MEANING OF AREA

Definition:
The area of a shape is the amount of surface it covers.
It helps us to measure how much space is inside a flat (two-dimensional) shape.

For example, if you draw a rectangle on paper, the area tells us how much space the rectangle covers on that paper.

Area is measured in square units, such as:

1. square centimetres (cm²)


2. square metres (m²)


3. square millimetres (mm²)


4. square kilometres (km²)

PLANE SHAPES

Plane shapes are flat shapes that have length and breadth, but no thickness.
Examples of plane shapes are:

1. Triangle


2. Rectangle


3. Square


4. Parallelogram


5. Trapezium


6. Circle

Now let us learn how to find the area of each one of them.

1. AREA OF A RECTANGLE

Definition:
A rectangle is a four-sided figure (quadrilateral) with opposite sides equal and parallel, and each angle is a right angle (90°).

Formula for area:
Area = Length × Breadth

Example 1:
Find the area of a rectangle whose length is 10 cm and breadth is 5 cm.
Solution:
Area = 10 × 5 = 50 cm²

Example 2:
Find the area of a rectangle of length 7 m and breadth 4 m.
Area = 7 × 4 = 28 m²

2. AREA OF A SQUARE

Definition:
A square is a special rectangle where all four sides are equal, and all angles are right angles.

Formula for area:
Area = side × side = side²

Example 1:
Find the area of a square whose side is 6 cm.
Area = 6 × 6 = 36 cm²

Example 2:
If each side of a square field is 20 m, find its area.
Area = 20 × 20 = 400 m²

3. AREA OF A PARALLELOGRAM

Definition:
A parallelogram is a four-sided shape with opposite sides equal and parallel.
The height of a parallelogram is the perpendicular distance between two opposite sides.

Formula for area:
Area = Base × Height

Example 1:
Find the area of a parallelogram with base 8 cm and height 5 cm.
Area = 8 × 5 = 40 cm²

Example 2:
A parallelogram has a base of 10 m and height of 7 m.
Find its area.
Area = 10 × 7 = 70 m²

4. AREA OF A TRIANGLE

Definition:
A triangle is a three-sided plane shape.

Formula for area:
Area = ½ × Base × Height

Example 1:
Find the area of a triangle whose base is 12 cm and height is 8 cm.
Area = ½ × 12 × 8 = 48 cm²

Example 2:
Find the area of a triangle with base 10 m and height 6 m.
Area = ½ × 10 × 6 = 30 m²

5. AREA OF A TRAPEZIUM

Definition:
A trapezium (or trapezoid) is a four-sided figure that has one pair of opposite sides parallel.
The two parallel sides are called the bases.

Formula for area:
Area = ½ × (Sum of parallel sides) × Height
That is, A = ½ (a + b)h
where a and b are the parallel sides, and h is the height.

Example 1:
Find the area of a trapezium whose parallel sides are 10 cm and 6 cm, and height is 5 cm.
Area = ½ × (10 + 6) × 5 = ½ × 16 × 5 = 40 cm²

Example 2:
Find the area of a trapezium with parallel sides 12 m and 8 m, and height 7 m.
Area = ½ × (12 + 8) × 7 = ½ × 20 × 7 = 70 m²

6. AREA OF A CIRCLE

Definition:
A circle is a plane shape with all points at an equal distance from a fixed point called the centre.
The distance from the centre to any point on the circle is called the radius (r).
The distance across the circle through the centre is called the diameter (d).

Formula for area:
Area = πr², where π (pi) = 3.142 or 22/7.

Example 1:
Find the area of a circle whose radius is 7 cm.
Area = 22/7 × 7 × 7 = 154 cm²

Example 2:
Find the area of a circle with radius 14 m (use π = 22/7).
Area = 22/7 × 14 × 14 = 616 m²

7. COMBINED SHAPES

Sometimes a shape can be made up of two or more plane shapes.
To find the total area, we divide the shape into smaller parts, find the area of each part, and add them together.

Example:
A figure is made up of a rectangle of length 8 cm and breadth 4 cm, joined to a semicircle of radius 2 cm at one end.
Find the total area.

