Solution:
$$\eqalign{ \frac{{2\left( {l + b} \right)}}{b} = \frac{5}{1} \cr \Rightarrow 2l + 2b = 5b \cr \Rightarrow 3b = 2l \cr b = \frac{2}{3}l \cr {\text{Then,}} \cr {\text{Area = }}216\,c{m^2} \cr \Rightarrow l \times b = 216 \cr \Rightarrow l \times \frac{2}{3}l = 216 \cr \Rightarrow {l^2} = 324 \cr l = 18\,cm \cr} $$

Solution:
Given base of triangle and its height is 15 cm and 12 cm respectively
Area of first triangle :
$$\eqalign{ = \frac{1}{2} \times {\text{Base}} \times {\text{Height}} \cr = \frac{1}{2} \times 15 \times 12 \cr = 90{\text{ sq}}{\text{. cm}} \cr} $$
According to the question,
Let height of triangle be h cm
Area of new triangle = 180 sq. cm
Base = 20 sq. cm
$$\eqalign{ \Rightarrow 180 = \frac{1}{2} \times 20 \times {\text{Height}} \cr \Rightarrow {\text{h}} = \frac{{2 \times 180}}{{20}} \cr \Rightarrow {\text{h}} = 18{\text{ cm}} \cr} $$

Solution:
Perimeter = Distance covered in 8 min.
Perimeter = $$\left( {\frac{{12000}}{{60}} \times 8} \right){\text{m}}$$
Perimeter = 1600 m
Let length = 3x metres and breadth = 2x metres.
Then, 2(3x + 2x) = 1600 or x = 160
Therefore Length = 480 m and Breadth = 320 m
Therefore Area = (480 x 320) m² = 153600 m²

Solution:
Let the original radius be R
New radius = (100 + 200)% of R = 300% of R = 3R
Original area = $$\pi $$R²
New area = $$\pi $$ × (3R)² = 9$$\pi $$R²
Increase in area :
$$\eqalign{ = \left( {9\pi {{\text{R}}^2} - \pi {{\text{R}}^2}} \right) \cr = 8\pi {{\text{R}}^2} \cr} $$
∴ Increase %:
$$\eqalign{ = \left( {\frac{{8\pi {{\text{R}}^2}}}{{\pi {{\text{R}}^2}}} \times 100} \right)\% \cr = 800\% \cr} $$

Solution:
We have :
(l - b) = 23 and 2(l + b) = 206 or (l + b) = 103
Solving the two equations, we get :
l = 63 and b = 40
∴ Area = (l × b) = (63 × 40)m² = 2520 m²

Solution:
$$\eqalign{ {\text{Breadth}} = \frac{{{\text{Area}}}}{{{\text{Length}}}} \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{{2100}}{{60}}} \right)m \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 35m \cr \therefore {\text{Perimeter}} = 2\left( {60 + 35} \right)m \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 190\,m \cr} $$

Solution:
Since 3² + 4² = 5²,
So it is a right-angled triangle with Base = 3 cm and Height = 4 cm
∴ Area :
$$\eqalign{ = \left( {\frac{1}{2} \times 3 \times 4} \right)c{m^2} \cr = 6\,c{m^2} \cr} $$