Perimeter: Square & Rectangle

Learn and practice calculating the perimeter of squares, rectangles, composite shapes, and shapes with holes.

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Square & Rectangle

Use the workspace below. Write equations like P = 24 to solve for the perimeter.

📖 Perimeter Study Guide

The perimeter is the total length of the boundary of a two-dimensional shape. It represents the path that surrounds or outlines the shape.

1. The Full Boundary (Inner + Outer)

An important rule is that the perimeter is the full boundary of a shape. If a shape has a hole inside, walking along the entire boundary includes the outer boundary PLUS the inner boundary (the hole). Therefore, for shapes with holes:

2. Square and Rectangle Formulas

A rectangle has a width ww and a height hh. Opposite sides are equal, so the outer boundary has two sides of length ww and two sides of length hh. Summing them together gives the formula for the perimeter of a rectangle:
P=2w+2h=2(w+h)P = 2w + 2h = 2(w + h)

Variables:
• PP = Perimeter
• ww = width of the rectangle
• hh = height of the rectangle

Derivation:
1. Perimeter is the sum of all four side lengths: P=w+h+w+hP = w + h + w + h.
2. Group like terms: P=(w+w)+(h+h)=2w+2hP = (w + w) + (h + h) = 2w + 2h.
3. Factor out the common factor of 2: P=2(w+h)P = 2(w + h)
A square is a special case of a rectangle where all four sides are equal (w=h=sw = h = s). Substituting w=sw = s and h=sh = s into the rectangle formula, we derive:
P=2(s+s)=2(2s)=4sP = 2(s + s) = 2(2s) = 4s

Variables:
• PP = Perimeter
• ss = side length of the square

Derivation:
1. A square is a rectangle where width and height are equal: w=h=sw = h = s.
2. Substitute ss into the rectangle perimeter formula: P=2(s+s)=2(2s)=4sP = 2(s + s) = 2(2s) = 4s.
Learning Topics

Frequently Asked Questions

What is perimeter?

Perimeter is the total boundary of a two-dimensional shape. It includes both the outer boundary and any inner boundaries (like the edges of holes inside the shape).

How do you calculate the perimeter of a rectangle?

The perimeter of a rectangle is the sum of all its four sides: P = 2w + 2h or P = 2(w + h), where w is the width and h is the height.

Why is a square's perimeter formula P = 4s?

A square is a special case of a rectangle where width and height are equal (w = h = s). Substituting this into the rectangle formula gives P = 2(s + s) = 4s.

How does a hole affect the perimeter?

Since perimeter measures the entire boundary of a shape, a hole adds to the perimeter. The total perimeter is the outer perimeter plus the inner perimeter (the perimeter of the hole).

How do you find the perimeter of a right-angled triangle if one side is missing?

Since we've already learned the Pythagorean theorem (a² + b² = c²), we can calculate the missing side length first and then sum all three sides together to get the perimeter.

Perimeter: Square & Rectangle | SealMath