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Mathematics › ⚡ Grade 6 › Area of Triangles and Parallelograms: Cutting a Rectangle in Half
⚡ Grade 6 · Lesson 11 of 14

Area of Triangles and Parallelograms: Cutting a Rectangle in Half

Every triangle hides a parallelogram. Every parallelogram hides a rectangle.

Grade 6Middle School
Area of Triangles and Parallelograms: Cutting a Rectangle in Half — illustration
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The big idea: A triangle is exactly half of a parallelogram with the same base and height. A parallelogram with the same base and height as a rectangle has the same area. Put both ideas together: triangle area = ½ × base × height. The height must be perpendicular to the base — the slant side of a triangle or parallelogram does NOT work as the height.
🎯 By the end, you'll be able to
  • Calculate the area of a parallelogram using base × height
  • Calculate the area of a triangle using ½ × base × height
  • Identify the correct perpendicular height for any triangle orientation
  • Explain why two congruent triangles form a parallelogram
📎 You should already know
  • Area of rectangles (length × width)
  • Perpendicular lines and right angles

Parallelogram = rectangle rearranged

Take a parallelogram and cut a right-triangle off one end. Slide that triangle to the other end — suddenly it's a rectangle! The area didn't change: A = base × height. Note that the 'height' is the perpendicular distance between the two parallel sides — NOT the slant side.

This is why a squashed parallelogram and a tall one with the same base can have the same area: what matters is the perpendicular height.

Triangle = half a parallelogram

Take any triangle and make an exact copy of it. Flip the copy and attach it along the longest edge — you always get a parallelogram (or rectangle for right triangles). So the triangle is exactly half the parallelogram: A = ½ × base × height.

This works for all triangles: acute, obtuse, right. For obtuse triangles, the height falls outside the triangle (you draw it as a perpendicular from the vertex to the base line extended), but the formula is identical.

🎮 Triangle Area Builder LIVE
Adjust the base and height with the sliders. The dashed line shows the perpendicular height. Press 'Duplicate → Parallelogram' to see the second triangle snap into place, forming a complete parallelogram — its area is base × height, and the triangle is half.
⚠️ The height is NEVER the slant side

For a triangle with sides 5, 6, and 7 and a base of 6: the height is NOT 5 or 7. The height is the perpendicular distance from the opposite vertex down to the base line — you have to calculate or measure it separately.

A very common mistake: using a slant side as the height. Area = ½ × base × perpendicular height only.

🔑 Area formulas

Rectangle: A = length × width

Parallelogram: A = base × height (perpendicular height)

Triangle: A = ½ × base × height (perpendicular height)

These three are nested: rectangle is a special parallelogram; parallelogram area = 2 × triangle area (same base and height).

📝 Worked example: A parallelogram has a base of 9 cm and a perpendicular height of 4 cm. What is its area?
  1. A = base × height = 9 × 4 = 36 cm².
✓ <strong>36 cm²</strong>.
📝 Worked example: A triangle has a base of 10 m and a perpendicular height of 6 m. What is its area?
  1. A = ½ × base × height = ½ × 10 × 6 = 30 m².
✓ <strong>30 m²</strong>.
📝 Worked example: A triangle has vertices at (0, 0), (8, 0) and (3, 5) on a grid. Find its area.
  1. Base = horizontal distance = 8 units.
  2. Height = perpendicular distance from top vertex to the base = 5 units (straight up from y = 0 to y = 5).
  3. A = ½ × 8 × 5 = 20 square units.
✓ <strong>20 square units</strong>.

Check your understanding

1. A triangle has base 12 cm and perpendicular height 7 cm. What is its area?
A = ½ × 12 × 7 = 42 cm².
2. A parallelogram has base 5 m and slant side 8 m and perpendicular height 6 m. What is its area?
Area = base × perpendicular height = 5 × 6 = 30 m². The slant side (8 m) is not used.
3. Why does the triangle area formula have ½?
Two congruent triangles tile a parallelogram — so one triangle is half the parallelogram's area.
4. An acute triangle and an obtuse triangle both have base 6 cm and perpendicular height 4 cm. Do they have the same area?
Area depends only on base and perpendicular height: ½ × 6 × 4 = 12 cm² for both. Shape and angles do not affect area when base and height are fixed.
5. A square of side 10 cm is cut diagonally into two triangles. What is the area of each triangle?
Square area = 100 cm². Each triangle = 100 ÷ 2 = 50 cm². Alternatively: ½ × 10 × 10 = 50 cm².
✅ Key takeaways
  • Parallelogram area = base × perpendicular height (NOT the slant side).
  • Triangle area = ½ × base × perpendicular height — because two congruent triangles form a parallelogram.
  • The height is always perpendicular to the base; for obtuse triangles it falls outside the figure.
  • All triangles with the same base and height have equal area regardless of shape.
  • Nested relationship: rectangle is a special parallelogram; triangle = half parallelogram.