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This is the second of our many quizzes on Area using geometry. Click on this to access more such competitive quizzes on Geometry.

Area using geometry Quiz 2

mathematics

 A triangle's, circle's, and square's perimeters are equal to $135cm$. Which of these will have the largest surface area?

 Calculate the area of different figures by using the perimeter and then find the area by using the respective formula of the areas. 

  $\text{For equilateral triangle }\text{a =}\dfrac{135}{3} \\ \text{Area of }\Delta \text{=}\dfrac{\sqrt{3}}{4}{{a}^{2}}=\dfrac{\sqrt{3}}{4}{{\left( \dfrac{135}{3} \right)}^{2}} \\ \text{For circle, }\text{r = }\dfrac{135}{2\pi } \\ \text{Area of circle }\text{= }\pi {{\text{r}}^{2}}=\dfrac{{{(135)}^{2}}}{4\pi } \\ \text{For square, }\text{s =}\dfrac{135}{4} \\ \text{Area of square, }\text{= }{{\text{s}}^{2}}={{\left[ \dfrac{135}{4} \right]}^{2}} \\ \text{Clearly from above }\dfrac{{{(135)}^{2}}}{4\pi }\text{ is the largest,} \\ \text{To the circle has the largest area.}$

 Find the area of the shaded region. <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/b9835e91-6cd1-4212-9900-a3c70a638ac07376775115244956954.png"/>

 The area of the equilateral triangle is given by $\dfrac{\sqrt{3}}{4}{{a}^{2}}$ hence put the value of $a$ i.e., side in the formula and then find the area.

  $\text{Area of the equilateral triangle is given by }\dfrac{\sqrt{3}}{4}{{a}^{2}} \\ \text{Here }a=4.5cm \\  \Rightarrow \dfrac{\sqrt{3}}{4}(4.5\times 4.5) \\  \Rightarrow 8.77c{{m}^{2}}$

 What is half the area of a rectangle whose length is the side of a rhombus with height $\text{5 }cm$and area $170c{{m}^{2}}$ while the breadth of the rectangle is given as the height of a parallelogram with area $312c{{m}^{2}}$ and base$104cm$.

 The area of a rhombus is $side\times height$ and the area of parallelogram is $base\times height$ . Use the formulas and find the unknown values and substitute in the formula of area of the rectangle.  The area of the rectangle is$l\times b$. Hence use the width and the length and substitute in the given formula.

  $\text{Area of a rhombus is given by,} \\  \text{Area = Side } \times \text{ Height} \\  \Rightarrow 170c{{m}^{2}} = \text{ Side }\times \text{ 5} \\  \Rightarrow \text{Side }= \dfrac{170}{\text{5}}=34cm \\ \therefore \text{ The length of the rectangle will be }34cm. \\ \text{Area of a parallelogram is given by,} \\ \text{Area = Base }\times\text{ Height }\\ \Rightarrow 312cm=104cm\times \text{ Height }\\ \Rightarrow \text{Height } = \dfrac{312}{104}=3cm \\ \therefore \text{ The breadth of the rectangle will be }3cm \\ \text{The area of the rectangle is given by,} \\  \text{Area = Length }\times \text{ Breadth }\\  \Rightarrow \text{Area }= 34cm\times 3cm \\  \Rightarrow \text{Area }= 102c{{m}^{2}} \\ \therefore \text{Half of the area of this rectangle will be }\dfrac{1}{2} \times 102c{{m}^{2}}=51c{{m}^{2}}$.

 What is ${{\dfrac{3}{4}}^{th}}$of the area of the region that is not shaded? <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/c8d5ac08-5ad0-4a6c-bef7-ae0e637b52dd4266675555892634522.png"/>

 Since the area of the circle is given by $\pi {{r}^{2}}$ therefore use the radius values of both the circles and substitute in the formula to obtain the value of area and find the difference between them to get the area of region that is not shaded.

  $\text{The area of the circle is given by }\pi {{\text{r}}^{2}}. \\ \text{Here, area of the circle with radius 5cm is,} \\ \pi \times \text{5}\times \text{5=25}\pi  \\ \text{Area of the circle with radius 3cm is,} \\ \pi \times 3\times 3\text{=9}\pi \\ \text{Therefore, the area of the unshaded region is,} \\ \text{25}\pi -\text{9}\pi =\text{16}\pi \\ \text{Now, }{{\dfrac{3}{4}}^{th}}\text{ of the area will be,} \\ \dfrac{3}{4}\times \text{16}\pi =12\pi =12\times 3.14=\text{37}\text{.68 }c{{m}^{2}}$

 The parallel sides of a trapezium are given as $5.5\text{ cm and }10.7\text{ cm}$ respectively and the height is  $6.3cm$ . The diagonals of a rhombus are $\text{12}\text{.3 cm and 4}\text{.5 cm}$ . Which one of these two has a larger area and by how much?

 Find the area of rhombus and then the area of the trapezium using the formula and then compare the values.

  $\text{Area of the trapezium } = \dfrac{1}{2}\times \text{ distance between the parallel sides } \times  \text{ sum of the parallel sides} \\ \text{Therefore, the area of the trapezium is,} \\ \text{=}\dfrac{1}{2}\times (5.5+10.7)\times 6.3=51.03c{{m}^{2}} \\ \text{Area of the rhombus }=\dfrac{1}{2}\times \text{ diagonal 1 }\times \text{ diagonal 2} \\ \text{Therefore, the area of the rhombus is,} \\  \Rightarrow \dfrac{1}{2}\times 12.3\times 4.5 \\ \Rightarrow 27.075c{{m}^{2}} \\ \text{Therefore, the area of the trapezium is largest.} \\ \text{Difference = area of trapezium – area of rhombus = }51.03 – 27.075 = 23.956c{{m}^{2}}$.