# Area Questions

FACTS  AND  FORMULAE  FOR  AREA  QUESTIONS

FUNDAMENTAL CONCEPTS :

I. Results on Triangles:

1. Sum of the angles of  a triangle is ${180}^{o}$

2. The sum of any two sides of a triangle is greater than the third side.

3. Pythagoras Theorem : In a right - angled triangle,

${\left(Hypotenuse\right)}^{2}={\left(Base\right)}^{2}+{\left(Height\right)}^{2}$

4. The line joining the mid-point of a side of a triangle to the opposite vertex is called the median.

5. The point where the three medians of a triangle meet, is called Centroid. The centroid divides each of the medians in the ratio 2 : 1.

6. In an Isosceles triangle, the altitude from the vertex bisects the base.

7. The median of a triangle divides it into two triangles of the same area.

8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle.

1. The diagonals of a parallelogram bisect each other

2. Each diagonal of a parallelogram divides it into two triangles of the same area.

3. The diagonals of a rectangle are equal and bisect each other.

4. The diagonals of a square are equal and bisect each other at right angles

5. The diagonals of a rhombus are unequal and bisect each other at right angles

6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

IMPORTANT FORMULAE

I.

1. Area of a rectangle = (length x Breadth)

2. Perimeter of a rectangle = 2( length + Breadth)

II. Area of square = ${\left(side\right)}^{2}=\frac{1}{2}{\left(diagonal\right)}^{2}$

III. Area of 4 walls of a room = 2(Length + Breadth) x Height

IV.

1. Area of a triangle =$\frac{1}{2}×base×height$

2. Area of a triangle = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$, where a, b, c are the sides of the triangle and $s=\frac{1}{2}\left(a+b+c\right)$

3. Area of an equilateral triangle =$\frac{\sqrt{3}}{4}×{\left(side\right)}^{2}$

4. Radius of incircle of an equilateral triangle of side $a=\frac{a}{2\sqrt{3}}$

5. Radius of circumcircle of an equilateral triangle of side $a=\frac{a}{\sqrt{3}}$

6. Radius of incircle of a triangle of area $∆$ and semi-perimeter $s=\frac{∆}{s}$

V.

1. Area of a parallelogram = (Base x Height)

2. Area of a rhombus =

3. Area of a trapezium =

VI.

1. Area of a cicle = ${\mathrm{\pi R}}^{2}$, where R is the radius.

2. Circumference of a circle = $2\mathrm{\pi R}$.

3. Length of an arc = $\frac{2\mathrm{\pi R\theta }}{360}$, where $\theta$ is the central angle.

4. Area of a sector = $\frac{1}{2}\left(arc×R\right)=\frac{{\mathrm{\pi R}}^{2}\mathrm{\theta }}{360}$

VII.

1. Area of a semi-circle = $\frac{{\mathrm{\pi R}}^{2}}{2}$

2. Circumference of a semi - circle = $\mathrm{\pi R}$

Q:

A wire can be bent in the form of a circle of radius 56cm. If it is bent in the form of a square, then its area will be

 A) 7744 B) 8844 C) 5544 D) 4444

Explanation:

length of wire = $2πr$= 2 x (22/7 ) x 56 = 352 cm
side of the square = 352/4 = 88cm
area of the square = 88 x 88 = 7744sq cm

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77 41774
Q:

The length of a rectangular hall is 5m more than its breadth. The area of the hall is 750 sq.m. The length of the hall is

 A) 20 B) 25 C) 30 D) 35

Explanation:

Then, length = (x+5)m

Area of a rectangle = Length x Breadth

x(x+5) = 750

x² + 5x - 750= 0

(x+30)(x-25)= 0

x = 25 or x = -30

=> Length = x + 5 = 25 + 5 = 30m.

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71 41758
Q:

Find the ratio of the areas of the incircle and circumcircle of a square.

 A) 1:1 B) 1:2 C) 1:3 D) 1:4

Explanation:

Let the side of the square be x. Then, its diagonal = $2x2=2x$

Radius of incircle = $x2$

Radius of circum circle= $2×x2=x2$

Required ratio = $πx24:πx22=14:12=1:2$

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83 41665
Q:

A room of 5m 44cm long and 3m 74cm broad is to be paved with squre tiles. Find the least number of squre tiles required to cover the floor.

 A) 136 B) 146 C) 166 D) 176

Explanation:

area of the room = 544 x 374 sq.cm

size of largest square tile = H.C.F of 544cm and 374 cm= 34 cm

area of 1 tile = 34x34 sq cm

no. of tiles required = (544 x 374) / (34 x 34) = 176

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61 41415
Q:

The length of a rectangular plot is 20 metres more than its breadth. If the cost of fencing the plot @ Rs. 26.50 per metre is Rs. 5300, what is the length of the plot in metres?

 A) 20 B) 200 C) 300 D) 400

Explanation:

Let length of plot = L meters, then breadth = L - 20 meters

and perimeter = 2[L + L - 20] = [4L - 40] meters

[4L - 40]  * 26.50 = 5300

[4L - 40] = 5300 / 26.50 = 200

4L = 240

L = 240/4= 60 meters.

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29 39258
Q:

One side of a rectangular field is 15m and one of its diagonal is 17m. Find the area of field?

 A) 110 B) 120 C) 130 D) 140

Explanation:

Other side = [(17 x 17) - (15 x 15)] = (289 - 225) = 8m
Area = 15 x 8 =120 sq. m

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73 37647
Q:

If the circumference of a circle is 22 cm, then what must be its area (in cm²)?

 A) 77 B) 38.5 C) 31.5 D) 63

Explanation:

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1 35833
Q:

The altitude drawn to the base of an isosceles triangle is 8cm and the perimeter is 32cm. Find the area of the triangle?

 A) 50 B) 60 C) 70 D) 80

Explanation:

let ABC be the isosceles triangle, the AD be the altitude

Let AB = AC = x then BC= 32-2x       [because parameter = 2 (side) + Base]

since in an isoceles triange the altitude bisects the base so

BD = DC = 16-x

In a triangle ADC, $AC2=AD2+DC2$

$x2=82+16-x2$ $⇒x=10$

BC = 32-2x = 32-20 = 12 cm

Hence, required area = $12*BC*AD$$12*12*10$ = 60 sq cm