Find rectangle area, perimeter and missing sides

Learning goal: Develop efficient strategies for rectangle area and perimeter, solve for an unknown side and distinguish covering from boundary length. Area measures how much surface a region covers using square units. Perimeter measures the distance around its outside boundary using length units. A rectangle has four right angles, with opposite sides equal in length.

The illustration shows a rectangle tiled with four rows of six unit squares. Counting each square gives twenty-four square units. Counting six squares per row in four equal rows gives 4 × 6 = 24 more efficiently. Each square covers one unit by one unit without gaps or overlaps. This arrangement explains why multiplying a rectangle's length by its width finds its area.

If the side lengths are six meters and four meters, the area is twenty-four square meters. The perimeter follows all four edges: 6 + 4 + 6 + 4 = 20 meters. Pairing opposite sides gives 2 × 6 + 2 × 4, or doubling the sum of adjacent sides gives 2 × (6 + 4). All three expressions count the same complete boundary.

The units distinguish the questions. Floor covering would concern square meters. Trim around the edges would concern meters. Writing twenty-four meters for area omits the square unit and changes what is being counted. A correct numeral with an incorrect unit may hide confusion about the task, so explain the selected operation using a drawing.

Sometimes area and one side are known. A rectangular garden covers thirty-six square meters and is nine meters long. Let w be its width. Since 9 × w = 36, division gives w = 4 meters. Picture four rows of nine square-meter tiles. Check by multiplying the two side lengths and recovering the given area. Then the perimeter is 9 + 4 + 9 + 4 = 26 meters.

A missing side can also be found from perimeter. A rectangle has perimeter twenty-two centimeters and length seven centimeters. The two long sides total fourteen centimeters. The two equal widths must total 22 − 14 = 8 centimeters, so each width is four centimeters. Check all four sides: 7 + 4 + 7 + 4 = 22. Its area is twenty-eight square centimeters.

Rectangles can share an area but have different perimeters. A three-by-eight rectangle and a four-by-six rectangle both cover twenty-four square units. Their perimeters are twenty-two and twenty length units. Rearranging the same number of tiles changes the boundary. Conversely, a two-by-eight rectangle and a four-by-six rectangle both have perimeter twenty units, but their areas are sixteen and twenty-four square units.

Read a real-world problem before choosing a formula. A garden needs fencing around its boundary, but an opening might not need fencing. For a six-meter by four-meter garden with a one-meter unfenced opening, the full perimeter is twenty meters and the required fence length is nineteen meters. The opening changes the requested material amount, not the rectangle's geometric perimeter.

Worked missing factor: A rectangular display has area forty square centimeters and one side eight centimeters. The unknown side solves 8 × s = 40, so s = 5 centimeters. Its perimeter is 2 × (8 + 5) = 26 centimeters. Division finds the side; multiplication and addition verify the two different measurements.

Worked critique: A learner uses 6 × 4 to buy edging for a six-by-four garden. That multiplication counts square-meter coverage. Edging follows four sides, so the required full-boundary length is 6 + 4 + 6 + 4 = 20 meters.

Guided practice: Build two rectangles from twenty-four square tiles. Record each pair of side lengths, area and perimeter. Trace the outside boundary without counting interior tile edges. Explain why the equal tile count does not force equal perimeter.

Independent evidence: Measure a rectangular paper with a real ruler, record sides with units and find both area and perimeter. Solve a missing-side problem using an equation and a contextual boundary problem with an opening. Require a drawing that distinguishes surface and edge, and record measurement accuracy separately from computational accuracy.