Trace perimeters and describe polygons by their properties

Learning goal: Find polygon perimeters, determine missing side lengths and describe shapes by their sides and square corners. A polygon is a closed flat shape made of straight line segments that meet only at their ends and do not cross one another. Its perimeter is the total length around its boundary. Unlike area, perimeter measures length rather than surface coverage, so it uses units such as centimeters, not square centimeters.

For a rectangle with sides four and seven centimeters, the opposite sides have matching lengths. Add all four sides: 4 + 7 + 4 + 7 = 22 centimeters. Trace the complete boundary as you add. Counting just four plus seven omits two sides. The same rectangle has area twenty-eight square centimeters, but that measures a different feature.

An unknown side can be found from a known perimeter. A four-sided garden has perimeter twenty-four meters and three sides of seven, five and six meters. Their sum is eighteen meters. The missing side is 24 − 18 = 6 meters. This reasoning does not require assuming that every four-sided shape is a rectangle; use only the given side information unless the shape's properties are established.

Rectangles can share a perimeter while having different areas. A one-by-five rectangle has perimeter twelve units and area five square units. A two-by-four rectangle also has perimeter twelve units, but area eight square units. Moving the same length of boundary to enclose a different arrangement can change the covered area.

Rectangles can also share an area while having different perimeters. A one-by-six rectangle and a two-by-three rectangle both cover six square units. Their perimeters are fourteen and ten units. Area therefore does not determine a unique perimeter, and perimeter does not determine a unique area. Compare actual side lengths instead of guessing from a single measurement.

Describe polygons using properties that do not depend on the drawing's orientation. A triangle has three straight sides, a quadrilateral four, a pentagon five, a hexagon six, a heptagon seven and an octagon eight. A turned square is still a square. A shape with a curved boundary is not a polygon, even if someone gives it four corners in an informal drawing.

A square corner is also called a right angle. Within quadrilaterals, a rectangle has four square corners. A rhombus has four equal side lengths. A square has both properties: four equal sides and four square corners. Thus a square fits both descriptions. A long rectangle need not have four equal sides, and a slanted rhombus need not have square corners. Use a paper corner or suitable drawing tool to check corners rather than relying on the name “diamond.”

Triangles are described by their corners too. A triangle with one square corner is called a right triangle; the corner of a sheet of paper fits that corner exactly. Many triangles have no square corner at all. A triangle cannot have two square corners, because the two sides leaving those corners would run side by side and never meet to close the shape. To draw a right triangle, trace the two edges of a paper corner and join their ends with a third straight side. The bottom row of the illustration shows one triangle with a square corner and one without. Turning either triangle does not add or remove a square corner.

Worked perimeter comparison: Build rectangles measuring one by five and two by four using unit squares. Walk a finger around each outside edge, counting twelve unit lengths for either boundary. Then count the interior squares: five versus eight. The direct model proves equal perimeter does not guarantee equal area.

Worked classification: A quadrilateral has four equal sides and no square corners. It meets the rhombus description but not the square or rectangle description used here. Rotating it cannot create square corners. Properties of the shape remain the same when the paper turns.

Guided performance: Draw a triangle with a square corner, a triangle without one, and polygons with four through eight sides. Count straight segments and check that the boundary closes without crossing itself in these examples. Draw a square, a nonsquare rectangle and a nonsquare rhombus; annotate equal sides and square corners. Include shapes in different orientations.

Independent evidence: Solve a missing-side perimeter problem, construct one pair with equal perimeter and different area, and one with equal area and different perimeter. Explain the units and the counted feature. Describe and draw an unfamiliar polygon from stated properties. If a learner confuses boundary and interior, trace the outside with string and cover the inside with tiles to separate the two measurements.