Mensuration - Exercise 23(C)

Find the perimeter and area of a rectangle having:

(i) Length = 16 cm, Breadth = 12 cm

(ii) Length = 9.6 cm, Breadth = 1.5 cm

(iii) Length = 8 m, Breadth = 9 dm

(iv) Length = 6 hm, Breadth = 8 dam

(v) Length = 1 m 25 cm, Breadth = 80 cm

Answer

(i) Length = 16 cm, Breadth = 12 cm

Perimeter of a rectangle = 2(Length + Breadth)

= 2(16 cm + 12 cm)

= 2(28 cm)

= 56 cm

Area of a rectangle = (Length × Breadth)

= (16 cm × 12 cm)

= 192 cm²

Perimeter = 56 cm, Area = 192 cm².

(ii) Length = 9.6 cm, Breadth = 1.5 cm

Perimeter of a rectangle = 2(Length + Breadth)

= 2(9.6 cm + 1.5 cm)

= 2(11.1 cm)

= 22.2 cm

Area of a rectangle = (Length × Breadth)

= (9.6 cm × 1.5 cm)

= 14.4 cm²

Perimeter = 22.2 cm, Area = 14.4 cm².

(iii) Length = 8 m, Breadth = 9 dm

First, let us convert length into dm:

1 m = 10 dm

∴ 8 m = 80 dm

Perimeter of a rectangle = 2(Length + Breadth)

= 2(80 dm + 9 dm)

= 2(89 dm)

= 178 dm

Area of a rectangle = (Length × Breadth)

= (80 dm × 9 dm)

= 720 dm²

Perimeter = 178 dm, Area = 720 dm².

(iv) Length = 6 hm, Breadth = 8 dam

First, let us convert length and breadth into m:

1 hm = 100 m

∴ 6 hm = 600 m

1 dam = 10 m

∴ 8 dam = 80 m

Perimeter of a rectangle = 2(Length + Breadth)

= 2(600 m + 80 m)

= 2(680 m)

= 1360 m

Area of a rectangle = (Length × Breadth)

= (600 m × 80 m)

= 48000 m²

Perimeter = 1360 m, Area = 48000 m².

(v) Length = 1 m 25 cm, Breadth = 80 cm

First, let us convert length to cm:

1 m = 100 cm

∴ 1 m 25 cm = (100 + 25) cm = 125 cm

Perimeter of a rectangle = 2(Length + Breadth)

= 2(125 cm + 80 cm)

= 2(205 cm)

= 410 cm

Area of a rectangle = (Length × Breadth)

= (125 cm × 80 cm)

= 10000 cm²

Perimeter = 410 cm, Area = 10000 cm²

Find the perimeter and area of a square whose side measures:

(i) 14 cm

(ii) 3.5 cm

(iii) 1 m 20 cm

Answer

(i)

Given:

Side = 14 cm

Perimeter of a square = 4 × side

= 4 × 14 cm

= 56 cm

Area of a square = (side)²

= (14 cm)²

= 196 cm²

Perimeter = 56 cm, Area = 196 cm²

(ii)

Given:

Side = 3.5 cm

Perimeter of a square = 4 × side

= 4 × 3.5 cm

= 14 cm

Area of a square = (side)²

= (3.5 cm)²

= 12.25 cm²

Perimeter = 14 cm, Area = 12.25 cm²

(iii)

Given:

Side = 1 m 20 cm

First, let us convert side into m:

100 cm = 1 m

∴ 1 m 20 cm = (1 + 0.2) m = 1.2 m

Perimeter of a square = 4 × side

= 4 × 1.2 m

= 4.8 m

Area of a square = (side)²

= (1.2 m)²

= 1.44 m²

Perimeter = 4.8 m, Area = 1.44 m²

The perimeter of a rectangular plot of land is 240 m and its length is 63 m. Find the breadth and area of the plot.

Answer

Given:

Perimeter = 240 m

Length = 63 m

We know the formula,

Perimeter of a rectangle = 2(Length + Breadth)

240 m = 2(63 m + Breadth)

240 m = (2 × 63 m + 2 × Breadth)

240 m = 126 m + (2 × Breadth)

240 m - 126 m = 2 × Breadth

114 m = 2 × Breadth

⇒ Breadth = $\dfrac{114}{2}$ m

⇒ Breadth = 57 m

Area of a rectangle = (Length × Breadth)

= (63 m × 57 m)

= 3591 m²

Breadth = 57 m, Area = 3591 m²

The perimeter of a rectangular grassy plot is 189 m and its breadth is 10.5 m. Find the length and area of the plot.

Answer

Given:

Perimeter = 189 m

Breadth = 10.5 m

We know the formula,

Perimeter of a rectangle = 2(Length + Breadth)

189 m = 2(Length + 10.5 m)

189 m = (2 × Length + 2 × 10.5 m)

189 m = (2 × Length) + 21 m

189 m - 21 m = 2 × Length

168 m = 2 × Length

⇒ Length = $\dfrac{168}{2}$ m

⇒ Length = 84 m

Area of a rectangle = (Length × Breadth)

= (84 m × 10.5 m)

= 882 m²

Length = 84 m, Area = 882 m²

A rectangular garden is 175 m long and 96 m broad. Find the cost of fencing it at ₹ 17.50 per metre. Also, find the cost of ploughing it at ₹ 4.50 per square metre.

Answer

This problem requires two different measurements: Perimeter for fencing and Area for ploughing.

Given:

Length = 175 m

Breadth = 96 m

Rate of fencing = ₹ 17.50 per metre

Rate of ploughing = ₹ 4.50 per square metre

We know the formula,

Perimeter of a rectangle = 2(Length + Breadth)

= 2(175 m + 96 m)

= 2(271 m)

= 542 m

Total cost of fencing = Perimeter × Rate of fencing

= 542 m × 17.50

= ₹ 9485

Area of a rectangle = (Length × Breadth)

= (175 m × 96 m)

= 16800 m²

Total cost of ploughing = Area × Rate of ploughing

= 16800 m² × ₹ 4.50

= ₹ 75600

Cost of fencing = ₹ 9485, Cost of ploughing = ₹ 75600.

Square tiles of side 20 cm are to be laid on the floor of a room 10 m by 4.5 m. How many tiles will be needed? Find the cost of putting the tiles at ₹ 131.40 per tile.

Answer

To find the number of tiles, we divide the total area of the floor by the area of a single tile.

Given:

Rate of cost of putting the tiles = ₹ 131.40 per tile.

Floor Length = 10 m = 1000 cm

Floor Breadth = 4.5 m = 450 cm

Side of square tile = 20 cm

Tile is in square shape,

∴ Area of tile = (Side)²

= (20 cm)²

= 400 cm²

Floor is in rectangular shape,

∴ Area of floor = Floor Length × Floor Breadth

= 1000 cm × 450 cm

= 450000 cm²

Number of tiles = $\dfrac{\text{Area of floor}}{\text{Area of tile}}$

= $\dfrac{450000}{400}$

= 1125

Cost of putting the tiles = Number of tiles × Rate

= 1125 × ₹ 131.40

= ₹ 147825

Number of tiles = 1125, Cost of putting the tiles = ₹ 147825.

The area of a rectangular park is 1560 m² and its breadth is 24 m. Find the length and perimeter of the park. Also, find the cost of fencing it at ₹ 22.50 per metre.

Answer

Given:

Area = 1560 m²

Breadth = 24 m

Rate of cost of fencing = ₹ 22.50 per metre

We know the formula,

Area of a rectangle = Length × Breadth

1560 m² = Length × 24 m

⇒ Length = $\dfrac{1560}{24}$ m

⇒ Length = 65 m

Perimeter of a rectangle = 2(Length + Breadth)

= 2(65 m + 24 m)

= 2(89 m)

= 178 m

Cost of fencing = Perimeter × Rate

= 178 m × ₹ 22.50

= ₹ 4005

Length = 65 m, Perimeter = 178 m, Cost of fencing = ₹ 4005.

Find the perimeter of a square whose area is 196 cm².

Answer

Given:

Area = 196 cm²

We know the formula,

Area of a square = (Side)²

196 cm² = (Side)²

⇒ Side = $\sqrt{196}$ cm

⇒ Side = 14 cm

Perimeter of a square = 4 × Side

= 4 × 14 cm

= 56 cm

The perimeter of a square = 56 cm

The area of a square field is 1 hectare. What is its perimeter?

Answer

Given:

Area = 1 hectare

Converting area into m²:

1 hectare (ha) = 10000 m²

We know that,

Area of a square = (Side)²

10000 m² = (Side)²

⇒ Side = $\sqrt{10000}$ m

⇒ Side = 100 m

Perimeter of a square = 4 × Side

= 4 × 100 m

= 400 m

The perimeter of a square field = 400 m.

It costs ₹ 5400 to fence a square field at ₹ 13.50 per metre. Find

(i) the length of the side of the field

(ii) the area of the field.

Answer

Given:

Total cost = ₹ 5400

Rate of fencing = ₹ 13.50 per metre

(i) the length of the side of the field

Perimeter of the field = $\dfrac{\text{Total cost of fencing}}{\text{Rate per metre}}$

= $\dfrac{5400}{13.50}$ m

= 400 m

We know the formula,

Perimeter = 4 × Side

⇒ Side = $\dfrac{\text{Perimeter}}{4}$

= $\dfrac{400}{4}$ m

= 100 m

The length of the side of the field = 100 m.

(ii) the area of the field

Area of a square = (Side)²

= (100 m)²

= 10000 m²

The area of the field = 10000 m².

A rectangular plot of land is 50 m long. The cost of levelling the plot at ₹ 12.50 per m² is ₹ 20000. Find

(i) the area of the plot;

(ii) the breadth of the plot;

(iii) the perimeter of the plot.

