Area of Triangles and Parallelograms
What Is This?
In Topic 17 you learned to find the area of rectangles and squares. Now we extend that to two more shapes: parallelograms (including rectangles as a special case) and triangles. Both formulas connect directly to what you already know about rectangles — making them easier to understand and remember.
Why Does It Matter?
Triangles and parallelograms appear constantly in architecture, design, engineering, and nature. Any time you need to find the space inside a slanted shape — a roof panel, a sail, a ramp, a floor tile — you are using one of these formulas. They also build the foundation for calculating surface area and volume at higher grades.
Key Vocabulary
| Term | Meaning |
|---|---|
| Base (b) | Any side of the shape chosen as the reference side |
| Height (h) | The perpendicular distance from the base to the opposite vertex or side — always measured at a right angle to the base |
| Perpendicular | At a right angle (90°) |
| Parallelogram | A quadrilateral with two pairs of parallel sides |
| Vertex | A corner point of a shape |
⚠️ The height is not the slanted side — it is always the straight-up-and-down (perpendicular) measurement from the base.
Part 1 — Parallelograms
What Is a Parallelogram?
A parallelogram is any four-sided shape where both pairs of opposite sides are parallel and equal in length. Rectangles and squares are special parallelograms where all angles are 90°.
___________
/ /
/ / h (height — perpendicular)
/___________/
b (base)
Why Does the Formula Work?
Imagine cutting a triangle off one end of a parallelogram and moving it to the other end. The result is a rectangle with the same base and height. So:
Area of parallelogram = base × height A = b × h
The slanted side length does not appear in the formula — only the base and the perpendicular height.
Examples
Example 1: A parallelogram with base 8 cm and height 5 cm.
- A = 8 × 5 = 40 cm² ✓
Example 2: A parallelogram with base 12 m and height 7 m.
- A = 12 × 7 = 84 m² ✓
Example 3: A parallelogram with base 9.5 cm and height 4 cm.
- A = 9.5 × 4 = 38 cm² ✓
Part 2 — Triangles
Why Does the Formula Work?
Every triangle is exactly half of a parallelogram with the same base and height. If you take any triangle and make a copy of it, you can always fit the two copies together to form a parallelogram.
/\ /\/\
/ \ → / \
/____\ /______\
triangle parallelogram (2 × triangle)
So the area of one triangle is half the area of the matching parallelogram:
Area of triangle = ½ × base × height A = ½ × b × h (also written as A = (b × h) ÷ 2)
Finding the Height of a Triangle
The height of a triangle is the perpendicular distance from the base to the opposite vertex. It may fall inside the triangle (for acute triangles) or outside it (for obtuse triangles — shown as a dotted line extended from the base).
Acute triangle: Obtuse triangle:
/\ /\
/ \ / \
/ h \ h / \
/ | \ / \
/___|____\ ---------/________\
b b
In both cases, the formula is the same: A = ½ × b × h.
Examples
Example 1: A triangle with base 10 cm and height 6 cm.
- A = ½ × 10 × 6 = ½ × 60 = 30 cm² ✓
Example 2: A triangle with base 14 m and height 9 m.
- A = ½ × 14 × 9 = ½ × 126 = 63 m² ✓
Example 3: A right triangle with legs 8 cm and 5 cm.
- One leg is the base; the other is the height (they meet at a right angle).
- A = ½ × 8 × 5 = ½ × 40 = 20 cm² ✓
Comparing the Three Formulas
| Shape | Formula | Notes |
|---|---|---|
| Rectangle | A = l × w | Special parallelogram with 90° angles |
| Parallelogram | A = b × h | Height is perpendicular — not the slant side |
| Triangle | A = ½ × b × h | Exactly half a parallelogram |
Finding a Missing Dimension
If you know the area and one dimension, you can find the other.
Example: A triangle has area 24 cm² and base 8 cm. What is the height?
- 24 = ½ × 8 × h → 24 = 4h → h = 6 cm ✓
Example: A parallelogram has area 60 m² and height 6 m. What is the base?
- 60 = b × 6 → b = 10 m ✓
Worked Examples
Worked Example 1
A triangular sail has base 6 m and height 9 m. What is the area of the sail?
- A = ½ × 6 × 9 = ½ × 54 = 27 m² ✓
Worked Example 2
A parallelogram-shaped tile has base 15 cm and height 8 cm. What is its area?
- A = 15 × 8 = 120 cm² ✓
Worked Example 3
A triangular garden has area 45 m². Its base is 10 m. What is its height?
- 45 = ½ × 10 × h → 45 = 5h → h = 9 m ✓
Worked Example 4 (Compound Shape)
A shape consists of a rectangle 12 cm × 5 cm with a triangle on top, base 12 cm and height 4 cm. What is the total area?
- Rectangle: 12 × 5 = 60 cm²
- Triangle: ½ × 12 × 4 = 24 cm²
- Total: 60 + 24 = 84 cm² ✓
Worked Example 5 (Word Problem)
A rooftop is shaped like a triangle with base 20 m and height 6 m. Roofing tiles cost $35 per m². What is the total cost?
- Area = ½ × 20 × 6 = 60 m²
- Cost = 60 × $35 = $2,100 ✓
Common Mistakes
Mistake 1: Using the slant side as the height In a parallelogram, the slant (oblique) side is longer than the perpendicular height. Always use the right-angle measurement. If a parallelogram has base 8, slant side 6, and height 5 — use 5, not 6.
Mistake 2: Forgetting the ½ for triangles The most common triangle mistake. A triangle is half a parallelogram — always divide b × h by 2.
Mistake 3: Confusing base and height The base can be any side. The height must be perpendicular to whichever side you chose as the base. They always form a 90° angle with each other.
Mistake 4: Wrong units Area is always in square units — cm², m², etc. Never write cm or m for area.
Key Takeaways
- Parallelogram area: A = b × h (height is perpendicular to the base — not the slant side)
- Triangle area: A = ½ × b × h (a triangle is always half a matching parallelogram)
- For a right triangle, the two legs are the base and height
- To find a missing dimension: rearrange the formula
- Area is always in square units
Practice and Resources
Ready to practise? Try our Grade 3-5 Area of Triangles and Parallelograms Worksheet covering both shapes, missing dimensions, and compound areas. Test yourself with the Grade 3-5 Math Practice Test.