The area of a disc, composite shapes and irregular shapes

The plane and space, grade 10 (3 units) Lesson 9 of 15Learning

The area of a disc, composite shapes and irregular shapes

The area of a disc

A disc is a circle with all that is inside it. Its area is $\displaystyle \pi$ times the square of the radius:

\displaystyle A

\displaystyle =

\displaystyle \pi r^2

$\displaystyle r^2$ is $\displaystyle r\cdot r$ . When the diameter is given, first divide it by 2.
Example. A pizza with a diameter of 30 cm: $\displaystyle {r}={15}$ , $\displaystyle {A}={3.14\cdot15\cdot15}={706.5}$ cm².

Exercise 1

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

A round table with a radius of 0.6 m, in m²

/ – 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

A pizza with a diameter of 40 cm, in cm²

/ – 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

A stage shaped like a half disc with a diameter of 10 m, in m²

/ – 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 pizzeria has two offers at the same price, 60 ₪:
one large pizza, with a diameter of 40 cm;
or two small pizzas, each with a diameter of 28 cm.

The area of the two small pizzas together, in cm²

/ – 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

Which offer gives more pizza? Use the area you worked out in Exercise 1.

A composite shape

The area of a composite shape is found in parts:

cutting – cut it into shapes you can work out, and add the areas;
completing – complete it to a bigger shape, and take away what was added.

Example: the field inside a running track. The measurements in the drawing are in metres.

The dashed lines cut the field into a rectangle and two half discs.
The rectangle: $\displaystyle {100\cdot64}={6{,}400}$ m².
The two half discs make a whole disc, radius 32 m: $\displaystyle {3.14\cdot32\cdot32}={3{,}215.36}$ m².
The whole field: $\displaystyle {6{,}400+3{,}215.36}={9{,}615.36}$ m².

Exercise 3

A rectangular lawn has a round pond in it. The measurements in the drawing are in metres.
Calculate the area of the grass, without the pond, in m².

/ – 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

An irregular shape: estimating

A lake has no shape with a formula. Its area is estimated with a grid of squares: each square on the map is 10 m by 10 m, that is 100 m².

Count the squares the lake covers completely: 11.
Count the squares the lake covers only partly: 21. Some hold a lot of water and some a little, so count each as half a square: $\displaystyle {21\div2}={10.5}$ .
Together: $\displaystyle {11+10.5}={21.5}$ squares, that is about $\displaystyle {21.5\cdot100}={2{,}150}$ m².
This is an estimate, not an exact result. On a finer grid – smaller squares – the estimate is closer.

Exercise 4

Each square on the map is 10 m by 10 m.
Estimate the area of the lake, in m². Write how many full and how many partial squares you counted.

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

Open as a page

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