AQA Level 2 Further Maths · Level 2 Certificate · Chapter 6: Geometry I · Lesson 1

Mensuration, Pythagoras' theorem and angle facts

About 80 minutesNo calculator61 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/further-maths-mensuration-pythagoras-theorem-and-angle-facts/

Date

Do Now

The first two are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A right-angled triangle has shorter sides $4$ cm and $6$ cm. Work out the length of the hypotenuse. Give your answer as a surd in its simplest form.
[2 marks]
Answer
A2
Needed today
Work out the size of each interior angle of a regular polygon with $18$ sides.
[2 marks]
Answer
A3
Last lesson
The point $P (5, - 2)$ lies on the circle $x^{2} + y^{2} = 29$ . Work out the gradient of the tangent to the circle at $P$ .
[2 marks]
Answer
A4
From chapter 2
Rearrange $V = \frac{1}{3} π r^{2} h$ to make $r$ the subject, where $r > 0$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

The trapezium has an area of $60$ cm². Work out the value of $x$ .

Not drawn accurately

Hint Area

= \frac{1}{2} (a + b) h

. Set it equal to

60

B1
The parallelogram has an area of $50$ cm². Work out the value of $x$ .
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$

Example 2

The sides of a right-angled triangle are $x$ cm, $(x + 7)$ cm and $(x + 9)$ cm. Work out the perimeter of the triangle.

Not drawn accurately

Hint The longest side is the hypotenuse. Use Pythagoras' theorem, expand and collect terms.

B2
The sides of a right-angled triangle are $x$ cm, $(2 x + 10)$ cm and $(2 x + 11)$ cm. Work out the area of the triangle.
[4 marks]
Not drawn accurately
Answer $\dots\dots$ cm²

Example 3

A rectangle measures $43$ cm by $26$ cm. Work out the length of its diagonal. Give your answer as a surd in its simplest form.

Not drawn accurately

Hint

(43)^{2} = 16 \times 3

. Add the two squares, then simplify the square root.

B3
A right-angled triangle has hypotenuse $217$ cm and one shorter side $25$ cm. Work out the area of the triangle. Give your answer in the form $a b$ , where $a$ and $b$ are integers.
[3 marks]
Not drawn accurately
Answer $\dots\dots$ cm²

Example 4

A cone has a volume of $108 π$ cm³. Its height and base radius are in the ratio $h : r = 3 : 2$ . Work out the base radius.

Volume of a cone $= \frac{1}{3} π r^{2} h$

Hint Write

h

in terms of

r

h = \frac{3}{2} r

. Then the volume is a multiple of

r^{3}

B4
A sphere has a surface area of $196 π$ cm². Work out its volume. Give your answer as a multiple of $π$ .
Surface area of a sphere $= 4 π r^{2}$ Volume of a sphere $= \frac{4}{3} π r^{3}$
[3 marks]
Answer $\dots\dots$ cm³

Example 5

The pentagon $A B C D E$ has the angles shown. Work out the size of the largest angle.

Not drawn accurately

Hint The angles of a pentagon add up to

(5 - 2) \times 180 = 54 0^{\circ}

B5
$A B C D$ is a kite with $A B = A D$ . Work out the size of angle $B C D$ .
[3 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$

Practice

Level 2

C1
A washer is the region between two circles with the same centre. The radii are $x$ cm and $(x + 2)$ cm. The area of the washer is $36 π$ cm². Work out the value of $x$ .
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$
C2
A prism is $12$ cm long. Its cross-section is a right-angled triangle with shorter sides $x$ cm and $(x + 4)$ cm. The volume of the prism is $270$ cm³. Work out the value of $x$ .
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$

C3
A triangle has sides of length $(n^{2} - 1)$ , $2 n$ and $(n^{2} + 1)$ , where $n > 1$ . Prove that the triangle is right-angled.
[3 marks]
Not drawn accurately
C4
A pyramid has a square base of side $2 x$ cm and a vertical height of $3 x$ cm. Its volume is $108$ cm³. Work out the value of $x$ .
Volume of a pyramid $= \frac{1}{3} \times$ base area $\times$ height
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$

Exam-style questions

AQA exam style

D1
Mia says, "A triangle with sides $9$ cm, $40$ cm and $42$ cm is right-angled."
Show that Mia is wrong.
[2 marks]
Not drawn accurately

D2
This shape is made from two rectangles. All lengths are in centimetres.
The perimeter of the shape is $72$ cm.
Not drawn accurately
(a)
Show that $y = 36 - 5 x$ .
[2 marks]
(b)
The area of the shape is $A$ cm². Show that $A = 144 x - 18 x^{2}$ .
[2 marks]
(c)
Write $144 x - 18 x^{2}$ in the form $p - q (x - r)^{2}$ . Hence work out the greatest possible area of the shape.
[3 marks]
Answer $\dots\dots - \dots\dots (x - \dots\dots)^{2}$ greatest area $\dots\dots$ cm²
Total for D2: 7 marks
D3
A cone has a slant height of $17$ cm. Its curved surface area is $136 π$ cm². Work out the volume of the cone. Give your answer as a multiple of $π$ .
Curved surface area of a cone $= π r l$ Volume of a cone $= \frac{1}{3} π r^{2} h$
[4 marks]
Not drawn accurately
Answer $\dots\dots$ cm³
D4
$D A E$ is a straight line parallel to $B C$ . Work out the size of angle $A C B$ . Give a reason for each step of your working.
[4 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$
D5
$A B C D$ is a square. The diagonal $A C$ is $(4 + 22)$ cm. Work out the area of $A B C D$ . Give your answer in the form $a + b 2$ , where $a$ and $b$ are integers.
[3 marks]
Not drawn accurately
Answer $\dots\dots + \dots\dots 2$ cm²

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A solid cone and a solid hemisphere have the same base radius $r$ . Their total surface areas are equal. Work out the ratio volume of the cone : volume of the hemisphere. Give your answer in the form $a : b$ , where $a$ and $b$ are integers.
Curved surface area of a cone $= π r l$ Surface area of a sphere $= 4 π r^{2}$
[5 marks]
Hint 1 · What to try
Write both total surface areas using $r$ and the slant height $l$ .
Hint 2 · The first line
Hemisphere: $3 π r^{2}$ . Cone: $π r l + π r^{2}$ . So $l = 2 r$ .
Hint 3 · The full method
$h = 3 r$ . Volumes $\frac{1}{3} 3 π r^{3}$ and $\frac{2}{3} π r^{3}$ ; divide both by $\frac{1}{3} π r^{3}$ .
Answer $\dots\dots : \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use area and volume formulae with algebra and $π$ .
I can use Pythagoras' theorem with algebra and surds.
I can use angle facts to form and solve equations.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers