The circumference and a running track

The plane and space, grade 10 (3 units) Lesson 2 of 15Learning

The circumference and a running track

The circumference

In every circle, the circumference is $\displaystyle \pi$ times the diameter. $\displaystyle \pi$ is a number whose digits after the point never end: $\displaystyle 3.14159\ldots$ , and in calculations we use $\displaystyle \pi\approx3.14$ .

$\displaystyle d$ – the diameter, $\displaystyle r$ – the radius; the diameter is twice the radius: $\displaystyle {d}={2r}$ .

\displaystyle C

\displaystyle =

\displaystyle \pi d

\displaystyle C

\displaystyle =

\displaystyle 2\pi r

Example. A round pool has a radius of 5 m, and a safety fence goes around it.
The length of the fence: $\displaystyle {2\cdot3.14\cdot5}={31.4}$ m.

A semicircle: an arched window

A window is shaped like a rectangle with a half disc on top. The measurements in the drawing are in metres.
The example below works out how many metres of frame go around the window.

The dashed line is not part of the frame: it only separates the parts.
The three sides of the rectangle: $\displaystyle {1.5+1+1.5}={4}$ m.
The arch is half the circumference of a circle with a diameter of 1 m: $\displaystyle {\frac{3.14\cdot1}{2}}={1.57}$ m.
The whole frame: $\displaystyle {4+1.57}={5.57}$ m.

A running track

A running track in a stadium is made of two straight parts and two semicircles at the ends. The measurements in the drawing are in metres, along the inner line of the track.

The two straight parts: $\displaystyle {2\cdot100}={200}$ m.
The two semicircles make one whole circle with a diameter of 64 m: $\displaystyle {3.14\cdot64}={200.96}$ m.
One lap: $\displaystyle {200+200.96}={400.96}$ m – very close to 400 m.

The outer lane is longer

A track has several lanes side by side. A runner in an outer lane runs on a bigger circle at the ends.

The straight parts are the same in every lane.
If the outer lane is 1 m away from the inner one on each side, its diameter is 2 m bigger: 66 m instead of 64.
A lap in it is longer by $\displaystyle {3.14\cdot2}={6.28}$ m.
That is why runners in a one-lap race do not start on the same line: the runner in the outer lane starts a little ahead of the others.

Exercise 1

Calculate the length ( $\displaystyle \pi\approx3.14$ ).

a

The circumference of a round tray with a diameter of 50 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

b

The circumference of a round fountain with a radius of 1.5 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

c

An edging around a flower bed shaped like a half disc with a diameter of 4 m – the arch and the diameter

/ – 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

Around a school’s sports field there is a running track.
The measurements in the drawing are in metres, along the inner line of the track.

a

The length of one lap, in metres

/ – 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 many metres longer is a lap in a lane whose diameter is 6 m bigger?

/ – 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

The circumference of a tree trunk is 157 cm.
Find the diameter of the trunk, in cm: solve the equation.

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
$\displaystyle \alpha_1+\alpha_2+\ldots=360^{\circ}$