Key points
The total distance around a shape. is a length measured in either metric units such as millimetres, centimetres and metres, or imperial units such as inches and miles.
The perimeter is found by adding together the lengths of each side. It may be necessary to convert between metric units to ensure that the added lengths are measured in the same units.
For a A closed 2D shape bounded by straight lines. All sides are equal in length. All angles are equal. the perimeter can be calculated by multiplication. There are two different formulae for finding the perimeter of a rectangle. Using formulae and changing the subject of a formula are useful skills for solving perimeter problems.
How to find the perimeter of simple shapes
The perimeter may be found by counting squares along all the edges of a A shape made up of straight lines and right angles. shape, or by adding up all lengths of the sides of any shape or by using a formula for a specific shape.
To find the perimeter of aA regular quadrilateral. All four sides are equal in length and all four angles are right angles. or a A quadrilateral with opposite pairs of sides that are both equal in length and parallel. All four angles are right angles. drawn on a grid:
- Pick a starting The point at which two or more lines intersect (cross or overlap). The corner of a shape. The plural form is vertices. (corner).
- Count the squares along each side, going around the whole shape back to the starting point.
To find the perimeter of a shape where the lengths are given for each side:
- Write down all the lengths, in the same units, and add them up.
To find the perimeter of a shape using a formula:
- Choose the correct formula
- For a square \(P\) = 4\(s\) where \(s\) is the length of one side
- For a rectangle either \(P\) = 2\(l\) + 2\(w\) or \(P\) = 2(\(l\) + \(w\)), where \(l\) is the length and \(w\) the width of the rectangle.
- Substitute the values into the formula and complete the calculation.
- Choose the correct formula
Once the perimeter has been worked out, write the answer with the correct units.
Examples
Image caption, The rectangle is drawn on a centimetre square grid. Count squares to find the perimeter of the square.
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Question
Find the perimeter of a rectangle with a length of 120 m and a width of 45 m.
Substitute \(l\) = 120 and \(w\) = 45 into the formula.
Multiply the length by two. 2 Ă 120 = 240
Multiply the width by two. 2 Ă 45 = 90
Add the two values together. 240 + 90 = 330
The perimeter of the rectangle is 330 m.
Adding together the length and the width, 120 + 45 = 165, and then multiplying by 2, 165 Ă 2 = 330, gives the same answer.
How to solve perimeter problems
Perimeter problems may involve working out a length or width or finding the perimeter of a different shape. A problem may be solved by:
A numerical method
Rearranging a formula
Using known facts to simplify the problem
Using a numerical method to find a length, given the total perimeter:
- Add up all the given side lengths.
- Subtract from the perimeter.
Rearranging a formula to find a length:
- Substitute the given values into the formula.
- Use inverse operations to undo each step to find the missing length.
To solve a perimeter problem when information is given in an unfamiliar way:
- Use the known facts to build understanding.
Examples
Image caption, The perimeter of the triangle is 44 cm. Find the length of side AB.
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Questions
Question 1: The perimeter of the rectangle is 80 mm. The length of the rectangle is 15 mm. Find the length of the rectangle.
The perimeter of the rectangle is \(P\) = 2\(l\) + 2\(w\). The perimeter is 80, \(P\) = 80. The width is 15 mm, 2\(w\) = 30
Substitute the given values, \(P\) = 80 and 2\(w\) = 30, into the formula.
80 = 2\(l\) + 30
The inverse of + 30 is - 30. Subtract 30 on both sides of the equation.
50 = 2\(l\)
The inverse of Ă 2 is Ă· 2. Divide both sides of the equation by 2
25 = \(l\)
The length of the rectangle is 25 mm.
Question 2: The large rectangle is made up of congruent small rectangles. The large rectangle has a length of 24 cm and a width of 10 cm. Find the perimeter of one small rectangle.
The length of the large rectangle is made up of 4 lengths of the small rectangle.24 Ă· 4 = 6The length of the small rectangle is 6 cm.
The length of the large rectangle is made up of 6 widths of the small rectangle.24 Ă· 6 = 4The width of the small rectangle is 4 cm.
The perimeter of the small rectangle is the sum of two lengths and two widths.2 Ă 6 + 2 Ă 4 = 20The perimeter of the small rectangle is 20 cm.
How to find the perimeter of a compound rectilinear shape
A A shape formed by combining two or more shapes. A shape made up of straight lines and right angles. shape is made up of two or more rectangles.
The perimeter may be found by counting squares along all the edges a shape or by adding up all lengths of the sides of the shape.
For a rectilinear compound shape drawn on a grid:
- Pick a starting vertex (corner).
- Count the squares along each side, going around the whole shape back to the starting point.
For a shape where the lengths are given for each side:
- Write down all the lengths, in the same units, and add them up.
For a shape where lengths are missing:
- Work out the missing lengths, using Always equidistant (at equal distances). Parallel lines, curves and planes never meet however far they extend. edges.
- Write down all the lengths, in the same units, and add them up.
- If the lengths cannot be worked out, use known facts to find the perimeter.
Examples
Image caption, Find the perimeter of the compound rectilinear shape.
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Questions
Question 1: Find the perimeter of the compound rectilinear shape.
There is one missing horizontal length. It is equal to the sum of the opposite parallel lengths. 12 + 7 = 19
The missing length is 19 cm.
The perimeter of the shape is the sum of all the sides.
19 + 11 + 7 + 3 + 12 + 8 = 60
The perimeter of the shape is 60 cm.
Question 2: Find the perimeter of the compound rectilinear shape.
There are four missing lengths. It is not possible to work out each specific length.
The shape is a rectangle with a rectangle cut from two corners. The perimeter of the shape is the same as the âoriginalâ rectangle. The total horizontal and the vertical lengths remain the same.
The total horizontal length is 21 m. The total vertical length is worked out by adding the given vertical lengths, 6 + 12, 18 m.
The perimeter is the sum of the two horizontal lengths and the two vertical lengths. 2 Ă 21 + 2 Ă 18 = 78The perimeter of the compound shape is 78 m.
Practise finding the perimeter of rectangles and compound shapes
Quiz
Practise finding the perimeter of rectangles and compound shapes in this quiz. You may need a pen and paper to help you with your answers.
Real-life maths
Fencing companies install perimeter fences for different clients, such as around businesses, private properties, parks and gardens.
A perimeter fence is part of a security system. The fencing company will measure the lengths around the site, adding them up to find the total length of perimeter fencing that is needed.
This then affects how much they will charge for the amount of fencing their client needs.
Game - Divided Islands
Play the Divided Islands game! game
Using your maths skills, help to build bridges and bring light back to the islands in this free game from ±«Óătv Bitesize.
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