MathsCBSE Class 8 · Area

Area of Rhombus, Trapezium, Polygons and Area Units for Class 8

Updated 17 October 2026 · Fact-checked

The area of a rhombus is (1/2) x d1 x d2, of a trapezium is (1/2) x (a + b) x h with a and b the parallel sides, and the area of any polygon is found by breaking it into triangles, rectangles and trapeziums. Areas are measured in square units: 1 sq in is 6.4516 sq cm and 1 acre is 43,560 sq ft. Exams ask for composite figures, finding a missing side or height and unit conversions.

Rhombus area uses half the product of its diagonals; trapezium area uses half the sum of the parallel sides times the perpendicular height.

Illustration drawn for QuestionBuddy.

Understand Area of Rhombus, Trapezium, Polygons and Area Units

Rhombus. A rhombus is a parallelogram, so its area is $base \times height$ . A second formula comes from its diagonals: the diagonals are perpendicular bisectors of each other, so they cut the rhombus $A B C D$ into 4 congruent right triangles with legs $\frac{d _{1}}{2}$ and $\frac{d _{2}}{2}$ . Splitting along one diagonal into two isosceles triangles and rearranging gives a rectangle $W X Y Z$ with $X W = d_{1}$ (one diagonal) and $W Z = \frac{d _{2}}{2}$ (half the other). Therefore
$ $area of a rhombus = d_{1} \times \frac{d _{2}}{2} = \frac{1}{2} d_{1} d_{2} .$ $
The same follows from the areas of triangles $A D B$ and $C D B$ (each with base $d_{1}$ and height $\frac{d _{2}}{2}$ ). Example: diagonals 16 cm and 30 cm give $\frac{1}{2} \times 16 \times 30 = 240$ cm $^{2}$ .

Trapezium. In a trapezium $W X Y Z$ with $W X ∥ Z Y$ , let the parallel sides be $a$ and $b$ and the height $h$ . Drop perpendiculars $W M$ and $X N$ to $Z Y$ : this creates a rectangle $W X N M$ and two right triangles, with $M Z = x$ , $N Y = y$ , $W M = X N = h$ . Then $b = x + y + a$ , so $x + y = b - a$ and
$ $area = ah + \frac{1}{2} x h + \frac{1}{2} y h = \frac{1}{2} h (2 a + x + y) = \frac{1}{2} h (a + b) .$ $
So **area of a trapezium $= \frac{1}{2} \times (sum of the parallel sides) \times height$ .** Other derivations: join two copies of the trapezium (one rotated) along a non-parallel side to form a parallelogram with base $a + b$ and height $h$ ; its area $(a + b) h$ is twice the trapezium's. The formula also holds for a trapezium with obtuse angles (use a difference of areas). An isosceles trapezium can be dissected into a rectangle.

Polygons. The area of any polygon is obtained by dividing it into triangles, rectangles, trapeziums or other figures whose area we know and adding. For a quadrilateral with a diagonal $d$ and perpendicular distances $h_{1}, h_{2}$ of the other two vertices from it, area $= \frac{1}{2} d (h_{1} + h_{2})$ .

Equal-area transformations. A trapezium can be converted to a rectangle of equal area by dissection, a rectangle to a rhombus of equal area, and a trapezium of a given area can be constructed by reversing the process.

Units of area. Area is measured in square units. The chapter gives: $1 in^{2} = 6.4516$ cm $^{2}$ (so $10 in^{2} = 64.516$ cm $^{2}$ ), $1$ acre $= 43, 560$ ft $^{2}$ , and notes that larger areas are measured in km $^{2}$ and local units such as bigha, gaj, katha, dhur, cent and ankanam. An A4 sheet is 21 cm by 29.7 cm, so its area is $623.7$ cm $^{2}$ . To change units, multiply or divide by the conversion factor; since areas multiply two lengths, the factor for square units is the square of the length factor.

Area of rectangles, paths, triangles and parallelograms is on the previous topic page of this chapter.