Area of rectangle = 8 × 4 = 32 cm²
Area of semicircle = ½ × π × r² = ½ × 22/7 × 2² = ½ × 22/7 × 4 = 6.2857 cm² ≈ 6.3 cm²
Total area = 32 + 6.3 = 38.3 cm²

SUMMARY TABLE OF FORMULAS

Shape	Formula for Area
Rectangle	Length × Breadth
Square	Side × Side = side²
Parallelogram	Base × Height
Triangle	½ × Base × Height
Trapezium	½ × (Sum of parallel sides) × Height
Circle	π × r²

IMPORTANCE OF KNOWING AREA

1. It helps us to Know how much land or space we have.


2. It helps us to Measure surfaces when building houses or painting walls.


3. It helps us to Calculate sizes of fields, playgrounds, and farmlands.


4. It helps us to Plan materials such as tiles, carpets, or roofing sheets.

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PRACTICE QUESTIONS & STEP-BY-STEP SOLUTIONS — AREA OF PLANE SHAPES

RECTANGLE

1. Question 1:
   A rectangular classroom measures 12 m by 8 m. Find its area and the cost to tile the floor at ₦1,500 per square metre.
   
   Solution:
   1. Shape: Rectangle.
   
   
   2. Formula: Area = Length × Breadth.
   
   
   3. Substitute: Area = 12 × 8.
   
   
   4. Area = 96 m².
   
   
   5. Cost per m² = ₦1,500. Total cost = 96 × 1,500 = ₦144,000.
   
   


2. Question 2:
   A rectangular garden has perimeter 60 m. If its length is 17 m, find its breadth and area.
   
   Solution:
   1. Perimeter of rectangle = 2(Length + Breadth) = 60.
   
   
   2. So Length + Breadth = 30. Given Length = 17, so Breadth = 30 − 17 = 13 m.
   
   
   3. Area = Length × Breadth = 17 × 13 = 221 m².
   
   


3. Question 3:
   A carpet of area 45 m² is rectangular and its length is three times its breadth. Find the length and breadth.
   
   Solution:
   1. Let breadth = b. Then length = 3b.
   
   
   2. Area = length × breadth = 3b × b = 3b² = 45.
   
   
   3. So b² = 45 ÷ 3 = 15. b = √15 ≈ 3.873 m. Length = 3 × 3.873 ≈ 11.619 m.
   
   
   4. Exact form: breadth = √15 m, length = 3√15 m. Approximate values given above.
   
   


4. Question 4:
   A rectangular plot has length 2.5 times its breadth. The difference between length and breadth is 30 m. Find the area.
   
   Solution:
   1. Let breadth = b. Then length = 2.5b.
   
   
   2. Length − breadth = 2.5b − b = 1.5b = 30, so b = 30 ÷ 1.5 = 20 m.
   
   
   3. Length = 2.5 × 20 = 50 m. Area = 50 × 20 = 1,000 m².
   
   


5. Question 5:
   One side of a rectangle is increased by 20% and the adjacent side is decreased by 10%. If original sides were 25 m and 16 m, find the new area and the percentage change in area (to two decimal places).
   
   Solution:
   1. Original area = 25 × 16 = 400 m².
   
   
   2. New sides: 25 increased by 20% → 25 × 1.20 = 30 m. 16 decreased by 10% → 16 × 0.90 = 14.4 m.
   
   
   3. New area = 30 × 14.4 = 432 m².
   
   
   4. Change = 432 − 400 = 32 m². Percentage change = (32 ÷ 400) × 100 = 8.00% increase.
   
   

SQUARE

1. Question 1:
   A square playground has area 2,025 m². Find the length of one side.
   
   Solution:
   1. Area = side² = 2,025. side = √2,025 = 45 m.
   
   


2. Question 2:
   A square tile measures 0.6 m on a side. How many such tiles are needed to cover a square room of side 6 m? Assume no wastage.
   
   Solution:
   1. Area of room = 6 × 6 = 36 m².
   
   
   2. Area of one tile = 0.6 × 0.6 = 0.36 m².
   
   
   3. Number of tiles = 36 ÷ 0.36 = 100 tiles.
   
   


3. Question 3:
   The diagonal of a square is 10√2 cm. Find the area of the square.
   
   Solution:
   1. Diagonal d = side × √2. So side = d ÷ √2 = (10√2) ÷ √2 = 10 cm.
   
   
   2. Area = side² = 10² = 100 cm².
   
   


4. Question 4:
   A square garden has side x m. If its area is increased by 44 m² to give a new square garden, and the new side is x + 2 m, find x.
   
   Solution:
   1. Original area = x². New area = (x + 2)² = x² + 4x + 4.
   
   
   2. Increase = (x² + 4x + 4) − x² = 4x + 4 = 44.
   
   
   3. So 4x + 4 = 44 ⇒ 4x = 40 ⇒ x = 10 m.
   
   
   4. Original area = 100 m², new area = 144 m².
   
   


5. Question 5:
   A square field is fenced around. If the fencing cost is ₦600 per metre and total cost is ₦28,800, find the area of the field.
   
   Solution:
   1. Total cost = ₦28,800, cost per metre = ₦600. Perimeter = 28,800 ÷ 600 = 48 m.
   
   
   2. Perimeter of square = 4 × side = 48 ⇒ side = 12 m.
   
   
   3. Area = side² = 12² = 144 m².
   
   

PARALLELOGRAM

1. Question 1:
   A parallelogram has base 14 cm and height 9 cm. Find its area.
   
   Solution:
   1. Formula: Area = Base × Height = 14 × 9 = 126 cm².
   
   


2. Question 2:
   A parallelogram and a rectangle have equal areas. The rectangle measures 12 m by 7 m. If the parallelogram has base 14 m, find its height.
   
   Solution:
   1. Area of rectangle = 12 × 7 = 84 m². Parallelogram area = 84 m².
   
   
   2. Area = Base × Height ⇒ Height = Area ÷ Base = 84 ÷ 14 = 6 m.
   
   


3. Question 3:
   A parallelogram has adjacent sides 15 m and 10 m with included angle 30°. Find its area. (Use exact value: sin 30° = 1/2.)
   
   Solution:
   1. Area = product of adjacent sides × sin(included angle) = 15 × 10 × sin 30°.
   
   
   2. sin 30° = 1/2. Area = 15 × 10 × 1/2 = 150 × 1/2 = 75 m².
   
   


4. Question 4:
   The area of a parallelogram is 240 cm². If its base is increased by 20% to 24 cm, find the original base and height.
   
   Solution:
   1. Let original base = b and height = h. Given b × h = 240.
   
   
   2. New base = 24 cm so 24 is 120% of b ⇒ b = 24 ÷ 1.20 = 20 cm.
   
   
   3. Height h = 240 ÷ b = 240 ÷ 20 = 12 cm.
   
   


5. Question 5:
   A parallelogram has area 360 m² and height 9 m. Find the base. Then find the area if the base is doubled while height remains the same.
   
   Solution:
   1. Base = Area ÷ Height = 360 ÷ 9 = 40 m.
   
   
   2. If base is doubled to 80 m, new area = 80 × 9 = 720 m².
   
   

TRIANGLE

1. Question 1:
   A triangle has base 18 cm and height 10 cm. Find its area.
   
   Solution:
   1. Formula: Area = ½ × Base × Height = ½ × 18 × 10 = 9 × 10 = 90 cm².
   
   


2. Question 2:
   The area of a triangle is 300 m². If its base is 25 m, find the height.
   
   Solution:
   1. Area = ½ × base × height ⇒ height = (2 × Area) ÷ base = (2 × 300) ÷ 25 = 600 ÷ 25 = 24 m.
   
   


3. Question 3:
   An isosceles triangle has equal sides 13 cm and base 10 cm. Find its area.
   
   Solution:
   1. Drop perpendicular from apex to base bisecting base into 5 and 5. Height h satisfies 13² = 5² + h² ⇒ 169 = 25 + h² ⇒ h² = 144 ⇒ h = 12 cm.
   
   
   2. Area = ½ × base × height = ½ × 10 × 12 = 60 cm².
   
   


4. Question 4:
   A right triangular flag has legs 9 m and 12 m. Find its area and the length of the hypotenuse.
   
   Solution:
   1. Area = ½ × product of legs = ½ × 9 × 12 = 54 m².
   
   
   2. Hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15 m.
   
   


5. Question 5:
   A triangle has sides 13 cm, 14 cm and 15 cm. Find its area using Heron formula.
   
   Solution:
   1. Semi perimeter s = (13 + 14 + 15) ÷ 2 = 42 ÷ 2 = 21 cm.
   
   
   2. Area = √[s(s − a)(s − b)(s − c)] = √[21(21 −13)(21 −14)(21 −15)] = √[21 × 8 × 7 × 6].
   
   
   3. Compute product: 21 × 8 =168; 168 × 7 =1,176; 1,176 × 6 =7,056. Area = √7,056 = 84 cm².
   
   

TRAPEZIUM

1. Question 1:
   A trapezium has parallel sides 12 m and 8 m with height 5 m. Find its area.
   
   Solution:
   1. Area = ½ × (sum of parallel sides) × height = ½ × (12 + 8) × 5 = ½ × 20 × 5 = 10 × 5 = 50 m².
   
   


2. Question 2:
   The area of a trapezium is 210 cm². If the parallel sides are 18 cm and 12 cm, find the height.
   
   Solution:
   1. Area = ½ × (18 + 12) × h = ½ × 30 × h = 15h = 210 ⇒ h = 210 ÷ 15 = 14 cm.
   
   


3. Question 3:
   A trapezium has parallel sides a and b with a − b = 6 m and height 4 m. If its area is 84 m² find a and b given that a + b = 21 m.
   
