Areas of Shapes

The formula for the area is \frac{1}{2} \times b \times h, where b = 11.5 \text{ and } h = 12. So, we get:

 

\text{Area } = \dfrac{1}{2} \times 11.5 \times 12 = 69\text{cm}^2

To work out the area, we will need to find the perpendicular height by forming a right-angled triangle,

 

The hypotenuse of the right-angled triangle is 5cm and the base is 3cm (8cm-5cm=3cm), so we get the perpendicular height of the trapezium as,

 

\text{Perpendicular height} = \sqrt{5^2 - 3^2} = \sqrt{16} = 4\text{cm}

 

Now we know the perpendicular height, we can calculate the area.

 

\text{Area} = \dfrac{1}{2}(a + b)h = \dfrac{1}{2}(5 + 8) \times 4 = 26\text{cm}^2

Area of a parallelogram is given by the formula,

 

\text{Area}=\text{base}\times\text{height}.

 

Therefore,

 

\text{Area}=8 \times 15 =120\text{cm}^2 

As we need to find a missing side-length rather than the area, we’re going to have to set up an equation and rearrange it to find x. The formula for the area we’ll need here is

 

\dfrac{1}{2}ab \sin(C),

 

so, our equation is

 

\dfrac{1}{2} \times 2.15 \times x \times \sin(26) = 1.47

 

Simplifying the left-hand-side, we get

 

1.075 \sin (26) \times x = 1.47

 

Finally, dividing through by 1.075\sin(26), and putting it into a calculator, we get

 

x = \dfrac{1.47}{1.075\sin(26)} = 3.12\text{m (2 d.p.)}