Perimeter: fencing a plot and frames

The plane and space, grade 10 (3 units) Lesson 1 of 15Learning

Perimeter: fencing a plot and frames

Perimeter

The perimeter of a shape is the length of the line around it: the sum of the lengths of all its sides.

A perimeter is measured in units of length: cm, metres, km.
To fence a plot, frame a picture or put a strip along a wall, you need the perimeter.
A rectangle has two pairs of equal sides, $\displaystyle a$ and $\displaystyle b$ :

\displaystyle P

\displaystyle =

\displaystyle 2(a+b)

In a square all four sides are equal:

\displaystyle P

\displaystyle =

\displaystyle 4a

Example: fencing a plot

A plot is shaped like a step, and all its corners are right angles. The measurements in the drawing are in metres.
The two sides marked ? were not measured.
The example below finds the missing sides, and then the perimeter.

The missing horizontal side: the bottom is 30, and the top already has 12. So $\displaystyle {30-12}={18}$ m.
The missing vertical side: the left side is 22, and the right side already has 12. So $\displaystyle {22-12}={10}$ m.
The perimeter: $\displaystyle {30+12+18+10+12+22}={104}$ m.
Notice: this is exactly the perimeter of a 30 by 22 rectangle: $\displaystyle {2\cdot(30+22)}={104}$ . The sides of the step, moved out, complete the rectangle.

A fence with a gate, and a price per metre

The plot from the example is to be fenced.
The fence will have a gate 4 m wide, and the gate is bought separately.
A metre of fence costs 85 ₪.

The length of the fence: $\displaystyle {104-4}={100}$ m.
The price of the fence: $\displaystyle {100\cdot85}={8{,}500}$ ₪.

Posts for a fence

A fence stands on posts, at equal distances from each other.

A closed fence – around a plot: there are as many posts as gaps. A square plot of side 20 m, a post every 4 m: the perimeter is $\displaystyle {4\cdot20}={80}$ m, so $\displaystyle {80\div4}={20}$ gaps and 20 posts.
An open fence – in a straight line: there is one post more than the gaps, since there is a post at the end too. A straight fence of 20 m, a post every 4 m: $\displaystyle {20\div4}={5}$ gaps, and 6 posts.

Exercise 1

Calculate the perimeter.

a

A rectangular plot, 25 m by 14 m

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

A square tile of side 7.5 cm

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

A triangle with sides 13 cm, 14 cm and 15 cm

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A yard is shaped like stairs, and all its corners are right angles.
The measurements in the drawing are in metres.
Calculate the perimeter of the yard.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A rectangular plot is 40 m by 25 m.
A fence is built around it, with a gate 5 m wide.
A metre of fence costs 120 ₪.

a

How many metres of fence are needed?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How much will the fence cost, in shekels?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A rectangular garden is 12 m by 8 m.
A fence is put up around it, with a post every 2 m.
How many posts are needed?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 10 (3 units)

Perimeters

The perimeter of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2(a+b)$
The perimeter of a square – $\displaystyle a$ the side
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4a$
Pi
$\displaystyle \pi\approx3.14$
The circumference – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$

Routes: Pythagoras, distance, speed and time

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2$
The hypotenuse
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2}$
A leg
$\displaystyle a$ $\displaystyle =$ $\displaystyle \sqrt{c^2-b^2}$
Speed – $\displaystyle s$ the distance, $\displaystyle t$ the time
$\displaystyle v$ $\displaystyle =$ $\displaystyle \frac{s}{t}$
Distance
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$
Time
$\displaystyle t$ $\displaystyle =$ $\displaystyle \frac{s}{v}$

Areas

The area of a rectangle
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$
A square metre in square centimetres (a dunam is 1000 m²)
$\displaystyle 1\text{ m}^2=100\cdot100\text{ cm}^2=10000\text{ cm}^2$

Tilings

The sum of the angles of a polygon with $\displaystyle n$ sides
$\displaystyle 180^{\circ}\cdot(n-2)$
The angle of a regular polygon with $\displaystyle n$ sides
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \frac{180^{\circ}\cdot(n-2)}{n}$
The angles around a vertex of a tiling
$\displaystyle \alpha_1+\alpha_2+\ldots=360^{\circ}$