   Solution:
   1. We are given a + b = 21 and a − b = 6. Solve: add both ⇒ 2a = 27 ⇒ a = 13.5 m. Then b = 21 − 13.5 = 7.5 m.
   
   
   2. Check area: Area = ½ × (13.5 + 7.5) × 4 = ½ × 21 × 4 = 10.5 × 4 = 42 m². The area does not match 84; therefore if area must be 84 the given sum or height would differ. For this question use the given equations to find a and b as above. (This question models solving linear pairs.)
   
   


4. Question 4:
   A trapezium has parallel sides 20 m and 8 m. A rectangle of height equal to the trapezium height and width 8 m is placed beside it to form a larger rectangle of area 456 m². Find the height of the trapezium and its area.
   
   Solution:
   1. Let height = h. The larger rectangle width = (8 + 20) = 28 m and height = h. So area of larger rectangle = 28h = 456 ⇒ h = 456 ÷ 28 = 16.285714... ⇒ h = 16.285714... m. For exam style use exact fraction: h = 456/28 = 114/7 = 16 2/7 m.
   
   
   2. Trapezium area = ½ × (20 + 8) × h = ½ × 28 × h = 14h = 14 × (114/7) = 14 × 16.285714... = 228 m². (Exact: 14 × 114/7 = 2 × 114 = 228 m².)
   
   


5. Question 5:
   A trapezium is composed of a rectangle of width 10 m and a right triangle whose base is 6 m and height equals the rectangle height. If the trapezium parallel sides are 16 m and 10 m, find the common height and the area of the trapezium.
   
   Solution:
   1. Parallel sides a = 16 and b = 10. Area formula: A = ½(a + b)h = ½(26)h = 13h.
   
   
   2. The trapezium is rectangle (width 10) plus triangle (base 6, same height). Area = rectangle area + triangle area = 10h + ½ × 6 × h = 10h + 3h = 13h. This matches formula and height is not determined by that alone; any positive h gives consistent areas. If a value is needed, additional data required. For exam purposes show equivalence and express area = 13h. If the question supplied total area value one would solve for h.
   
   

CIRCLE

1. Question 1:
   Find the area of a circle with radius 7 m. Use π = 22/7.
   
   Solution:
   1. Area = πr² = 22/7 × 7 × 7 = 22 × 7 = 154 m².
   
   


2. Question 2:
   A circular cake has diameter 20 cm. Find its area. Use π = 3.142 and give answer to three decimal places.
   
   Solution:
   1. Radius r = diameter ÷ 2 = 10 cm.
   
   
   2. Area = πr² = 3.142 × 10² = 3.142 × 100 = 314.200 cm².
   
   


3. Question 3:
   A circular well has circumference 31.416 m. Find its radius and area. Use π = 3.1416.
   
   Solution:
   1. Circumference C = 2πr = 31.416. So r = C ÷ (2π) = 31.416 ÷ (2 × 3.1416) = 31.416 ÷ 6.2832 = 5 m.
   
   
   2. Area = πr² = 3.1416 × 5² = 3.1416 × 25 = 78.54 m².
   
   


4. Question 4:
   A circular park of radius 21 m is to be covered with grass. If each square metre costs ₦250 to lay grass, find the total cost. Use π = 22/7.
   
   Solution:
   1. Area = πr² = 22/7 × 21 × 21 = 22 × 63 = 1,386 m².
   
   
   2. Cost = 1,386 × 250 = ₦346,500.
   
   


5. Question 5:
   Two concentric circles form a ring. Outer radius is 10 m and inner radius is 6 m. Find the area of the ring. Use π = 22/7.
   
   Solution:
   1. Area of ring = Area outer − Area inner = π(R² − r²) = 22/7 × (10² − 6²) = 22/7 × (100 − 36) = 22/7 × 64 = 22 × 64 ÷ 7 = 1,408 ÷ 7 = 201.142857... m².
   
   
   2. Exact fraction: 1,408/7 m². Approximate = 201.142857... m².
   
   

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WORD PROBLEMS INVOLVING AREA OF PLANE SHAPES

MEANING

Word problems are questions that describe real-life situations using words.
They help us to apply the formulas for area to practical problems.

When solving:

1. we must Read the question carefully.


2. we must Identify the shape involved (rectangle, triangle, circle, etc.).


3. we must Write down the correct formula for the area of that shape.


4. we must Substitute the given values into the formula.


5. we must Solve and write the correct unit (cm², m², etc.).

Let us solve many examples together for better understanding.

EXAMPLE 1

A rectangular farm has a length of 50 m and a breadth of 30 m.
Find the area of the farm.

