AQA Level 2 Further Maths · Level 2 Certificate · Chapter 7: Geometry II · Lesson 1

The area of a triangle

About 80 minutesCalculator allowed68 marks

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Four questions to start. Two of them use what you need today.

A1
Needed today
Triangle $A B C$ has a right angle at $C$ . $A B = 14$ cm and angle $B A C = 3 7^{\circ}$ . Work out the length of $B C$ . Give your answer to 3 significant figures.
[2 marks]
Answer $\dots\dots$ cm
A2
Last lesson
Do not use a calculator. $sin x = \frac{2}{3}$ and $9 0^{\circ} < x < 18 0^{\circ}$ . Work out the exact value of $cos x$ .
[2 marks]
Answer $cos x = \dots\dots$
A3
Needed today
Solve $sin x = 0.8$ for $0^{\circ} \leq x \leq 18 0^{\circ}$ . Give your answers to 1 decimal place.
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A4
GCSE Higher
A sector of a circle has radius $6$ cm and angle $5 0^{\circ}$ . Work out its area. Give your answer to 3 significant figures.
[2 marks]
Answer $\dots\dots$ cm²

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it. Area of a triangle $= \frac{1}{2} ab sin C$ , where $C$ is the angle between the sides $a$ and $b$ . Give answers to 3 significant figures.

Example 1

Work out the area of triangle $P QR$ .

Hint The

4 8^{\circ}

angle is between the two sides you know, so use

\frac{1}{2} ab sin C

straight away.

B1
Work out the area of triangle $K L M$ .
[2 marks]
Answer $\dots\dots$ cm²

Example 2

Triangle $A B C$ is isosceles with $A B = A C = 8$ cm and angle $A B C = 3 5^{\circ}$ . Work out its area.

Hint You need the angle BETWEEN the two

8

cm sides. That is angle

B A C = 180 - 35 - 35

B2
Triangle $X Y Z$ is isosceles with $X Y = X Z = 9.5$ cm and angle $X Y Z = 6 2^{\circ}$ . Work out its area.
[3 marks]
Answer $\dots\dots$ cm²

Example 3

A regular octagon is drawn inside a circle of radius $5$ cm, with its corners on the circle. Work out the area of the octagon.

Hint Join each corner to the centre. This makes

8

equal triangles, each with two sides of

5

cm and an angle of

360 \div 8

at the centre.

B3
A regular nine-sided polygon is drawn inside a circle of radius $6$ cm, with its corners on the circle. Work out the area of the polygon.
[3 marks]
Answer $\dots\dots$ cm²

Example 4

Triangle $A B C$ has area $30$ cm². $C A = 8$ cm and $C B = 10$ cm. Work out the two possible sizes of angle $A C B$ . Give your answers to 1 decimal place.

Hint Find

sin C

from the area. Then remember that

sin (18 0^{\circ} - C) = sin C

: an obtuse angle gives the same area.

B4
Triangle $P QR$ has area $18$ cm². $P Q = 6$ cm and $P R = 9$ cm. Work out the two possible sizes of angle $QP R$ . Give your answers to 1 decimal place.
[3 marks]
Answer $\dots \dots^{\circ}$ or $\dots \dots^{\circ}$

Example 5

Triangle $A B C$ has area $40$ cm². $A B = 12$ cm and angle $B A C = 5 5^{\circ}$ . Work out the length $A C$ .

Hint Write the area formula with

A C = x

, then solve for

x

B5
An isosceles triangle has area $50$ cm². The angle between its two equal sides is $7 0^{\circ}$ . Work out the length of one of the equal sides.
[3 marks]
Not drawn accurately
Answer $\dots\dots$ cm

Practice

Level 2

C1
$A$ , $C$ and $D$ lie on a straight line. Work out the area of triangle $A B C$ .
[3 marks]
Answer $\dots\dots$ cm²
C2
A regular hexagon has sides of length $9$ cm. Work out its area.
[3 marks]
Answer $\dots\dots$ cm²