Answer

Given:

Length = 50 m

Rate of cost of levelling = ₹ 12.50 per m²

Total cost = ₹ 20000

(i) the area of the plot

Area = $\dfrac{\text{Total cost}}{\text{Rate}}$

= $\dfrac{20000}{12.50}$ m²

= 1600 m²

The area of the plot = 1600 m².

(ii) the breadth of the plot

We know the formula,

Area of a rectangle = Length × Breadth

1600 m² = 50 m × Breadth

⇒ Breadth = $\dfrac{1600}{50}$ m

⇒ Breadth = 32 m

The breadth of the plot = 32 m.

(iii) the perimeter of the plot

Perimeter of a rectangle = 2(Length + Breadth)

= 2(50 m + 32 m)

= 2(82 m)

= 164 m

The perimeter of the plot = 164 m.

Find the area and perimeter of the shaded part in each of the following figures.

(i)

(ii)

(iii)

(iv)

Answer

(i)

Let us partition the given figure into suitable rectangles by drawing the dotted lines, as shown below:

Thus, we obtain the rectangles ABKL, CDIJ and EFGH.

∴ Area of shaded figure = Area rectangle ABKL + Area rectangle CDIJ + Area rectangle EFGH

= (AB × AL) + (CD × DI) + (EF × FG)

= (9 m × 2 m) + (2.5 m × (1.5 m + 1 m + 4.5 m)) + (18 m × 1 m)

= 18 m² + (2.5 m × 7 m) + 18 m²

= 18 m² + 17.5 m² + 18 m²

= 53.5 m²

Perimeter of shaded figure = AB + BC + CD + DE + EF + FG + GH + HI + IJ + JK + KL + LA

= 9 m + 2.5 m + 2.5 m + 1.5 m + 18 m + 1 m + 18 m + 4.5 m + 2.5 m + 2.5 m + 9 m + 2 m

= 73 m

Area = 53.5 m², Perimeter = 73 m

(ii)

Let us partition the given figure into suitable rectangles by drawing the dotted lines, as shown below:

Thus, we obtain the rectangles ABKJ, BCDE and FGHI.

∴ Area of shaded figure = Area rectangle ABKJ + Area rectangle BCDE + Area rectangle FGHI

= (BJ × KJ) + (CD × BC) + (GH × IH)

= ((2 + 4 + 2 + 6) m × 2 m) + (2 m × 5 m) + (2 m × 4 m)

= (14 m × 2 m) + 10 m² + 8 m²

= 28 m² + 18 m²

= 46 m²

Perimeter of shaded figure = AK + KJ + JI + IH + HG + GF + FE + ED + DC + CA

= 14 m + 2 m + 6 m + 4 m + 2 m + 4 m + 4 m + 5 m + 2 m + 7 m

= 50 m

Area = 46 m², Perimeter = 50 m

(iii)

Let us partition the given figure into suitable rectangles by drawing the dotted lines, as shown below:

Thus, we obtain the rectangles ABQR, EFNO, IJKL, CDPQ and GHLM.

∴ Area of shaded figure = Area rectangle ABQR + Area rectangle EFNO + Area rectangle IJKL + Area rectangle CDPQ + Area rectangle GHLM

= (AR × AB) + (EO × EF) + (JK × IJ) + (CD × CQ) + (GH × GM)

= (9 m × 2 m) + ((9 + 2) m × 2 m) + (9 m × 2 m) + (2 m × 3 m) + (2 m × 3 m)

= 18 m² + (11 m × 2 m) + 18 m² + 6 m² + 6 m²

= 18 m² + 22 m² + 18 m² + 6 m² + 6 m²

= 70 m²

Perimeter of shaded figure = AR + RP + PO + ON + NM + MK + KJ + JI + IH + HG + GF + FE + ED + DC + CB + BA

= 9 m + 4 m + 2 m + 2 m + 2 m + 4 m + 9 m + 2 m + 6 m + 2 m + 6 m + 2 m + 6 m + 2 m + 6 m + 2 m

= 66 m

Area = 70 m², Perimeter = 66 m

(iv)

Let us partition the given figure into suitable rectangles by drawing the dotted lines, as shown below:

Thus, we obtain the rectangles ABCL, IFGH and KDEJ.

∴ Area of shaded figure = Area rectangle ABCL + Area rectangle IFGH + Area rectangle KDEJ

= (AB × BC) + (IF × FG) + (KJ × KD)

= (15 m × 3 m) + ((8 + 8 + 5) m × 2 m) + (8 m × 5 m)

= 45 m² + (21 m × 2 m) + 40 m²

= 85 m² + 42 m²

= 127 m²

Perimeter of shaded figure = AB + BC + CD + DE + EF + FG + GH + HI + IJ + JK + KL + LA

= 15 m + 3 m + 5 m + 8 m + 8 m + 2 m + 21 m + 2 m + 8 m + 8 m + 5 m + 3 m

= 88 m

Area = 127 m², Perimeter = 88 m