Practice: circle, cylinder and cone

Circle and solids, grade 8 Lesson 5 of 7Practice

Practice: circle, cylinder and cone

1

Find ( $\displaystyle \pi\approx3.14$ ):

a

The circumference of a circle 12 cm across

/ – 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

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b

The area of a disc of radius 8 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

The area of a half disc 10 cm across

/ – 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

2

A water tank on a roof is a cylinder.
Its radius is 1 m, and its height is 2 m.
How many m³ of water fit in it? And how many litres? (1 m³ = 1000 litres, $\displaystyle \pi\approx3.14$ )

m³

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Litres

/ – 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

3

A paint roller is a cylinder of radius 3 cm and length 20 cm.
Each turn of the roller paints a strip of the wall.
How many cm² does it paint in one turn?
How many whole turns does it take to paint 1 m²? ( $\displaystyle \pi\approx3.14$ )

/ – 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

4

A cone’s radius is 9 cm, and its height is 12 cm.
Find ( $\displaystyle \pi\approx3.14$ ):

a

its slant height

/ – 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

its curved surface area

/ – 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

its volume

/ – 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

5

A cup in the shape of a cone, of radius 4 cm and height 9 cm, is full of water.
The water is poured into a cup in the shape of a cylinder, also of radius 4 cm.
How high will the water reach in the second cup?
Find the volume of the water first, then solve an equation ( $\displaystyle \pi\approx3.14$ ).

/ – 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

6

A grain silo is made of a cylinder with a roof in the shape of a cone on top.
The radius of both is 3 m.
The cylinder’s height is 10 m, and the roof’s height is 4 m.
What is the volume of the whole silo? ( $\displaystyle \pi\approx3.14$ )

/ – 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

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7

A cylinder’s volume is 785 cm³, and its radius is 5 cm.
What is its height? Write an equation and solve it ( $\displaystyle \pi\approx3.14$ ).

8

Answer without working out the volumes themselves, and explain.

A cylinder’s radius grows 2 times, and its height stays the same. How many times as big is the volume?
A cylinder’s height grows 2 times, and its radius stays the same. How many times as big is the volume?
A cylinder’s volume is 600 cm³. What is the volume of a cone with the same base and the same height?

Radius ×2

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Height ×2

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The cone

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Formulas: circle and solids, grade 8

The circle and the disc

The number pi – the value to calculate with
$\displaystyle \pi\approx3.14$
The circumference of a circle – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference of a circle – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$
The area of a disc
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$

The cylinder

The curved surface area – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle L$ $\displaystyle =$ $\displaystyle 2\pi r h$
The surface area – the two bases and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2\pi r^2+2\pi r h$
The volume – the base area times the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2 h$

The cone

The slant height $\displaystyle l$ – the hypotenuse of a triangle whose legs are the radius $\displaystyle r$ and the height $\displaystyle h$
$\displaystyle l^2$ $\displaystyle =$ $\displaystyle r^2+h^2$
The curved surface area
$\displaystyle L$ $\displaystyle =$ $\displaystyle \pi r l$
The surface area – the base and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2+\pi r l$
The volume – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2 h}{3}$

Pythagoras' theorem in space

The diagonal of a box $\displaystyle d$ – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three edges
$\displaystyle d^2$ $\displaystyle =$ $\displaystyle a^2+b^2+c^2$
In a square pyramid – $\displaystyle k$ the height of a side face, $\displaystyle h$ the pyramid’s height, $\displaystyle a$ the side of the base
$\displaystyle k^2$ $\displaystyle =$ $\displaystyle h^2+\left(\frac{a}{2}\right)^2$