are congruent when corresponding sides and angles are equal — same shape and same size.
• Equal areas do not make shapes congruent: 6 × 4 and 12 × 2 both give 24 cm².
• Doubling both sides multiplies the area by 4, not by 2.

Key words: perimeter  ·  area  ·  square centimetre  ·  congruent  ·  formula  ·  dimension

• A compound shape is built from rectangles, so it can always be handled with the rectangle formula.
• Method 1 — split. Cut it into rectangles and add the areas.
• Method 2 — subtract. Take the missing piece away from the enclosing rectangle.
• Both methods must give the same answer. Doing one and checking with the other is free insurance.
• Write every side length on the diagram first, including the ones you have to work out by subtracting.
• For the perimeter, walk right round the outline. Inside joins are not part of it.
• Cutting a rectangular corner out changes the area but not the perimeter.
• A border of width b round an l by w picture makes the outside (l + 2b) by (w + 2b).

Key words: compound shape  ·  split  ·  enclosing rectangle  ·  L-shape  ·  border

• An area conversion factor is the square of the length one.
• 1 cm = 10 mm, so 1 cm² = 100 mm².
• 1 m = 100 cm, so 1 m² = 10 000 cm².
• 1 km = 1000 m, so 1 km² = 1 000 000 m².
• 1 hectare = 10 000 m² — a square 100 m by 100 m.
• 1 km² = 100 hectares.
• Big unit → small unit: multiply. Small → big: divide.
• Choose a unit that keeps the numbers sensible: mm² for a stamp, hectares for a farm.
• When a shape has mixed units, convert before multiplying.

Key words: square millimetre  ·  square centimetre  ·  square metre  ·  hectare  ·  square kilometre  ·  convert

• Area of a triangle = ½ × base × perpendicular height.
• It works because two copies of the triangle make a parallelogram of base × height.
• The height must be perpendicular to the base — not a sloping side.
• In an obtuse triangle the height falls outside the triangle. Extend the base and measure to it; the formula is unchanged.
• Triangles with the same base and height have the same area, however different they look.
• The answer may end in .5 — that is real, not a rounding error.
• Area of a parallelogram = base × height, with no halving.
• To find a missing base or height: double the area first, then divide.
• Split a compound shape into rectangles and triangles and add.

Key words: base  ·  perpendicular height  ·  triangle  ·  parallelogram  ·  derive

• Volume of a cuboid = l × w × h.
• It is also base area × height — one layer, stacked h times.
• Volume of a cube = s × s × s = s³.
• Volume is measured in cubic units, because three lengths are multiplied.
• 1 litre = 1000 cm³, a cube 10 cm on each edge.
• 1 m³ = 1 000 000 cm³ — the length factor is cubed.
• For a compound solid, split it into cuboids and add the volumes; for a hole, subtract.
• To find a missing dimension, divide the volume by the other two.
• Doubling every edge multiplies the volume by 8, not by 2.

Key words: volume  ·  cuboid  ·  cube  ·  cubic centimetre  ·  capacity  ·  litre

• Surface area is the total area of all the faces. It is measured in cm², not cm³.
• A cuboid has 6 faces, 12 edges and 8 vertices.
• The six faces come in three matching pairs.
• Surface area of a cuboid = 2(lw + lh + wh).
• Surface area of a cube = 6s².
• The net lays all six faces out flat, so none can be forgotten.
• For an open box, work out the closed one and subtract the missing face.
• The total edge length of a cuboid is 4(l + w + h).
• The front, side and top views are flat rectangles: l × h, w × h and l × w.
• Equal volume does not mean equal surface area — a cube is the most economical box.

Key words: surface area  ·  face  ·  edge  ·  vertex  ·  net  ·  front view