RS Aggarwal Class 8 Mathematics Solutions Chapter 18 Area of a Trapezium and a Polygon

Exercise 18A

Question 1. A trapezium has parallel sides of 24 cm and 20 cm, with a perpendicular distance of 15 cm between them. Find its area.
Answer: Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

Area = (1/2) × (24 + 20) × 15 cm²

= (1/2) × 44 × 15 cm²

= 22 × 15 cm²

= 330 cm²

Therefore, the area of the trapezium is 330 cm².
In simple words: Add the two parallel sides, multiply by the height, then divide by 2 to get the area.

Exam Tip: Always remember to add both parallel sides first before multiplying by the height and dividing by 2 - forgetting the division by 2 is a common error.

Question 2. A trapezium has parallel sides measuring 38.7 cm and 22.3 cm, with a perpendicular distance of 16 cm. Calculate its area.
Answer: Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

Area = (1/2) × (38.7 + 22.3) × 16 cm²

= (1/2) × 61 × 16 cm²

= 61 × 8 cm²

= 488 cm²

Therefore, the area of the trapezium is 488 cm².
In simple words: Add the lengths of the two parallel sides, multiply by the perpendicular distance, then divide the final result by 2.

Exam Tip: When the parallel sides are given in decimals, add them first to get a cleaner number before proceeding with multiplication and division.

Question 3. The top surface of a table shaped like a trapezium has parallel sides of 1 m and 1.4 m. If the perpendicular distance between these sides is 0.9 m, find the area of the table's top surface.
Answer: Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

Area = (1/2) × (1 + 1.4) × 0.9 m²

= (1/2) × 2.4 × 0.9 m²

= 1.2 × 0.9 m²

= 1.08 m²

Therefore, the area of the top surface of the table is 1.08 m².
In simple words: Add the two parallel sides to get 2.4 m. Multiply this by the height of 0.9 m, then divide by 2.

Exam Tip: Decimal calculations must be handled carefully - multiply accurately and divide by 2 to avoid rounding errors.

Question 4. A trapezium with parallel sides of 55 cm and 35 cm has an area of 1080 cm². Find the perpendicular distance between the parallel sides.
Answer: Let the perpendicular distance between the parallel sides be x.

Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

1080 = (1/2) × (55 + 35) × x cm²

1080 = (1/2) × 90 × x cm²

1080 = 45x cm²

x = 1080 ÷ 45

x = 24 cm

Therefore, the perpendicular distance between the parallel sides is 24 cm.
In simple words: If you know the area and the parallel sides, you can work backwards to find the height by dividing the area by half the sum of the parallel sides.

Exam Tip: When finding the height, divide the given area by the value obtained from (1/2) × (sum of parallel sides) to isolate the unknown distance.

Question 5. A trapezium has one parallel side measuring 84 m and an area of 1586 m². If the other parallel side is unknown, find its length.
Answer: Let the length of the other parallel side be x cm.

Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

1586 = (1/2) × (84 + x) × 26 m²

1586 = (1092 + 13x) m²

1092 + 13x = 1586

13x = 1586 - 1092

13x = 494

x = 494 ÷ 13

x = 38 m

Therefore, the length of the other parallel side is 38 m.
In simple words: Set up the area formula with one unknown parallel side, substitute all known values, and solve for the missing side length.

Exam Tip: Always expand the equation completely before collecting like terms and isolating the variable to avoid algebraic mistakes.

Question 6. The parallel sides of a trapezium are in the ratio 4:5, the perpendicular distance between them is 18 cm, and the area is 405 cm². Find the lengths of both parallel sides.
Answer: Let the lengths of the parallel sides be 4x cm and 5x cm.

Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

405 = (1/2) × (4x + 5x) × 18 cm²

405 = (1/2) × 9x × 18 cm²

405 = 81x cm²

x = 405 ÷ 81

x = 5 cm

Length of one side = 4 × 5 = 20 cm

Length of the other side = 5 × 5 = 25 cm

Therefore, the parallel sides measure 20 cm and 25 cm respectively.
In simple words: When sides are in a given ratio, use variables (like 4x and 5x) to represent them, substitute into the area formula, and solve for x first.

