Perimeter

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

Perimeter Study Guide

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

Practice Topics

1. Square & Rectangle

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

2. Right-angled Triangle

Learn and practice calculating the perimeter of right-angled triangles using the Pythagorean theorem.

Learning Guide

1. Square & Rectangle

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:

Total Perimeter (with Holes)

P_{\text{total}} = P_{\text{outer}} + P_{\text{inner}}

2. Square and Rectangle Formulas

A rectangle has a width

w

and a height

h

. Opposite sides are equal, so the outer boundary has two sides of length

w

and two sides of length

h

. Summing them together gives the formula for the perimeter of a rectangle:

P = 2w + 2h = 2(w + h)

Variables:
•

P

= Perimeter
•

w

= width of the rectangle
•

h

= height of the rectangle

Derivation:
1. Perimeter is the sum of all four side lengths:

P = w + h + w + h

.
2. Group like terms:

P = (w + w) + (h + h) = 2w + 2h

.
3. Factor out the common factor of 2:

P = 2(w + h)

Perimeter of a Rectangle

P = 2(w + h)

A square is a special case of a rectangle where all four sides are equal (

w = h = s

). Substituting

w = s

and

h = s

into the rectangle formula, we derive:

P = 2(s + s) = 2(2s) = 4s

Variables:
•

P

= Perimeter
•

s

= side length of the square

Derivation:
1. A square is a rectangle where width and height are equal:

w = h = s

.
2. Substitute

s

into the rectangle perimeter formula:

P = 2(s + s) = 2(2s) = 4s

Perimeter of a Square

P = 4s

2. Right-angled Triangle

3. Right-angled Triangle Perimeter

For any triangle with sides

a

b

, and

c

, the perimeter is simply the sum of all three sides:

P = a + b + c

For a right-angled triangle with perpendicular legs

a

and

b

, we can use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the hypotenuse

c = \sqrt{a^2 + b^2}

, and then calculate the perimeter.

Variables:
•

P

= Perimeter
•

a

= first perpendicular leg
•

b

= second perpendicular leg
•

c

= hypotenuse

Derivation:
1. The perimeter of any triangle is the sum of its three sides:

P = a + b + c

.
2. For a right-angled triangle, if the hypotenuse

c

is unknown, calculate it using the Pythagorean theorem:

c = \sqrt{a^2 + b^2}

.
3. Substitute

c

back to find the perimeter.

Perimeter of a Right-angled Triangle

P = a + b + \sqrt{a^2 + b^2}

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.