Area of Rhombus, Trapezium, Polygons and Area Units: formulas to remember

Rhombus: $A = \frac{1}{2} d_{1} d_{2}$; d1 and d2 are the diagonals; also A = base x height.
Trapezium: $A = \frac{1}{2} (a + b) h$; a and b are the parallel sides.
Quadrilateral from a diagonal: $A = \frac{1}{2} d (h_{1} + h_{2})$; h1 and h2 are the distances of the other vertices from the diagonal.
Square inch and square centimetre: $1 in^{2} = 6.4516 cm^{2}$; Quoted in the chapter.
Acre: $1 acre = 43, 560 ft^{2}$; Also 1 hectare = 10,000 sq m.

How to solve Area of Rhombus, Trapezium, Polygons and Area Units questions

Routine for rhombus, trapezium and polygon areas.

1Identify the figure and which measurements are given: diagonals, parallel sides and height, or side lengths.
2If a measurement is missing, use properties (right triangles in a rhombus, perpendicular distance between parallel lines) to find it.
3Apply the formula: rhombus (1/2) d1 d2, trapezium (1/2)(a + b) h.
4For composite figures, split into triangles, rectangles and trapeziums, compute each area and add (or subtract).
5For unit conversion, write the factor and decide whether to multiply or divide.
6Round sensibly and give square units.

Area of Rhombus, Trapezium, Polygons and Area Units in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Area of a trapezium plot

Suppose a plot is a trapezium with parallel sides 40 m and 24 m, and the distance between them is 15 m.

Find its area in square metres and in hectares, using 1 hectare = 10,000 sq m.

Try it first, then show the solutionSolution

Area $= \frac{1}{2} \times (40 + 24) \times 15 = 480$ m $^{2}$ .
In hectares: $480 \div 10000 = 0.048$ hectare.

Answer: 480 square metres, which is 0.048 hectare

Memory hook: Average of the parallel sides times the height.

Unit conversion

Imagine a phone screen that measures 2.5 inches by 5 inches and a seller who quotes its size in square centimetres.

Find the screen area in square inches and in square centimetres (use 1 sq in = 6.4516 sq cm).

Try it first, then show the solutionSolution

Area $= 2.5 \times 5 = 12.5$ in $^{2}$ .
In cm $^{2}$ : $12.5 \times 6.4516 = 80.645$ cm $^{2}$ .

Answer: 12.5 sq in, about 80.6 sq cm

Memory hook: Compare areas in the same unit first.

Tricks to remember Area of Rhombus, Trapezium, Polygons and Area Units

Short memory aids. Say each one aloud once, then use it on a question the same day.

Rhombus: half the diagonals
Half of the product of the two diagonals.
Use it when: Rhombus area questions.
Trapezium: average width times height
The average of the parallel sides times the height.
Use it when: Trapezium area questions.
Cut it into easy pieces
Any polygon is a set of triangles and trapeziums; add them.
Use it when: Irregular fields.
Square the factor for area
If 1 in = 2.54 cm, then 1 sq in = 2.54 squared sq cm.
Use it when: Unit conversions of area.

Area of Rhombus, Trapezium, Polygons and Area Units: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Area of a rhombus · 1 mark · MCQBoard level

The area of a rhombus with diagonals 18 cm and 24 cm is: (a) 42 sq cm (b) 108 sq cm (c) 216 sq cm (d) 432 sq cm

Try it first, then show the solutionSolution

$\frac{1}{2} \times 18 \times 24 = 216$ .

Answer: (c) 216 sq cm

Example 2: Area of a trapezium · 2 marks · Short answerBoard level

A window pane is a trapezium with parallel sides 14 cm and 22 cm, which are 9 cm apart. Find its area.

Try it first, then show the solutionSolution

Area $= \frac{1}{2} (14 + 22) \times 9$ . (1 mark)
$= \frac{1}{2} \times 36 \times 9 = 162$ sq cm. (1 mark)

Answer: 162 sq cm

Example 3: Rhombus from a side and a diagonal · 3 marks · Short answerHarder board level

The side of a rhombus is 17 cm and one of its diagonals is 16 cm. Find the other diagonal and the area of the rhombus.