Exam Tip: For ratio-based problems, always express both quantities in terms of the same variable and set up the equation using all known values.

Question 7. A trapezium has parallel sides of x cm and (x + 6) cm with a perpendicular height of 9 cm. If the area is 180 cm², find the lengths of the parallel sides.
Answer: Let the lengths of the parallel sides be x cm and (x + 6) cm.

Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

180 = (1/2) × (x + x + 6) × 9 cm²

180 = (1/2) × (2x + 6) × 9 cm²

180 = 4.5(2x + 6) cm²

180 = (9x + 27) cm²

9x + 27 = 180

9x = 180 - 27

9x = 153

x = 153 ÷ 9

x = 17 cm

Therefore, the lengths of the parallel sides are 17 cm and 23 cm (that is, 17 + 6 cm).
In simple words: Express the unknown sides in terms of a single variable, substitute into the area formula, and solve the resulting equation.

Exam Tip: Expand and simplify the right side completely before isolating the variable to reduce the chance of arithmetic errors.

Question 8. A trapezoidal field has parallel sides of x cm and 2x cm with a perpendicular distance of 84 m between them. If the area is 9450 m², determine the lengths of both parallel sides and the longer side.
Answer: Let the lengths of the parallel sides be x m and 2x m.

Using the trapezium area formula: Area = (1/2) × (Sum of parallel sides) × (Distance between them)

9450 = (1/2) × (x + 2x) × 84 m²

9450 = (1/2) × 3x × 84 m²

9450 = 42 × 3x m²

9450 = 126x m²

x = 9450 ÷ 126

x = 75 m

Therefore, the lengths of the parallel sides are 75 m and 150 m (that is, 2 × 75 m), and the length of the longer side is 150 m.
In simple words: When one side is a multiple of the other, set them up as x and kx, substitute into the formula, and solve for x to find both sides.

Exam Tip: State which side is the longer one in your final answer to ensure you have fully addressed what the question asks.

Question 9. A trapezoidal field ABCD has sides AD, DC, and CB measuring 54 m, 19 m, and 42 m respectively. If the total perimeter is 130 m, calculate the area of the field where the shape is formed by connecting points in a trapezium arrangement.
Answer: First, find the length of side AB:

AB = 130 - (54 + 19 + 42) m

AB = 130 - 115 m

AB = 15 m

For the trapezoidal field with parallel sides AD and BC, and perpendicular distance AB:

Area of the trapezium = (1/2) × (AD + BC) × AB

= (1/2) × (54 + 42) × 15 m²

= (1/2) × 96 × 15 m²

= 48 × 15 m²

= 720 m²

Therefore, the area of the field is 720 m².
In simple words: Use the perimeter to find the missing side length, then apply the trapezium area formula with the parallel sides and the perpendicular distance.

Exam Tip: Always verify that the shape described matches a trapezium before applying the trapezium area formula - check which sides are actually parallel.

Question 10. In a trapezium ABCD where angle ABC is 90°, the sides AC and BC measure 41 cm and 40 cm respectively. The parallel sides are AD and BC with AD = 16 cm. If the perpendicular distance is 9 cm, calculate the area.
Answer: First, find the length of AB using the right triangle ABC:

From right triangle ABC: AB² = AC² - BC²

AB² = (41)² - (40)²

AB² = 1681 - 1600

AB² = 81

AB = √81

AB = 9 cm

Length AB = 9 cm

Now, for the trapezium with parallel sides AD and BC, and perpendicular distance AB:

Area of the trapezium = (1/2) × (AD + BC) × AB

= (1/2) × (16 + 40) × 9 cm²

= (1/2) × 56 × 9 cm²

= 28 × 9 cm²

= 252 cm²

Therefore, the area of the trapezium is 252 cm².
In simple words: Use the Pythagorean theorem in the right triangle to find the missing side, then use this length as the perpendicular distance in the trapezium area formula.

