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The perimeter
The students created polygons using a \latex{ 2 } \latex{ m }-long folding ruler. How long is the segmented line bordering the polygons?
The perimeter of a polygon is the sum of the lengths of the sides.
The symbol of perimeter is \latex{ P }.
Since perimeter is the length of the bordering line of a shape, its units of measurement are the units of length.
Since perimeter is the length of the bordering line of a shape, its units of measurement are the units of length.
\latex{1} \latex{ mm } \latex{\underset{\times10}{\lt}1} \latex{ cm } \latex{\underset{\times10}{\lt}1} \latex{ dm } \latex{\underset{\times10}{\lt}1} \latex{ m }\latex{\underset{\times1,000}{\lt}1} \latex{ km }
The perimeter of every polygon in the previous example equals the length of the folding ruler (\latex{ P } \latex{= 2} \latex{ m }).
The perimeter of quadrilaterals
Roll a quadrilateral along a line and mark the places where its vertices touch the line.
\latex{ A }
\latex{ D }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ A }
\latex{ a }
\latex{ b }
\latex{ c }
\latex{ d }
\latex{ d }
\latex{ a }
\latex{ c }
The horizontal line shows the perimeter of the quadrilateral: \latex{P = a + b + c + d}.
\latex{P=a+b+c+d}
The perimeter of the quadrilateral is the sum of the lengths of its sides.
The perimeter of rectangles
The lengths of a rectangle's opposite sides are equal. One of the lengths is indicated by the letter \latex{ a }, while the other is indicated by the letter \latex{ b }. The perimeter of a rectangle, that is, the sum of the lengths of its sides, can be calculated in several ways.
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ a }
- Add the lengths of the sides one after the other:
\latex{P=a+b+a+b}.
- Add the length of the opposite sides first:
\latex{P=(a+a)+(b+b)=2\times a + 2 \times b}
- Add the lengths of the adjacent sides first:
\latex{P=(a+b)+(a+b)=2\times (a + b)}
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ b }
\latex{ b }
\latex{ b }
\latex{ b }
\latex{ a+b }
\latex{ a+b }
\latex{ a }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{P=a+b+a+b}
\latex{P=2\times a+2\times b}
\latex{P=2\times (a+b)}
The perimeter of a rectangle is twice the sum of the lengths of the two adjacent sides. The perimeter of a rectangle with sides \latex{a} and \latex{b} is:
\latex{P = 2 \times a + 2 \times b = 2 \times (a + b)}.
The perimeter of squares
A square is a rectangle whose sides are equal in length. The length of its sides is marked with the letter \latex{a}.
The perimeter of a square, that is, the sum of the lengths of its sides, is:
\latex{P = a + a + a + a = 4 \times a}.
\latex{P = a + a + a + a = 4 \times a}.
The perimeter of a square is four times the length of one side. The perimeter of a square with side \latex{a} is:
\latex{P = 4 \times a}.
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
Exercises
{{exercise_number}}. Create polygons with a perimeter of a) \latex{ 8 } matchsticks and b) \latex{ 15 } matchsticks. Make a drawing of the polygons you create.
{{exercise_number}}. Using matchsticks, create polygons with a perimeter of a) \latex{12}; b) \latex{16}; c) \latex{20} units. Can you make a rectangle using \latex{ 9 } matchsticks without breaking them?
{{exercise_number}}. Dan breeds German shepherd dogs and wants to build a fence around the square-shaped kennel. The side of the kennel measures \latex{ 6 } \latex{ m } in length. How long will the fence be with the door included?
{{exercise_number}}. Frank is making a kite. How much adhesive tape will he need to tape the sides of the kite? (→)
{{exercise_number}}. When renovating the floor in the classroom, the skirting boards were also replaced. How much skirting is needed if the classroom is rectangular, the sides measure \latex{ 9 } and \latex{ 7 } \latex{ m } in length, and the door is \latex{ 1 } \latex{ m } wide? (The bottom of the door does not include skirting.)
