Practice Maths

Area and Perimeter of Quadrilaterals

Key Ideas

Key Terms

perimeter
the total distance around the boundary of a shape; found by adding all side lengths. Measured in length units (cm, m).
area
the amount of surface inside a shape; measured in square units (cm², m²).
rectangle
a quadrilateral with four right angles; A = l × w, P = 2(l + w).
square
a rectangle with all four sides equal; A = s², P = 4s.
parallelogram
a quadrilateral with two pairs of parallel sides; A = base × perpendicular height, P = 2(base + slant side).
perpendicular height
the vertical distance between two parallel sides, measured at right angles; used in the area formula for parallelograms (not the slant side).
slant side
the non-perpendicular side of a parallelogram; used for perimeter but NOT for area.
Hot Tip Perimeter uses the actual side lengths (including slant sides). Area of a parallelogram uses the perpendicular height — the vertical distance between the parallel sides — NOT the slant side.

Worked Example

Question: A parallelogram has base 9 cm, perpendicular height 5 cm, and slant side 6 cm. Find its area and perimeter.

Area: A = b × h = 9 × 5 = 45 cm²

Perimeter: P = 2(9 + 6) = 2 × 15 = 30 cm

Perimeter: The Distance Around the Outside

Perimeter is the total distance around the boundary of a shape. To find it, add the lengths of all the sides. Perimeter is measured in length units (cm, m, km).

For any quadrilateral: Perimeter = sum of all four sides.

Special cases make this quicker:
• Rectangle: P = 2(l + w) where l = length and w = width
• Square: P = 4s where s = side length
• Parallelogram: P = 2(a + b) where a and b are the two different side lengths (like a rectangle, opposite sides are equal)

Area Formulas for Quadrilaterals

Area is measured in square units (cm2, m2).

Rectangle: A = length × width = l × w

Square: A = side × side = s2

Parallelogram: A = base × height = b × h
Important: the height must be the perpendicular height — the vertical distance between the two parallel sides, not the slant side length.

Why the Parallelogram Formula Works

Imagine cutting a triangle from one end of a parallelogram and sliding it to the other end. The parallelogram becomes a rectangle with the same base and the same height. So the area formula is the same: base × height. The slant side is irrelevant for area — only the perpendicular height matters.

This is a classic visual proof. If you remember the "cut and slide" idea, you'll never mix up the formula.

Worked Examples

Rectangle: A garden bed is 8 m long and 3.5 m wide.
Area = 8 × 3.5 = 28 m2
Perimeter = 2(8 + 3.5) = 2 × 11.5 = 23 m

Parallelogram: A parallelogram has base 12 cm, perpendicular height 7 cm, and slant sides 9 cm.
Area = 12 × 7 = 84 cm2
Perimeter = 2(12 + 9) = 42 cm (use the slant side, not the height, for perimeter)

Square: A square tile has side length 15 cm.
Area = 152 = 225 cm2. Perimeter = 4 × 15 = 60 cm.

Practical Context: Flooring, Fencing and Painting

Area tells you how much surface is covered — useful for floor tiles, paint, carpet, turf, or fabric. Perimeter tells you the length of the boundary — useful for fencing, framing, or border tape.

Example: You want to carpet a rectangular room 6 m by 4.5 m. Carpet costs $35/m2. Area = 27 m2. Cost = 27 × $35 = $945.

Example: A parallelogram-shaped garden bed has base 5 m, perpendicular height 3 m, and slant sides 3.6 m. Topsoil is needed (area problem): 5 × 3 = 15 m2. Garden edging is needed (perimeter problem): 2(5 + 3.6) = 17.2 m.

Key tip: For the parallelogram, the height in the area formula is the perpendicular height — drawn at a right angle from the base to the opposite side. This is often shown as a dashed line inside the shape. Never use the slant side length in the area formula — that's only used for perimeter.

Mastery Practice

  1. Calculate the perimeter of each quadrilateral. Fluency

    1. Rectangle: length 8 cm, width 5 cm
    2. Square: side length 11 m
    3. Rectangle: length 14.5 cm, width 6 cm
    4. Square: side length 2.4 m
    5. Parallelogram: base 10 cm, slant side 7 cm
    6. Rectangle: length 3.8 m, width 1.5 m
    7. Square: side length 8.5 mm
    8. Parallelogram: base 12 m, slant side 9 m

  2. Calculate the area of each rectangle or square. Fluency

    1. Rectangle: length 9 cm, width 4 cm
    2. Square: side length 7 m
    3. Rectangle: length 12.5 cm, width 8 cm
    4. Square: side length 1.5 km
    5. Rectangle: length 30 mm, width 24 mm
    6. Square: side length 11.2 m
    7. Rectangle: length 6.4 m, width 3.5 m
    8. Square: side length 0.9 cm

