Practice Maths

Area of Parallelograms

Key Terms

Parallelogram
A quadrilateral with two pairs of parallel sides and equal opposite sides.
Base (b)
The length of a chosen side of the parallelogram (usually the bottom).
Perpendicular height (h)
The vertical distance between the two parallel sides, measured at 90° to the base. This is NOT the slant side.
Area
The amount of space inside a shape, measured in square units (cm², m², mm²).
base b h ✓ (use this) slant side ✗

Always use the perpendicular height (green dashed) — not the slant side.

Area Formula

The area of a parallelogram is:

A = base × height    (A = b × h)

  • The HEIGHT must be perpendicular to the base — the vertical height, NOT the slant side.
  • The base and height must be in the same units.
  • A rectangle is a special parallelogram where the perpendicular height equals the side length.
Hot Tip
Students often use the slant side instead of the perpendicular height. The height is always the perpendicular distance between the two parallel sides — draw it as a right angle.

Worked Example 1

Parallelogram with base 8 cm and perpendicular height 5 cm.

A = 8 × 5 = 40 cm²

Worked Example 2

Parallelogram with base 12 m and height 7 m.

A = 12 × 7 = 84 m²

Worked Example 3

Base 6.5 cm, height 4 cm.

A = 6.5 × 4 = 26 cm²

Parallelogram → Rectangle

The red triangle slides from one end to the other — forming a rectangle with the same base and height.

What Is a Parallelogram?

A parallelogram is a quadrilateral (4-sided shape) where opposite sides are parallel and equal in length. Rectangles and rhombuses are special types of parallelograms. The slanted sides of a parallelogram make it look like a "pushed-over" rectangle.

Examples of parallelograms in real life: a slanted door frame, the faces of a Toblerone box, a diamond shape in a tile pattern.

The Formula: A = b × h

The area of a parallelogram is: A = base × perpendicular height

The perpendicular height (h) is the vertical distance between the two parallel sides — measured at 90° to the base. It is not the slant side length.

  • A parallelogram with base 10 cm and perpendicular height 6 cm: A = 10 × 6 = 60 cm²
  • A parallelogram with base 8 m and perpendicular height 4.5 m: A = 8 × 4.5 = 36 m²

Why Does the Formula Work?

If you cut a triangular piece from one end of a parallelogram and move it to the other end, you form a rectangle with the same base and the same perpendicular height. Since the area of a rectangle is l × w = base × height, the area of the parallelogram is the same.

This is a great way to understand the formula — you're just rearranging the shape, not changing its area.

Distinguishing Base, Height, and Slant Side

This is the most common source of errors. In a parallelogram diagram:

  • The base is the bottom side (or any side you choose as the base).
  • The perpendicular height is shown as a dotted vertical line inside the shape, meeting the base at 90°. It is usually labelled h.
  • The slant side is the diagonal side — do NOT use this as the height!

If the slant side is 8 cm and the perpendicular height is 6 cm, use 6 cm in your formula.

Key tip: The height in A = b × h is always the perpendicular height — the straight vertical distance between the two parallel sides. Using the slant side instead of the perpendicular height is the most common mistake with parallelogram area. Always look for the right angle symbol or the dotted height line in the diagram.

Mastery Practice

  1. Calculate the Area Fluency

    Find the area of each parallelogram. Use A = b × h.

    1. b = 10 cm h = 6 cm
    2. b = 8 m h = 5 m
    3. b = 15 cm h = 4 cm
    4. b = 3.5 m h = 8 m
    5. b = 9 cm h = 9 cm

  2. Find the Missing Dimension Fluency

    Use A = b × h to find the missing value (shown as ?). Area is given above each parallelogram.

    1. A = 48 cm² b = 8 cm h = ?
    2. A = 90 m² b = ? h = 6 m
    3. A = 35 cm² b = 7 cm h = ?
    4. A = 144 mm² b = ? h = 12 mm
    5. A = 63 m² b = 9 m h = ?

  3. Units and Conversion Understanding

    1. A parallelogram has base 50 mm and height 30 mm. Find the area in cm².
    2. A parallelogram has base 2 m and height 80 cm. Convert to the same units and find the area in m².
    3. A parallelogram has base 1.2 m and height 45 cm. Find the area in cm².
    4. Find the area in mm² of a parallelogram with base 5 cm and height 3 cm.

