Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Rectangles are easy: length times width. Sixth grade asks students to find the area of everything else in the polygon family, and the method is to rearrange or split shapes into ones whose areas are already known. A right triangle is half of a rectangle, which is why its area is half of base times height. Any triangle can be boxed inside a rectangle or split into two right triangles, so the same formula, A = 1/2 bh, works for all of them.
Parallelograms work by cutting a triangle off one end and sliding it to the other, which makes a rectangle with the same base and height. Trapezoids can be split into a rectangle and triangles, or doubled into a parallelogram. Irregular polygons, like the floor plan of an L-shaped room, are decomposed into rectangles and triangles whose areas are added, or composed into a larger rectangle with pieces subtracted. Students apply these methods to real problems, so knowing which length is the height (always perpendicular to the base) matters more than memorizing formulas.
For a parallelogram with base 8 and slanted side 5, students compute 8 × 5. The height is the perpendicular distance between the bases, which is shorter than the slanted side.
Students who multiply base by height for a triangle get double the area. Drawing the surrounding rectangle shows the triangle fills only half of it.
When splitting an L-shape into two rectangles, some students use overlapping pieces. The pieces must not overlap and must cover the whole shape.
Area is measured in square units. An answer of 24 cm instead of 24 cm² suggests the student is thinking about length.
A garden is shaped like a trapezoid with parallel sides 10 m and 6 m, and a height of 4 m. Find its area by decomposing it.
Answer: The garden has an area of 32 square meters.
Let students cut paper shapes and rearrange them before writing formulas: a parallelogram becomes a rectangle, two identical triangles become a parallelogram, two identical trapezoids form a parallelogram. Formulas then become summaries of something they have seen.
Assessments frequently show a composite figure with labeled lengths and ask for its area, sometimes including extra lengths that are not needed. Practice choosing the right height is essential.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 42 (also accepted: 42 cm²)
A = 1/2 × 12 × 7 = 42 square centimeters.
Answer: C) 36 in²
Area of a parallelogram = base × height = 9 × 4 = 36 square inches. The slanted side is not the height.
Answer: 20
The legs are perpendicular, so they serve as base and height: 1/2 × 5 × 8 = 20 square feet.
Answer: A) 55 m²
Add the two rectangles: 10 × 4 = 40 and 3 × 5 = 15, so 40 + 15 = 55 square meters.
Answer: 54
Two copies make a parallelogram with base 7 + 11 = 18 and height 6, area 108. Half of that is 54 square centimeters.
Answer: C) Because two identical triangles fit together to make a parallelogram with that base and height
Two copies of any triangle form a parallelogram with area b × h, so one triangle has half that area.
A full lesson with slides, activities and an exit ticket on area of triangles and polygons, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.G.A.1 with an answer key, ready in about a minute.
Make a worksheet →Turn area of triangles and polygons into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Right triangles, other triangles, special quadrilaterals such as parallelograms, trapezoids and rhombuses, and polygons that can be split into these shapes.
The standard emphasizes composing and decomposing shapes. Many students learn the formula, but they should be able to find the area by splitting the shape even without it.