Try it first, then show the solutionSolution

The diagonals bisect each other at right angles; half of the given diagonal is 8 cm, forming a right triangle with hypotenuse 17 cm. (1 mark)
The other half-diagonal $= 1 7^{2} - 8^{2} = 225 = 15$ cm, so the other diagonal is 30 cm. (1 mark)
Area $= \frac{1}{2} \times 16 \times 30 = 240$ sq cm. (1 mark)

Answer: Other diagonal 30 cm; area 240 sq cm

Example 4: Missing parallel sides · 3 marks · Short answerHarder board level

A trapezium has height 12 cm, area 330 sq cm, and its parallel sides differ by 5 cm. Find the parallel sides.

Try it first, then show the solutionSolution

$\frac{1}{2} (a + b) \times 12 = 330$ gives $a + b = 55$ . (1 mark)
$a - b = 5$ . Adding: $2 a = 60$ , $a = 30$ . (1 mark)
$b = 25$ . Check: $\frac{1}{2} (30 + 25) \times 12 = 330$ . (1 mark)

Answer: 30 cm and 25 cm

Example 5: Land area units · 4 marks · Long answerHarder board level

A plot is a trapezium with parallel sides 150 ft and 90 ft and height 100 ft. Find its area in square feet, in acres to two decimal places, and in square metres to the nearest square metre (use 1 acre = 43,560 sq ft and 1 sq m = 10.764 sq ft).

Try it first, then show the solutionSolution

Area $= \frac{1}{2} (150 + 90) (100) = 12000$ sq ft. (1 mark)
Acres $= 12000 \div 43560 \approx 0.2755 \approx 0.28$ acre. (1 mark)
Square metres $= 12000 \div 10.764 \approx 1114.8 \approx 1115$ sq m. (1 mark)
The conversion factors are rounded, so the last digits are approximate. (1 mark)

Answer: 12,000 sq ft; about 0.28 acre; about 1115 sq m

Where students lose marks in Area of Rhombus, Trapezium, Polygons and Area Units

The slips examiners see most often, and what to write instead.

Mistake: Using the full diagonal product for a rhombus
The area is half the product of the diagonals.
Wrong working
"Rhombus with diagonals 16 and 30: area = 480."
Right working
$\frac{1}{2} \times 16 \times 30 = 240$ .
Do this: Write 1/2 first.
Usually costs: 1 mark.
Mistake: Using a non-parallel side as the height of a trapezium
The height is the perpendicular distance between the parallel sides.
Wrong working
"Trapezium with parallel sides 12 and 18 and slanting side 7: area = (1/2)(30)(7)."
Right working
Use the perpendicular height; the slanting side is longer than the height.
Do this: Mark the right angle.
Usually costs: The answer.
Mistake: Multiplying by the length factor for area units
Area units need the square of the length factor, or the area factor given.
Wrong working
"1 ft^2 = 12 in^2."
Right working
$1 ft^{2} = 12 \times 12 = 144 in^{2}$ .
Do this: Square the length factor.
Usually costs: 2 marks.
Mistake: Dividing instead of multiplying when converting to a smaller unit
A smaller unit gives a bigger number.
Wrong working
"10 in^2 = 10 / 6.4516 cm^2."
Right working
$10 \times 6.4516 = 64.516$ cm $^{2}$ .
Do this: Ask: should the number grow or shrink?
Usually costs: 1 mark.

Exam tips

Write the formula, then substitute with units, then compute.
For a rhombus with a side and a diagonal, find the other diagonal first by the theorem.
In composite figures, draw the dissection lines and label each piece.
For unit conversions state the conversion factor you use.
Check that area is in square units.