  4. Compare and Reason Understanding

    1. Two parallelograms both have base 10 cm. The first has perpendicular height 6 cm and a slant side of 8 cm. The second has perpendicular height 8 cm. Which has greater area? By how much?
    2. A parallelogram and a rectangle both have base 12 cm and the same area of 84 cm². What is the height of the parallelogram?
    3. Explain why two parallelograms with the same base and same slant side but different heights will have different areas.
    4. A parallelogram is rearranged into a rectangle. The rectangle is 9 cm × 7 cm. What was the area of the original parallelogram?

  5. Problem Solving Problem Solving

    1. A garden bed is shaped like a parallelogram with base 6 m and height 3.5 m. Find the area.
    2. A parallelogram-shaped window has base 120 cm and perpendicular height 80 cm. Find its area in m².
    3. 1 litre of paint covers 8 m². How many litres are needed to paint a parallelogram wall with base 5 m and height 4 m?
    4. A path through a park is parallelogram-shaped, base 25 m and height 4 m. Paving costs $12 per m². What is the total cost?
    5. A parallelogram has area 56 cm². Its height is three-quarters of its base. Find the base and height.

  6. More Calculate the Area Fluency

    Find the area of each parallelogram. Use A = b × h.

    1. b = 7 cm h = 11 cm
    2. b = 14 m h = 6 m
    3. b = 2.5 cm h = 8 cm
    4. b = 11 mm h = 9 mm
    5. b = 6 m h = 6 m

  7. Spot the Error Understanding

    Each calculation below contains an error. Identify the mistake and give the correct answer.

    b = 10 cm h = 4 cm slant 6 cm
    (a)
    b = 8 m h = 5 m
    (b)
    b = 15 mm h = 4 mm
    (c)
    l = 9 cm w = 6 cm
    (d)
    1. A parallelogram with base 10 cm, perpendicular height 4 cm, and slant side 6 cm. A student writes: A = 10 × 6 = 60 cm².
    2. A parallelogram with base 8 m and height 5 m. A student writes: A = ½ × 8 × 5 = 20 m².
    3. A parallelogram with base 15 mm and height 4 mm. A student writes: A = 15 + 4 = 19 mm².
    4. A rectangle 9 cm × 6 cm. A student says: “This is a parallelogram so I need to use a different formula.”

  8. Area and Perimeter Together Understanding

    The diagram below shows the parallelogram in part (a) with its three measurements (base, slant side, and perpendicular height). Use these for parts (a) and (d).

    b = 12 cm h = 6 cm s = 8 cm
    1. A parallelogram has base 12 cm, slant side 8 cm, and perpendicular height 6 cm. Find (i) the perimeter and (ii) the area. Are they the same? Why not?
    2. Two parallelograms have the same perimeter (40 cm). The first has base 12 cm and slant side 8 cm, height 5 cm. The second has base 10 cm and slant side 10 cm, height 8 cm. Which has the greater area?
    3. A parallelogram has base b and slant side s and perpendicular height h. Write a formula for the perimeter in terms of b and s.
    4. A parallelogram with base 9 cm and slant side 7 cm has perpendicular height 5 cm. Find the area. Find the perimeter.

  9. Parallelograms in Composite Figures Problem Solving

    20 cm 12 cm removed (b=8, h=4)
    Part (a): rectangle with parallelogram-shaped piece removed (red dashed)
    1. A large rectangle is 20 cm × 12 cm. A parallelogram-shaped section has been removed from one corner with base 8 cm and height 4 cm. What is the remaining area?
    2. A floor is made of parallelogram-shaped tiles. Each tile has base 30 cm and height 20 cm. The floor area is 24 m². How many tiles are needed?
    3. A garden is divided into three parallelogram-shaped plots. The plots have base 5 m each and heights of 3 m, 4 m, and 5 m respectively. What is the total garden area?
    4. A parallelogram has area 48 cm². Its base is twice its height. Find the base and height.
    5. Two identical parallelograms are placed side by side to form a larger parallelogram. Each small parallelogram has base 7 cm and height 5 cm. What is the area of the large parallelogram?

  10. Extended Investigation Problem Solving

    1. A parallelogram and a rectangle both have base 10 cm. The parallelogram has perpendicular height 8 cm and a slant side of 12 cm. The rectangle has width 10 cm. If their areas are equal, find the rectangle’s height.
    2. A parallelogram has base 6 m and area 42 m². Find the height. If the slant side is 8 m, find the perimeter.
    3. A patchwork quilt is made of 48 identical parallelogram-shaped pieces. Each piece has base 15 cm and height 10 cm. Find the total fabric area needed in m².
    4. The area of a parallelogram is given by the formula A = bh. Explain why this formula also works for a rectangle. What does h represent in a rectangle?
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