Solution:
Shape: Rectangle
Formula: Area = Length × Breadth
Area = 50 × 30 = 1,500 m²

EXAMPLE 2

A square carpet has each side measuring 8 m.
Find its area.

Solution:
Shape: Square
Formula: Area = side × side
Area = 8 × 8 = 64 m²

EXAMPLE 3

A triangular piece of land has a base of 12 m and a height of 10 m.
Find its area.

Solution:
Shape: Triangle
Formula: Area = ½ × base × height
Area = ½ × 12 × 10 = 6 × 10 = 60 m²

EXAMPLE 4

A parallelogram has a base of 15 cm and height of 10 cm.
Find its area.

Solution:
Shape: Parallelogram
Formula: Area = base × height
Area = 15 × 10 = 150 cm²

EXAMPLE 5

A trapezium has parallel sides of 14 cm and 10 cm, and height 8 cm.
Find its area.

Solution:
Shape: Trapezium
Formula: Area = ½ × (sum of parallel sides) × height
Area = ½ × (14 + 10) × 8 = ½ × 24 × 8 = 12 × 8 = 96 cm²

EXAMPLE 6

Find the area of a circular flower bed of radius 7 m (use π = 22/7).

Solution:
Shape: Circle
Formula: Area = πr²
Area = 22/7 × 7 × 7 = 154 m²

EXAMPLE 7

A rectangular floor is 20 m long and 15 m wide.
How many square tiles of size 1 m × 1 m are required to cover the floor?

Solution:
Area of floor = 20 × 15 = 300 m²
Area of one tile = 1 × 1 = 1 m²
Number of tiles = 300 ÷ 1 = 300 tiles

EXAMPLE 8

A square lawn has a side of 12 m. It is surrounded by a path 1 m wide.
Find the area of the path.

Solution:
Outer square (including path): side = 12 + 2(1) = 14 m
Inner square (lawn only): side = 12 m
Area of outer square = 14² = 196 m²
Area of inner square = 12² = 144 m²
Area of path = 196 - 144 = 52 m²

EXAMPLE 9

A circular pond has a radius of 10 m.
Find the cost of fencing it round at ₦150 per metre.

Solution:
The cost of fencing depends on the perimeter (circumference), not the area.
However, it is part of word problems involving shapes.

Circumference = 2πr
= 2 × 22/7 × 10 = 440/7 = 62.86 m
Cost = 62.86 × 150 = ₦9,429

EXAMPLE 10

A trapezium has parallel sides of 20 m and 10 m, and height 6 m.
If one square metre of land costs ₦100, find the total cost of the land.

Solution:
Area = ½ × (20 + 10) × 6 = ½ × 30 × 6 = 15 × 6 = 90 m²
Cost = 90 × 100 = ₦9,000

EXAMPLE 11

A triangular banner has a base of 15 cm and height of 8 cm.
If each square centimetre of the banner costs ₦5, find the cost of making it.

Solution:
Area = ½ × base × height = ½ × 15 × 8 = 7.5 × 8 = 60 cm²
Cost = 60 × 5 = ₦300

EXAMPLE 12

The length and breadth of a rectangular table are 2 m and 1.2 m respectively.
Find the area covered by 6 such tables.

Solution:
Area of one table = 2 × 1.2 = 2.4 m²
Area of 6 tables = 2.4 × 6 = 14.4 m²

EXAMPLE 13

A circular playground has a radius of 14 m.
Find its area. (Take π = 22/7).

Solution:
Area = πr² = 22/7 × 14 × 14 = 22/7 × 196 = 22 × 28 = 616 m²

EXAMPLE 14

A trapezium has parallel sides 18 m and 12 m, and height 10 m.
If it is painted at ₦50 per square metre, find the cost.

Solution:
Area = ½ × (18 + 12) × 10 = ½ × 30 × 10 = 15 × 10 = 150 m²
Cost = 150 × 50 = ₦7,500

EXAMPLE 15

A wall is 8 m long and 3 m high.
How many square tiles of side 0.5 m are needed to cover the wall?

Solution:
Area of wall = 8 × 3 = 24 m²
Area of one tile = 0.5 × 0.5 = 0.25 m²
Number of tiles = 24 ÷ 0.25 = 96 tiles

SUMMARY STEPS FOR SOLVING AREA WORD PROBLEMS

1. Read and understand the question properly.


2. Identify the shape involved.


3. Write down the formula for the area of that shape.


4. Substitute the given numbers into the formula.


5. Write your final answer with the correct unit (cm², m², etc.).


6. If money or cost is involved, multiply the area by the cost per square unit.