Question

. Calculate the area of a pentagon PQRST, in which PQ = QR = RS = ST = TP = 6 cm. Also PS = QS = 8 cm.​

Answer 1

Answer:

The area of the pentagon is $3\sqrt{55}+8\sqrt{5}$.

Step-by-step explanation:

When we draw the pentagon and the diagonals mentioned, we get three triangles.

1) $\triangle PQS$ with  sides $PQ=6cm,PS=QS=8cm$.

2)$\triangle QRS$ with sides $QR=RS=6cm, QS=8cm$.

3)$\triangle PST$ with sides $PT=TS=6cm, PS=8cm$

We can find the area of each of these triangles using Heron's formula.

(Heron's formula: Area of the triangle with sides $a,b$ and $c$ is $\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\dfrac{a+b+c}{2}$ )

1) Area of $\triangle PQS$

$Area = \sqrt{(\frac{6+8+8}{2})(11-6)(11-8)(11-8) } =\sqrt{11\times 5 \times 3\times 3}\\=3\sqrt{55}$

2) Area of $\triangle QRS$ and $\triangle PST$ are the same as they have same sides.

$Area =\sqrt{(\frac{6+6+8}{2})(10-6)(10-6)(10-8) } =\sqrt{10\times 4\times4\times 2} \\=8\sqrt{5}$

So area of the pentagon is $3\sqrt{55}+8\sqrt{5}$.

Answer 2

Given: A pentagon PQRST, in which PQ = QR = RS = ST = TP = 6 cm.

To find: The area of the pentagon.

Solution:

The area of a two dimensional figure is the measure of its surface. A pentagon is a two dimensional figure with 5 sides. The area of a pentagon can be calculated using the following formula.

$Area = \frac{1}{4} \sqrt{5(5+2\sqrt{5}) } a^{2}$

Here, a is the length of the side of the pentagon which is given as 6 cm in the question.

$Area = \frac{1}{4} \sqrt{5(5+2\sqrt{5}) } * (6)^{2}$

        $= 61.94 cm^{2}$

Therefore, the area of the pentagon is 61.94 cm².