Exam Tip: When a trapezium is formed from a right-angled triangle configuration, always find the perpendicular side length first using the Pythagorean theorem before calculating area.

Question 11. In trapezium ABCD where AB || DC, AB = 20 cm, DC = 10 cm, and AD = 13 cm. Points L and M are where perpendiculars from C and D meet AB respectively. Triangle CMB is isosceles with CM = DA = 13 cm. Draw the figure and find the area of the trapezium.
Answer: Given the trapezium ABCD with AB || DC, AB = 20 cm, DC = 10 cm, and AD = 13 cm.

Draw perpendiculars CL and CM to AB, meeting at L and M respectively. Since AMCD forms a parallelogram, AM = DC = 10 cm.

Therefore: MB = AB - AM = 20 - 10 = 10 cm

Since triangle CMB is isosceles with CM = DA = 13 cm, and MB = 10 cm, L is the midpoint of B.

From right triangle CLM: CL² = CM² - ML²

CL² = (13)² - (5)²

CL² = 169 - 25

CL² = 144

CL = √144

CL = 12 cm

Length of CL = 12 cm

Area of the trapezium = (1/2) × (AB + DC) × CL

= (1/2) × (20 + 10) × 12 cm²

= (1/2) × 30 × 12 cm²

= 15 × 12 cm²

= 180 cm²

Therefore, the area of the trapezium is 180 cm².
In simple words: Construct perpendiculars from the upper vertices to the lower base, use the isosceles triangle property to find the perpendicular height, then calculate the area.

Exam Tip: Always use the perpendicular distance as the height when applying the trapezium area formula - this is the key to getting the correct answer.

Question 12. Trapezium ABCD has AB || DC, AB = 25 cm, CD = 11 cm, AD = 13 cm, and BC = 15 cm. Draw perpendiculars CL and CM to meet AB at L and M. Triangle CMB has sides CM = 13 cm, MB, and BC = 15 cm. Find the area of the trapezium.
Answer: Given trapezium ABCD with AB || DC, AB = 25 cm, CD = 11 cm, AD = 13 cm, and BC = 15 cm.

Draw perpendiculars CL and CM to AB, meeting at L and M respectively. Since AMCD forms a parallelogram, MC = AD = 13 cm and AM = DC = 11 cm.

Therefore: MB = AB - AM = 25 - 11 = 14 cm

In triangle CMB, we have: CM = 13 cm, MB = 14 cm, BC = 15 cm.

Using Heron's formula, let s = (13 + 14 + 15) ÷ 2 = 42 ÷ 2 = 21 cm

(s - a) = 21 - 13 = 8 cm

(s - b) = 21 - 14 = 7 cm

(s - c) = 21 - 15 = 6 cm

Area of triangle CMB = √[s(s-a)(s-b)(s-c)]

= √[21 × 8 × 7 × 6] cm²

= √7056 cm²

= 84 cm²

Since (1/2) × MB × CL = 84 cm²

(1/2) × 14 × CL = 84 cm²

CL = 84 ÷ 7

CL = 12 cm

Area of the trapezium = (1/2) × (AB + DC) × CL

= (1/2) × (25 + 11) × 12 cm²

= (1/2) × 36 × 12 cm²

= 18 × 12 cm²

= 216 cm²

Therefore, the area of the trapezium is 216 cm².
In simple words: Use Heron's formula to calculate the area of the triangle formed by three known sides, then work backwards to find the perpendicular height of the trapezium.

Exam Tip: When all four sides of a trapezium are given but only one base and height are known, Heron's formula helps find the height efficiently through the triangular portion.

Exercise 18B

Question 1. A quadrilateral ABCD is divided into triangles ADC and ACB by diagonal AC. Triangle ADC has base AC = 24 cm and height DM = 7 cm. Triangle ACB has base AC = 24 cm and height BL = 8 cm. Calculate the total area of the quadrilateral.
Answer: Area of quadrilateral ABCD = (Area of triangle ADC) + (Area of triangle ACB)

= (1/2) × AC × DM + (1/2) × AC × BL

= [(1/2) × 24 × 7 + (1/2) × 24 × 8] cm²

= (84 + 96) cm²

= 180 cm²

Therefore, the area of the quadrilateral is 180 cm².
In simple words: Find the area of each triangle formed by drawing a diagonal, then add them together to get the total quadrilateral area.

