Area of Triangles and Parallelograms

Unit 10: Area of Polygons and Circles · ~15–30 min

Base and height

Every area formula in this unit starts from the same idea: a shape's base is any side you choose to measure along, and its height is the perpendicular distance from that base to the opposite side or vertex — never a slanted side length. Height is always measured straight up and down from the base, even if the shape leans.

• Parallelogram: Area = base × height (A = bh).
• Triangle: Area = ½ × base × height (A = ½bh).

The triangle formula isn't a coincidence — any triangle is exactly half of a parallelogram with the same base and height. Copy a triangle, rotate the copy 180°, and slide it against the original: the two triangles tile a parallelogram perfectly. That's why the triangle formula carries the extra factor of ½.

Guided practice

• 1.Find the area of a parallelogram with base 11 m and height 6 m.
  Show answer

  A = bh = 11 × 6 = 66 m².
• 2.Find the area of a triangle with base 20 ft and height 7 ft.
  Show answer

  A = ½(20)(7) = ½(140) = 70 ft².
• 3.A triangle has area 84 cm² and height 12 cm. Find its base.
  Show answer

  84 = ½b(12) = 6b, so b = 14 cm.
• 4.Explain why a triangle and a parallelogram sharing the same base and height don't have the same area.
  Show answer

  A triangle is exactly half of a parallelogram with matching base and height — two congruent triangles tile the parallelogram — so the triangle's area formula carries an extra factor of ½ that the parallelogram's does not.
• 5.A parallelogram-shaped garden plot has base (3x) meters and height 10 meters, with area 240 m². Find x.
  Show answer

  240 = (3x)(10) = 30x, so x = 8.