Introduction to Area of Triangles & Parallelograms: Fifth Grade Mathematics
Understanding the area of parallelograms and triangles connects directly to the rectangle area formula through shearing and half-rectangle decomposition.
Area of a Parallelogram: Base Times Height
Any parallelogram can be sheared or sliced to form a rectangle of equal area:
- Slice a right triangle off one slanted side of the parallelogram.
- Slide that triangle to the opposite slanted side.
- The shape transforms into a rectangle with the same base (b) and perpendicular height (h).
- Formula: Area of Parallelogram = base * height (A = b * h).
+--------------------------------------------------------------------------+
| PARALLELOGRAM AREA TRANSFORMATION |
| |
| +---------------+ Slice and Shift: |
| /| / +---------------+ |
| / | h / | | |
| / | / ---> | h | |
| +---+-----------+ +---------------+ |
| b b |
| |
| Area = base * height (Always use perpendicular height, NOT slant!)|
+--------------------------------------------------------------------------+
Area of a Triangle: Half of a Parallelogram
Any triangle is exactly half of a parallelogram (or half of a rectangle) sharing the same base and height:
- Take any triangle with base b and height h.
- Duplicate the triangle, rotate it 180°, and attach it along their shared slanted side.
- Together, the two congruent triangles form a parallelogram of area b * h.
- Therefore, one triangle has half that area: Area of Triangle = (1/2) * base * height, or A = (b * h) / 2.
+--------------------------------------------------------------------------+
| TRIANGLE AS HALF OF A PARALLELOGRAM |
| |
| /|\ |
| / | \ |
| / |h \ |
| / | \ |
| +----+----+ |
| b |
| |
| Area = (base * height) / 2 |
+--------------------------------------------------------------------------+
The Perpendicular Height Rule
In both parallelograms and triangles, the height (h) MUST be perpendicular (forming a 90° angle) to the base. Never use the slanted side length as the height.
Exercises and Solutions
Exercise 1: Area of a Parallelogram. A parallelogram has a base of 14 cm, a perpendicular height of 8 cm, and a slanted side length of 10 cm. Find the area and explain which measurement must be excluded from the area calculation.
Solution 1: Apply the parallelogram area formula: Area = base * perpendicular height = b * h. Use the base of 14 cm and the perpendicular height of 8 cm: Area = 14 * 8 = 112 square centimeters. The slanted side length of 10 cm must be excluded from the area calculation because it is not perpendicular to the base; it would only be used if finding the perimeter.
Exercise 2: Area of a Triangle. A sail on a sailboat is shaped like a triangle with a base of 9 meters and a perpendicular height of 6 meters. Calculate the area of the sail in square meters.
Solution 2: Apply the triangle area formula: Area = (base * height) / 2. Multiply base by height: 9 * 6 = 54. Divide by 2: 54 / 2 = 27 square meters. The area of the sail is 27 square meters.