Exam Tip: Splitting a quadrilateral along a diagonal simplifies the calculation - find each triangle's area separately, then sum them.

Question 2. Quadrilateral ABCD is divided by diagonal BD into triangles ABD and BCD. Triangle ABD has base BD = 36 m and height AL = 19 m. Triangle BCD has base BD = 36 m and height CM = 11 m. Find the area of the field.
Answer: Area of quadrilateral ABCD = (Area of triangle ABD) + (Area of triangle BCD)

= (1/2) × BD × AL + (1/2) × BD × CM

= [(1/2) × 36 × 19 + (1/2) × 36 × 11] m²

= (342 + 198) m²

= 540 m²

Therefore, the area of the field is 540 m².
In simple words: Draw a diagonal to split the quadrilateral into two triangles, calculate each triangle's area using base and height, then combine them.

Exam Tip: Always choose a diagonal that creates triangles with clearly defined heights to simplify calculations.

Question 3. Pentagon ABCDE is divided into triangles and a trapezium. Triangle AEN has area calculated from AN = 6 cm and EN = 9 cm. Trapezium EDMN has parallel sides (9 + 12) = 21 cm and (14 - 6) = 8 cm with height 14 cm. Triangle DMC and triangle ACB complete the pentagon. Calculate the total area.
Answer: Area of pentagon ABCDE = (Area of triangle AEN) + (Area of trapezium EDMN) + (Area of triangle DMC) + (Area of triangle ACB)

= (1/2) × AN × EN + (1/2) × (EN + DM) × NM + (1/2) × MC × DM + (1/2) × AC × BL

= [(1/2) × 6 × 9 + (1/2) × (9 + 12) × (14 - 6) + (1/2) × (18 - 14) × 12 + (1/2) × 18 × 4]

= (27 + 84 + 24 + 36) cm²

= 171 cm²

Therefore, the area of the pentagon is 171 cm².
In simple words: Break the pentagon down into simpler shapes - triangles and trapeziums - calculate each area separately, and add them to get the total.

Exam Tip: For irregular polygons, always look for natural divisions into standard shapes (triangles, trapeziums, rectangles) to simplify area calculations.

Question 4. Hexagon ABCDEF is divided into triangles and trapeziums using parallel lines. Calculate the total area by finding the area of each component and summing them.
Answer: Area of hexagon ABCDEF = (Area of triangle AFP) + (Area of trapezium FENP) + (Area of trapezium ND×EN) + (Area of trapezium MD×CM) + (Area of trapezium CM×BL) + (Area of triangle MD×CM)

= [(1/2) × 6 × 8 + (1/2) × (8 + 12) × (2 + 8) + (1/2) × (2 + 3) × 12 + (1/2) × 3 × 6 + (1/2) × (6 + 8) × (8 + 2) + (1/2) × (6 + 2) × 8]

= (24 + 100 + 30 + 9 + 70 + 32) cm²

= 265 cm²

Therefore, the area of the hexagon is 265 cm².
In simple words: Divide the hexagon into triangles and trapeziums using perpendicular lines, calculate each shape's area, and add them together.

Exam Tip: For complex polygons, drawing perpendiculars from all vertices to a common baseline creates a systematic way to divide the shape into manageable pieces.

Question 5. Pentagon ABCDE has three sides with known lengths. Find its area by dividing it into triangle ABC, triangle ACD, and triangle ADE.
Answer: Area of pentagon ABCDE = (Area of triangle ABC) + (Area of triangle ACD) + (Area of triangle ADE)

= (1/2) × AC × BL + (1/2) × AD × CM + (1/2) × AD × EM

= [(1/2) × 10 × 3 + (1/2) × 12 × 7 + (1/2) × 12 × 5] cm²

= (15 + 42 + 30) cm²

= 87 cm²

Therefore, the area of the pentagon is 87 cm².
In simple words: Draw lines from one vertex to all non-adjacent vertices to create triangles, find each triangle's area, and add them.