C3
$A B C D$ is a rhombus with sides of $6.5$ cm and angle $D A B = 7 2^{\circ}$ . Work out its area.
[3 marks]
Answer $\dots\dots$ cm²
C4
Do not use a calculator. In triangle $D E F$ , $D E = 6$ cm, $D F = 82$ cm and angle $E D F = 13 5^{\circ}$ . Work out the area of the triangle. Give your answer as an integer.
[3 marks]
Answer $\dots\dots$ cm²
C5
A regular tetrahedron has four faces. Each face is an equilateral triangle with sides of $8$ cm. Work out the total surface area of the tetrahedron.
[3 marks]
Answer $\dots\dots$ cm²
C6
The circle has centre $O$ and radius $7$ cm. Angle $A O B = 11 0^{\circ}$ . Work out the area of the shaded segment.
[4 marks]
Answer $\dots\dots$ cm²

Exam-style questions

AQA exam style

D1
Sam says, "I drew a triangle with two sides of $10$ cm and $6$ cm. Its area is $35$ cm²."
Show that Sam must be wrong.
[2 marks]

D2
A regular hexagon has an area of $150$ cm². Work out the length of one side. Give your answer to 3 significant figures.
[3 marks]
Answer $\dots\dots$ cm
D3
The quadrilateral $A B C D$ is made from two triangles. Work out the area of $A B C D$ . Give your answer to 3 significant figures.
[4 marks]
Answer $\dots\dots$ cm²
D4
$A B C D$ is a parallelogram. $A B = 12$ cm and $B C = 9$ cm. The area of $A B C D$ is $86.4$ cm². Angle $A B C$ is obtuse.
Not drawn accurately
(a)
Show that $sin A B C = 0.8$
[2 marks]
(b)
Work out the size of angle $A B C$ . Give your answer to 1 decimal place.
[2 marks]
Answer $\dots \dots^{\circ}$
(c)
Write down the area of triangle $A B D$ .
[1 mark]
Answer $\dots\dots$ cm²
Total for D4: 5 marks
D5
Do not use a calculator. In triangle $P QR$ , $P Q = (2 x - 1)$ cm, $P R = (x + 3)$ cm and angle $QP R = 3 0^{\circ}$ . The area of the triangle is $18$ cm².
Not drawn accurately
(a)
Show that $2 x^{2} + 5 x - 75 = 0$
[3 marks]
(b)
Hence work out the length of $P Q$ .
[2 marks]
Answer $\dots\dots$ cm
Total for D5: 5 marks
D6
A tiler covers a wall of area $3$ m² with triangular tiles. Each tile is an isosceles triangle with two sides of $20$ cm and an angle of $5 0^{\circ}$ between them.
(a)
Work out the area of one tile. Give your answer to 3 significant figures.
[2 marks]
Answer $\dots\dots$ cm²
(b)
The tiler divides the area of the wall by the area of one tile and rounds up. What number does this give?
[1 mark]
Answer $\dots\dots$
(c)
Give one reason why the tiler is likely to need more tiles than this.
[1 mark]
Total for D6: 4 marks

Extension

Stretch

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

E1
Two sides of a triangle have lengths that add up to $20$ cm. The angle between them is $5 0^{\circ}$ . Work out the greatest possible area of the triangle. Give your answer to 3 significant figures.
[4 marks]
Hint 1 · What to try
Call one side $x$ cm. The other side is $(20 - x)$ cm.
Hint 2 · The first line
Area $= \frac{1}{2} x (20 - x) sin 5 0^{\circ}$ . Only $x (20 - x)$ changes as $x$ changes.
Hint 3 · The full method
$x (20 - x) = 100 - (x - 10)^{2}$ , which is greatest when $x = 10$ . Area $= \frac{1}{2} \times 10 \times 10 \times sin 5 0^{\circ} = 38.3$ cm².
Answer $\dots\dots$ cm²

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the area of any triangle from two sides and the angle between them.
I can find the area of a regular polygon or a segment by splitting it into triangles.
I can work backwards from an area to a side, or to the two possible angles.

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