Lesson 4 — Parallelograms and Trapezoids
Big idea: Parallelogram and trapezoid areas both come from triangle areas, using a base and the perpendicular altitude between parallel sides.
Words to know
| Word | Meaning |
|---|---|
| Parallelogram | A quadrilateral with both pairs of opposite sides parallel. |
| Trapezoid | A quadrilateral with exactly one pair of parallel sides. |
| Bases of a trapezoid | The two parallel sides. |
| Legs of a trapezoid | The two nonparallel sides. |
| Altitude of a quadrilateral | The perpendicular distance between the lines containing its parallel sides; also any perpendicular segment joining those lines. |
Concepts to know
- The altitude of a parallelogram or trapezoid is measured perpendicular to the parallel sides.
- Theorem 40: The area of a parallelogram is the product of any base and corresponding altitude.
- Theorem 41: The area of a trapezoid is half the product of its altitude and the sum of its bases.
- A diagonal splits a parallelogram into two congruent triangles, which is how Theorem 40 is proved.
- A diagonal splits a trapezoid into two triangles with the same altitude; their areas add to give Theorem 41.
- Either base of a parallelogram can be used, as long as you use the matching altitude.
Formulas / rules
- Parallelogram: A = bh
- Trapezoid: A = ½h(b₁ + b₂)
- In a trapezoid, b₁ and b₂ are the parallel sides, and h is the perpendicular distance between them.
- Area is always in square units.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find a parallelogram’s area | Multiply a base by the altitude perpendicular to it. |
| Find a trapezoid’s area | Add the two parallel bases, multiply by the height, then divide by 2. |
| Find a missing height | Substitute the area and base(s) into the formula and solve. |
| Find a missing base | Use A = bh or A = ½h(b₁ + b₂) and solve. |
| Choose the correct height | Use the perpendicular distance between the parallel sides, not a slanted leg. |
| Split a figure into familiar shapes | Divide it into rectangles, triangles, and trapezoids, then add areas. |
Watch out
- A slanted side of a parallelogram or trapezoid is not the altitude.
- In a trapezoid, the bases are the parallel sides, not the legs.
- Do not forget the ½ in the trapezoid formula.
- The altitude of a trapezoid is the distance between the parallel lines, even if the perpendicular segment falls outside the figure.