Exam Tip: Triangulation (dividing a polygon into triangles from a single vertex) is one of the most reliable methods for finding irregular polygon areas.

Question 6. A figure is composed of a trapezium FEDC and a square ABCF. The trapezium has parallel sides of (6 + 20) = 26 cm and 20 cm with a height of 8 cm. The square has a side length of 20 cm. Calculate the total area enclosed by the figure.
Answer: Area enclosed by the given figure = (Area of trapezium FEDC) + (Area of square ABCF)

= [(1/2) × (6 + 20) × 8 + (20 × 20)] cm²

= (104 + 400) cm²

= 504 cm²

Therefore, the area enclosed by the figure is 504 cm².
In simple words: Find the area of the trapezium part and the square part separately, then add them together to get the total area.

Exam Tip: For composite figures made of standard shapes, always identify each shape clearly, calculate areas separately, and sum them carefully.

Question 7. A figure ABCDEFGH contains a rectangle ADEH, triangles ABC and HGF, with specific dimensions. Find the length AC using the right triangles ABC and HGF where AC² - HF² = (6)² - (4)² = 25 - 16 = 9 cm, so AC - HF = 3 cm. Then calculate the total area.
Answer: First, find the length of AC:

From right triangles ABC and HGF: AC² - HF² = (6)² - (4)²

AC² - HF² = 25 - 16 = 9 cm

AC = HF = √9 = 3 cm

Area of the given figure ABCDEFGH = (Area of rectangle ADEH) + 2(Area of triangle ABC)

= (AD × DE) + 2(Area of triangle ABC)

= (AD × DE) + 2[(1/2) × BC × AC]

= [(3 + 4) × 8] + 2[(1/2) × 4 × 3] cm²

= (56 + 12) cm²

= 68 cm²

Therefore, the area of the given figure is 68 cm².
In simple words: Use the Pythagorean theorem to find any missing side lengths, then break the figure into a rectangle and triangles to calculate the total area.

Exam Tip: When a figure contains right triangles, use the Pythagorean theorem to find unknown dimensions before proceeding with area calculations.

Question 8. A regular hexagon has two parallel sides forming a trapezium at the top. The trapezium ADEF has parallel sides AD = 23 cm and EF = 13 cm with a perpendicular height FL = 12 cm. The bottom trapezium ABCD has sides equal to the sides of the top trapezium. Calculate the area of the regular hexagon.
Answer: Let AL = DM = x cm and LM = BC = 13 cm.

Since x + 13 + x = 23, we have: 2x + 13 = 23

=> 2x = 10

=> x = 5

Therefore, AL = 5 cm

From right triangle AFL: FL² = AF² - AL²

FL² = (13)² - (5)²

FL² = 169 - 25 = 144

FL = √144 = 12 cm

Therefore, FL = BL = 12 cm

Area of a regular hexagon = (Area of trapezium ADEF) + (Area of trapezium ABCD)

= 2(Area of trapezium ADEF)

= 2[(1/2) × (AD + EF) × FL]

= 2[(1/2) × (23 + 13) × 12] cm²

= 2[(1/2) × 36 × 12] cm²

= 2(216) cm²

= 432 cm²

Therefore, the area of the given regular hexagon is 432 cm².
In simple words: Use the given dimensions to find unknown side lengths, apply the Pythagorean theorem to find the height, then use the trapezium area formula twice.

Exam Tip: Regular hexagons often can be divided into two equal trapeziums, which simplifies the calculation by allowing you to find one area and multiply by 2.

Exercise 18C

Question 1. Find the area of a trapezium with parallel sides of 14 cm and 18 cm and a perpendicular distance of 9 cm between them.
Answer: Area of the trapezium = (1/2) × (14 + 18) × 9 cm²

= (1/2) × 32 × 9 cm²

= 144 cm²

The correct answer is (b) 144 cm²

Area = 144 cm²
In simple words: Add the two parallel sides, multiply by the height, and divide by 2 to find the area.