  3. Calculate the area of each parallelogram. Use the perpendicular height given. Fluency

    1. Base 8 cm, perpendicular height 5 cm
    2. Base 14 m, perpendicular height 9 m
    3. Base 6.5 cm, perpendicular height 4 cm
    4. Base 20 mm, perpendicular height 13 mm
    5. Base 11 m, perpendicular height 7.5 m
    6. Base 3.2 m, perpendicular height 2.6 m (answer in cm²)
    7. Base 25 cm, perpendicular height 18 cm
    8. Base 7 m, perpendicular height 4.4 m

  4. Find the missing dimension for each quadrilateral. Understanding

    1. A rectangle has area 72 cm² and length 9 cm. Find the width.
    2. A square has area 144 m². Find the side length.
    3. A rectangle has perimeter 38 cm and length 12 cm. Find the width.
    4. A parallelogram has area 90 cm² and base 15 cm. Find the perpendicular height.
    5. A square has perimeter 52 m. Find the side length and area.
    6. A rectangle has perimeter 46 m. Its length is three times its width. Find both dimensions.
    7. A parallelogram has area 48 m² and perpendicular height 6 m. Find the base.
    8. A square tile has area 256 cm². Find the side length and perimeter of the tile.

  5. Find the area of each composite shape made of rectangles. Understanding

    1. An L-shape: the full shape is 10 cm × 8 cm, with a 4 cm × 3 cm rectangle removed from one corner.
    2. A plus sign shape: a horizontal rectangle 12 cm × 3 cm and a vertical rectangle 3 cm × 12 cm overlapping in the centre.
    3. Two rectangles joined end-to-end: first is 8 m × 5 m, second is 6 m × 3 m.
    4. A T-shape: top bar is 14 cm × 4 cm, vertical stem is 4 cm × 8 cm attached at the centre of the top bar.
    5. An L-shape: outer dimensions 15 m × 10 m, with a 6 m × 4 m rectangle removed from one corner.
    6. A staircase shape: three steps, each step is 3 cm wide and 2 cm tall. Find the total area.

  6. Solve each problem, showing all working. Problem Solving

    1. A rectangular kitchen measures 4.8 m by 3.6 m.
      1. Find the area of the kitchen floor.
      2. Vinyl flooring costs $28 per m². Find the total cost to cover the floor.
      3. A tiler adds 10% extra for wastage. How many m² should they buy?
    2. A farmer wants to fence a rectangular paddock 120 m long and 85 m wide.
      1. Find the length of fencing needed.
      2. Fencing costs $18.50 per metre. Find the total cost.
      3. The paddock also has a diagonal internal fence from one corner to the opposite corner. Given the paddock is a rectangle, calculate the length of this diagonal fence (use Pythagoras: c² = 120² + 85², give answer to 1 decimal place).
    3. A parallelogram-shaped garden bed has a base of 6.5 m and a perpendicular height of 3.2 m.
      1. Find the area of the garden bed.
      2. Mulch is sold in bags that each cover 2 m². How many whole bags are needed?
      3. Each bag costs $8.99. What is the total cost?

  7. Find the area and perimeter of each shape shown. Understanding

    1. Find the area and perimeter of the rectangle below. 11 cm 4 cm
    2. Find the area and perimeter of the square below. 6 m 6 m
    3. Find the area of the parallelogram below. (The dashed line is the perpendicular height.) 15 cm h = 7 cm 9 cm

  8. Complete the table by filling in the missing values. Understanding

    ShapeDimensions givenAreaPerimeter
    Rectanglel = 9 cm, w = 5 cm______
    Squares = 8 m______
    Parallelogramb = 12 cm, h = 7 cm, slant = 9 cm______
    Rectanglel = 20 mm, w = 13 mm______
    Squares = 4.5 km______
    Parallelogramb = 8 m, h = 6 m, slant = 7 m______

  9. Each student made one error. Find it and write the correct answer. Understanding

    1. Sam calculated the area of a parallelogram with base 8 cm and slant side 5 cm (perpendicular height 4 cm) as A = 8 × 5 = 40 cm². What did Sam do wrong?
    2. Jordan found the perimeter of a rectangle (length 7 cm, width 3 cm) as P = 7 × 3 = 21 cm. What should the answer be?
    3. Alex said: “A square with side 5 m has area 20 m² because 4 × 5 = 20.” Explain the error.
    4. Riley found the area of a square with perimeter 24 cm as A = 24² = 576 cm². What went wrong?

  10. A builder is designing a house. Use the information below to answer each question. Show all working. Problem Solving

    The house has three rectangular rooms:

    RoomLengthWidth
    Bedroom4.2 m3.6 m
    Lounge6.5 m4.8 m
    Kitchen3.8 m3.0 m
    1. Find the floor area of each room.
    2. Find the total floor area of all three rooms combined.
    3. Carpet costs $42 per m² for the bedroom, and vinyl at $19 per m² for the other two rooms. Find the total flooring cost.
    4. Skirting board is needed around the perimeter of each room (assume no doorways). Find the total skirting board length needed.
    5. Skirting board costs $4.80 per metre. Find the total cost of skirting board.
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