To find the perimeter of any shape we can add up all of the side lengths, however the properties of some shapes result in special perimeter forumulas.

The perimeter, P, of a triangle is given by

P=a+b+c

where a,\,b,\, and c are the lengths of each side.

A parallelogram with two pairs of parallel sides and opposite sides are congruent. Sides are label with a and b.

A parallelogram is a quadrilateral with two pairs of parallel sides, its opposite sides are congruent.

We will use the base, b, and slant height, a, to find its perimeter.

\text{Perimeter} = b + a + b + a

\text{Perimeter} =2a+2b=2 \left(a+b \right)

A rectangle. Opposite sides are congruent with labels l and w.

Consider a rectangle, a special type of parallelogram. Its opposite sides are congruent.

If the width is w and the length is l, the perimeter is: \text{Perimeter} = l+w+l+w

\text{Perimeter} = 2l+2w= 2 \left(l+w \right)

A square. All sides are congruent with label l.

Consider a square, a special type of rectangle. All of its sides are congruent.

If each side has length l, the perimeter is:\text{Perimeter} = l+l+l+l

\text{Perimeter} = 4 l

Examples

Example 1

Find the perimeter of the isosceles triangle shown.

Isosceles triangle where the opposite sides are 12 centimeters and the base is 8 centimeters.

Worked Solution

Create a strategy

Use the perimeter of the triangle formula.

Apply the idea

Since the triangle is isosceles, we can let the opposite sides a=12 and b=12. The base is c=8.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle a+b+c	Write the formula
	\displaystyle =	\displaystyle 12+12+8	Substitute a=12,\,b=12,\,c=8
	\displaystyle =	\displaystyle 32 \text{ cm}	Evaluate

Example 2

Find the perimeter of an equilateral triangle with a side length of 5\text{ mm}.

Worked Solution

Create a strategy

All 3 sides in an equilateral triangle are congruent. If we call the side length s, we can write the perimeter forumula:

P=s+s+s

or P=3s

Apply the idea

We are given that \text{side length}=5.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 3s	Write the formula
	\displaystyle =	\displaystyle 3 \cdot 5	Substitute s=5
	\displaystyle =	\displaystyle 15 \text{ mm}	Evaluate the multiplication

Example 3

Find the side length indicated on the diagram if the perimeter of the shape is 69\text{ cm}.

A triangle with side length of 23 centimeters and 19 centimeters. One side length is missing.

Worked Solution

Create a strategy

Use the formula for the perimeter of a triangle and substitute the known values.

Apply the idea

The triangle has P=69. We can let the side lengths a=23, b=19, and the missing side c.

\displaystyle P	\displaystyle =	\displaystyle a+b+c	Write the formula
\displaystyle 69	\displaystyle =	\displaystyle 23+19+c	Substitute P=69,\,a=23,\,b=19
\displaystyle 69	\displaystyle =	\displaystyle 42+c	Evaluate the addition
\displaystyle 27	\displaystyle =	\displaystyle c	Subtract 42 from both sides of the equation
\displaystyle c	\displaystyle =	\displaystyle 27 \text{ cm}	Rewrite with c on the left

Example 4

In the rectangle ABCD, side AB has a length of 6 units.

The image shows rectangle ABCD. Side AB has a length of 6 units.

State the other side of ABCD which must have a length of 6 units.

Worked Solution

Create a strategy

The opposite sides of a rectangle are congruent.

Apply the idea

Since \overline{AB} has a length of 6 units, properties of rectangles tells us the opposite side, \overline{CD}, will also have a length of 6 units.

If another side of the rectangle measures 5 units, find the perimeter.

Worked Solution

Create a strategy

Use the formula for the perimeter of a rectangle, P=2\left(l+w\right). The fourth side must also be 5 units since the opposite sides of a rectangle are congruent.

Apply the idea

The length is 6 units and the width is 5 units.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2(l + w)	Formula for perimeter of a rectangle
\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2(6 + 5)	Substitute the length and width
	\displaystyle =	\displaystyle 2 \cdot 11	Evaluate the addition
	\displaystyle =	\displaystyle 22 \text{ units}	Evaluate the multiplication

Example 5

Jacob is helping his father build a garden with a fence around it, and they decide to make the garden in the shape of a parallelogram. The garden has one side that is 15 feet long and the side adjacent to it is 10 feet long. What is the total perimeter of the garden fence that Jacob and his father need to prepare?

Worked Solution

Create a strategy

Opposite sides of a parallelogram are equal in length. Use formula for perimeter of a parallelogram: P=2\left(a+b\right).

Apply the idea

The base is 15 feet and the slant height is 10 feet. Using the formula to find perimeter.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2(a+b)	Formula for perimeter of a parallelogram
\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2(10 + 15)	Substitute the slant height and base
	\displaystyle =	\displaystyle 2 \cdot 25	Evaluate the addition
	\displaystyle =	\displaystyle 50 \text{ feet}	Evaluate the multiplication

Reflect and check

If we couldn't remember the formula we could have just written down all of the side lengths and added them together:

10+10+15+15=50 \text{ feet}

Idea summary

To find the perimeter of a figure, add up all of its side lengths.

The perimeter of a triangle, with sides a,\,b,\, and c has the formula: P_\text{triangle} = a + b + c

The perimeter of a parallelogram, with slant height, a, and base, b, has the formula: P_\text{parallelogram} = 2 \left(a+b\right)

The perimeter of a rectangle, with length, l, and width, w, has the formula:P_\text{rectangle} = 2 \left(l+w\right)

The perimeter of a square, with side length, l, has the formula: P_\text{square} = 4 l

Outcomes

6.MG.2

The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles, and parallelograms.

6.MG.2b

Solve problems, including those in context, involving the perimeter and area of triangles, and parallelograms.

7.03 Perimeter of triangles and parallelograms

Ideas

Perimeter of triangles and parallelograms

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Outcomes

6.MG.2

6.MG.2b

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