Exam Tip: Always double-check that you have added the parallel sides correctly before multiplying by the height.

Question 2. A trapezium has parallel sides of 19 cm and 13 cm. If its area is 128 cm², find the perpendicular distance between the parallel sides.
Answer: The correct answer is (c) 8 cm

Let the perpendicular distance between the parallel sides be x cm.

Area of the trapezium = (1/2) × (19 + 13) × x cm²

128 = (1/2) × 32 × x cm²

128 = 16x cm²

But it is given that the area of the trapezium is 128 cm².

∴ 16x = 128

=> x = 128 ÷ 16

=> x = 8 cm

Therefore, the perpendicular distance between the parallel sides is 8 cm.
In simple words: Substitute the known values into the trapezium area formula and solve for the unknown height by dividing both sides by the coefficient.

Exam Tip: When finding height from area, always isolate the height term by dividing the area by half the sum of the parallel sides.

Question 3. A trapezium has parallel sides of 3x cm and 4x cm, with a perpendicular height of 12 cm, and an area of 630 cm². Find the shorter parallel side.
Answer: Let the lengths of the parallel sides be 3x cm and 4x cm.

Area of the trapezium = (1/2) × (3x + 4x) × 12 cm²

= (1/2) × 7x × 12 cm²

= 42x cm²

But it is given that the area of the trapezium is 630 cm².

∴ 42x = 630

=> x = 630 ÷ 42

=> x = 15 cm

Length of the parallel sides = (3 × 15) cm = 45 cm

(4 × 15) cm = 60 cm

Therefore, the shorter of the parallel sides is 45 cm.
In simple words: Express the sides in terms of a variable, substitute into the area formula, and solve for the variable to find both side lengths.

Exam Tip: When sides are given in ratio form, always solve for the variable first, then calculate each individual side length.

Question 4. A trapezium has parallel sides of x cm and (x + 6) cm with a perpendicular height of 9 cm. If the area is 180 cm², find the longer parallel side.
Answer: The correct answer is (b) 23 cm

Let the length of the parallel sides be x cm and (x + 6) cm respectively.

Area of the trapezium = (1/2) × (x + x + 6) × 9 cm²

= (1/2) × (2x + 6) × 9 cm²

= 4.5(2x + 6) cm²

= (9x + 27) cm²

But it is given that the area of the trapezium is 180 cm².

∴ 9x + 27 = 180

=> 9x = (180 - 27)

=> 9x = 153

=> x = 153 ÷ 9

=> x = 17

Therefore, the length of the parallel sides are 17 cm and (17 + 6) cm, which is equal to 23 cm.

Therefore, the length of the longer parallel side is 23 cm.
In simple words: Set up the area equation with the sides in terms of a variable, substitute the given area, and solve for the variable to find both sides.

Exam Tip: Make sure to give both sides when solving, and identify which one is the longer side if the question specifically asks for it.

Question 5. In a trapezium, DC = AL = 7 cm. From the given trapezium, find CL where CB = 10 cm and LB = 6 cm, using the right triangle CBL. Then calculate the area where AB || DC and AB = 13 cm, DC = 7 cm.
Answer: The correct answer is (c) 80 cm²

From the given trapezium, find the value of CL.

DC = AL = 7 cm [since DA ⊥ AB and CL ⊥ AB]

From the right triangle CBL: CL² = CB² - LB²

CL² = (10)² - (6)²

CL² = 100 - 36

CL² = 64

CL = √64

CL = 8 cm

Area of the trapezium = (1/2) × (7 + 13) × 8 cm²

= (1/2) × 20 × 8 cm²

= 80 cm²

Therefore, the area of the trapezium is 80 cm².
In simple words: Use the Pythagorean theorem in the right triangle to find the perpendicular height, then apply the trapezium area formula.

Exam Tip: Always look for right triangles within the trapezium to find the perpendicular height - this is essential for area calculations.