1 · The two formulas
Perimeter is the distance around: add all four sides, or 2 × (length + width). Area is the space inside: length × width.
A 12 cm by 4 cm rectangle: perimeter 2 × (12 + 4) = 32 cm. Area 12 × 4 = 48 square cm.
2 · Squares: one number tells all
A square has 4 equal sides. Side 7: perimeter 4 × 7 = 28, area 7 × 7 = 49.
Backwards: if the area is 49, the side is 7 (because 7 × 7 = 49), so the perimeter is 28. If the perimeter is 36, the side is 36 ÷ 4 = 9, so the area is 81.
3 · Perimeter to a missing side
A rectangle has perimeter 34 cm and length 10 cm. Two lengths use 20 cm. The rest, 34 − 20 = 14, is two widths, so the width is 7 cm.
Now the area: 10 × 7 = 70 square cm. Tests often ask you to go from the perimeter to the missing side, then to the area.
4 · Area to a missing side
Area 54, length 9: 9 × ? = 54, so the width is 54 ÷ 9 = 6. Perimeter: 2 × (9 + 6) = 30.
5 · Area minus perimeter
Tests often ask this: "A rectangle is 12 cm long and 4 cm wide. What is the difference between its area and its perimeter?" Area 48, perimeter 32, difference 48 − 32 = 16.
Work out both numbers first, then subtract, keeping area and perimeter separate.
6 · Odd shapes
Cut an L shape into two rectangles, find each area, and add. For the perimeter, walk every edge: opposite sides across a notch still add up to the full width and height.
An L that fits in a 9 by 9 square has the same perimeter as that square (36), but less area.
7 · Triangles and other polygons
Perimeter works for any shape: add every side. A triangle with sides 16, 13, and 10 has perimeter 39 cm.
A regular polygon has equal sides, so multiply: a regular hexagon with side 8 has perimeter 6 × 8 = 48.
8 · Angles: right, acute, obtuse, straight
A right angle is a square corner, 90°. Acute is less than 90°. Obtuse is more than 90° but less than a straight line. A straight angle is 180°, a straight line.
A full turn is 360°. A half turn is 180°. A quarter turn is one right angle, 90°.
9 · Parallel and perpendicular
Parallel lines run side by side and never meet, like train tracks. Perpendicular lines cross at a right angle, like a plus sign.
Lines that cross at any other angle are just intersecting.
10 · Naming triangles
By sides: equilateral (3 equal), isosceles (2 equal), scalene (none). By angles: right (one 90°), acute (all under 90°), obtuse (one over 90°).
A triangle can have two names: a right isosceles triangle has a square corner and two equal sides.
11 · Naming quadrilaterals
Square: 4 equal sides, 4 right angles. Rectangle: 4 right angles. Rhombus: 4 equal sides. Parallelogram: 2 pairs of parallel sides. Trapezoid: exactly 1 pair.
Every square is also a rectangle, a rhombus, and a parallelogram. A square follows the rules of all three.
12 · Symmetry
A line of symmetry folds a shape onto itself. A square has 4, a rectangle has 2, an equilateral triangle has 3, a circle has endless lines.
A parallelogram that is not a rectangle or rhombus has none, even though it looks balanced.
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