Practice questions on Area of Rhombus, Trapezium, Polygons and Area Units

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): The area of a rhombus is half the product of its diagonals.
Reason (R): The diagonals of a rhombus are perpendicular bisectors of each other, so it splits into four right triangles with legs d1/2 and d2/2.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Four triangles each with area (1/2)(d1/2)(d2/2).
Answer
(a). Area $= 4 \times \frac{1}{2} \times \frac{d _{1}}{2} \times \frac{d _{2}}{2} = \frac{1}{2} d_{1} d_{2}$ .
Q2 · 2 marks · Very short answerBoard level
Using 1 in^2 = 6.4516 cm^2 and 1 ft = 12 in, convert 40 in^2 to cm^2 and 1 ft^2 to cm^2.
Hint
1 ft^2 = 144 in^2.
Answer
$40 \times 6.4516 = 258.064$ cm $^{2}$ . $1 ft^{2} = 144 in^{2}$ , so $144 \times 6.4516 = 929.0304$ cm $^{2}$ .
Q3 · 3 marks · Short answerHarder board level
An A4 sheet measures 21 cm by 29.7 cm. Find its area. A tabletop is 1.2 m by 0.6 m: find how many A4 sheets would be needed to cover it if they could be cut and rearranged, and say how many whole sheets you must buy.
Hint
Convert the table to cm.
Answer
A4 area $= 21 \times 29.7 = 623.7$ cm $^{2}$ . Table: $120 \times 60 = 7200$ cm $^{2}$ . $7200 \div 623.7 \approx 11.54$ , so about 11.5 sheets of area; you must buy 12 whole sheets.
Q4 · 4 marks · Long answerHOTS / competency
Show two different ways that the area of a trapezium with parallel sides a and b and height h equals (1/2)(a + b)h: (i) by joining two copies to form a parallelogram, (ii) by cutting it into a rectangle and two triangles. Check with a = 9, b = 15, h = 8.
Hint
The two copies form a parallelogram of base a + b.
Answer
(i) Rotate a second copy through $18 0^{\circ}$ about the midpoint of a non-parallel side and join: the figure is a parallelogram of base $a + b$ and height $h$ , area $(a + b) h$ , which is twice the trapezium. So the trapezium has $\frac{1}{2} (a + b) h$ . (ii) Drop perpendiculars: a rectangle $a \times h$ and two right triangles with bases $x$ and $y$ , $x + y = b - a$ . Area $= ah + \frac{1}{2} x h + \frac{1}{2} y h = \frac{1}{2} h (2 a + b - a) = \frac{1}{2} h (a + b)$ . (iii) For $a = 9, b = 15, h = 8$ : (i) gives $\frac{1}{2} \times 24 \times 8 = 96$ and (ii) gives $72 + \frac{1}{2} \times 6 \times 8 = 72 + 24 = 96$ (with $x + y = 6$ ).
Q5 · 4 marks · Case-basedHOTS / competency
Case study: a field ABCDE is divided into three triangles by the diagonals AC and AD. Triangle ABC has area 60 sq m. Triangle ACD has base AC = 15 m and height 8 m. Triangle ADE has base AD = 13 m and height 6 m.
(i) Find the area of triangle ACD. (1 mark)
(ii) Find the area of triangle ADE. (1 mark)
(iii) Find the area of the field. (1 mark)
(iv) Land costs Rs 2,400 per sq m. Find the cost of the field. (1 mark)
Hint
Area of a triangle is (1/2) base x height.
Answer
(i) $\frac{1}{2} \times 15 \times 8 = 60$ sq m. (ii) $\frac{1}{2} \times 13 \times 6 = 39$ sq m. (iii) $60 + 60 + 39 = 159$ sq m. (iv) $159 \times 2400 = 3, 81, 600$ rupees.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is the area of a rhombus in terms of its diagonals?Show answer
Half the product of the diagonals.
2.What is the area of a trapezium?Show answer
Half x (sum of parallel sides) x height.
3.How many square centimetres in a square inch (chapter value)?Show answer
6.4516.
4.How many square feet in an acre?Show answer
43,560.
5.How do you find the area of a polygon?Show answer
Break it into triangles, rectangles, trapeziums and add.
6.What is the area of an A4 sheet?Show answer
21 cm x 29.7 cm = 623.7 square cm.

Area of Rhombus, Trapezium, Polygons and Area Units: frequently asked questions

Is the rhombus formula d1 d2 / 2 a new formula or a special case?

A rhombus is also a parallelogram, so base x height works; the diagonal formula is an alternative that uses the perpendicular diagonals.

Does the trapezium formula work for any shape?

It works for any trapezium with parallel sides a and b and height h, including obtuse ones.

How do I find the area of an irregular polygon?

Split it into triangles (using diagonals) or rectangles and trapeziums, find each area and add.

Where are local area units used?

The chapter notes units such as bigha, gaj, katha, dhur, cent and ankanam in different parts of India; their sizes vary, so check the local conversion.