{{exercise_number}}. The following polygons were drawn on squared graph paper. Calculate the perimeter if the unit is one side of the small squares. Compare the lengths of the polygons' perimeters.
a)
b)
c)
{{exercise_number}}. Which of the following polygons does not fit into the series? Write the letters of the polygons in ascending order according to the lengths of their perimeters.
a)
b)
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
{{exercise_number}}. The hexagon shown in the images was created using five congruent squares. Add one more congruent square so that the perimeter of the resulting shape is
- as short as possible;
- as long as possible.
How many \latex{ cm } is the smallest and the largest perimeter?
\latex{1} \latex{ cm }
{{exercise_number}}. Add one square to the shape in the image in such a way that the perimeter
a) remains unchanged; b) decreases; c) increases. (→)
{{exercise_number}}. How long is the perimeter of the rectangle if the lengths of its adjacent sides are
a) \latex{12} \latex{ cm } and \latex{26} \latex{ cm };
b) \latex{480} \latex{ mm } and \latex{2} \latex{ dm };
c) \latex{136} \latex{ mm } and \latex{14} \latex{ cm ?}
{{exercise_number}}. What is the perimeter of the square if the length of its sides is
a) \latex{12} \latex{ cm };
b) \latex{24} \latex{ mm };
c) \latex{125} \latex{ m ?}
{{exercise_number}}. How many \latex{ centimetres } is the perimeter of a square if one side is
a) \latex{\frac{4}{5}} \latex{ dm };
b) \latex{4\frac{1}{5}} \latex{ dm };
c) \latex{ 17 } and a half \latex{ millimetres? }
{{exercise_number}}. One side of a rectangle is \latex{\frac{2}{3}} \latex{ dm }, the other side is \latex{\frac{3}{4}} \latex{ dm } long. What is its perimeter?
{{exercise_number}}. What is the perimeter of a rectangle if one of its sides is \latex{\frac{5}{6}} \latex{ dm } long, while the other is
\latex{\frac{1}{4}} \latex{ dm } longer than the other?
{{exercise_number}}. What is the perimeter of the rectangle if one of its sides is \latex{ 120 } \latex{ mm } long and the difference between the lengths of its adjacent sides is \latex{ 10 } \latex{ mm ?}
{{exercise_number}}. How long are the sides of a square if its perimeter is \latex{\frac{2}{3}} \latex{ dm ?}
{{exercise_number}}. How long can the side of a square be if, measured in \latex{ centimetres }, it is a whole number, and its perimeter is less than \latex{ 2 } \latex{ dm ?}
{{exercise_number}}. The dimensions of the frame can be seen in the image. What is the perimeter of the frame? What is the perimeter of the picture in it? (→)
\latex{40} \latex{ cm }
\latex{5} \latex{ cm }
\latex{5} \latex{ cm }
\latex{26} \latex{ cm }
{{exercise_number}}. The perimeter of a rectangle is \latex{ 36 } \latex{ cm }. What is the sum of the lengths of two adjacent sides?
{{exercise_number}}. What is the perimeter of a rectangular envelope if one of its sides is \latex{ 16 } \latex{ cm } long and the sum of the lengths of its adjacent sides is \latex{ 27 } \latex{ cm ?}
{{exercise_number}}. How long are the sides of a rectangular piece of paper if its perimeter is \latex{ 102 } \latex{ cm } and one of its sides is \latex{ 9 } \latex{ cm } shorter than the other? When cut in half, what will the perimeter of the resulting pieces of paper be?
{{exercise_number}}. One side of a rectangular plot of land is \latex{ 14 } \latex{ metres } shorter than the other. The owner bought \latex{ 130 } \latex{ metres } of wire to build a fence around it. He used the wire to build a \latex{ 6 } \latex{ m } wide gate as well. When he was finished, he still had \latex{ 6 } \latex{ m } of wire left. What are the dimensions of the plot?
{{exercise_number}}. One of the sides of a rectangle is \latex{ 14 } \latex{ cm } long. If the opposite sides are increased by \latex{ 5 } \latex{ cm }, you get a square. How long is the other side of the rectangle? What is the perimeter of the resulting square?
{{exercise_number}}. How many different rectangles can be made using \latex{ 12 } matchsticks? (You cannot place the matchsticks on top of each other, right next to each other, and they cannot protrude either.)
Quiz
Copy the polygon into your notebook and cut it into three pieces along the gridlines, so that they form a square when assembled. What is the perimeter of the resulting square?
