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CBSE · Class 8 · Mathematics Part 2

Part 2, Chapter 7: Area

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Rectangles, squares and paths

From daily life

A farmer has two plots: 9 m by 5 m and 7 m by 6 m. To find which needs more seed, count the unit squares: 9 × 5 = 45 and 7 × 6 = 42. The first plot is larger.

Simple definition

Area is the number of unit squares that fill a region. Rectangle: length × width. Square: side × side. Perimeter is only the length of the boundary and cannot measure area. Area of a path = area of the outer shape − area of the inner shape.

16 × 820 × 12 − 16 × 8 = 11220 × 12

In the figure: Path: 240 − 128 = 112 square units (outer 20 × 12, inner 16 × 8).

Rule 1: Area and a missing side — Area = length × width. For a missing side divide the area by the known side.

Live figure

Rectangles, squares and paths

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In the figure: Change the sides and watch the area.

  • Easy: 12 m by 5 m: 12 × 5 = 60 m².
  • Medium: Area 84 m², length 12 m: width 84 ÷ 12 = 7 m.
  • Exam-type: Square of side 9 cm: 9 × 9 = 81 cm².

Rule 2: Area is not perimeter — Two shapes can have the same perimeter but different areas, or a longer perimeter and a smaller area.

  • Easy: 10 × 2: perimeter 24, area 20. 6 × 5: perimeter 22, area 30.
  • Medium: 11 × 1 and 6 × 6 both have perimeter 24, but areas 11 and 36.
  • Exam-type: Among rectangles of equal perimeter, the square has the largest area.

Rule 3: Area of a path — Outer rectangle minus inner rectangle. A path of width w outside adds w on every side.

  • Easy: Park 18 m by 10 m with a 1 m path outside: 20 × 12 − 18 × 10 = 60 m².
  • Medium: Plot 24 m by 16 m with a 1 m path inside: 384 − 22 × 14 = 76 m².
  • Exam-type: Square 10 m with a 2 m path outside: 14 × 14 − 100 = 96 m².
Remember

Area is measured in square units: 1 m² = 100 × 100 = 10000 cm². Convert before you compare.

Common mistake

Using the perimeter as the area, or writing cm instead of cm² for an area.

Shortcut

Area: lay the tiles (length × width). Path: big minus small. Memory clue: "Tiles inside, Big minus Small for a path".

Spot the difference

PointAreaPerimeter
Measuresspace insidelength of the boundary
Unitcm²cm
Rectangle 12 by 56034

Practice questions

Easy
  1. 1Find the area of a 14 m by 5 m rectangle.
  2. 2Find the area of a square of side 11 m.
  3. 3A rectangle has area 96 m² and length 12 m. Find its width.
Medium
  1. 4Convert 4 m² into cm².
  2. 5A park 10 m by 6 m has a 2 m wide path all round, outside. Find the area of the path.
  3. 6A floor 7 m by 5 m is tiled at ₹40 per m². Find the cost.
Challenging
  1. 7A rectangle has area 72 cm² and length 12 cm. Find its perimeter.
  2. 8The length of a rectangle is doubled and the width is tripled. The area becomes how many times?
Show answers (try them yourself first)
  1. 170
  2. 2121
  3. 38
  4. 440000 cm²
  5. 580 m²
  6. 6₹1400
  7. 736 cm
  8. 86 times

Area of a triangle

From daily life

A triangular signboard is exactly half of a rectangular board with the same base and height. So its paint is half as much.

Simple definition

Area of a triangle = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex. Any side can be the base. Triangles on the same base and between the same parallel lines have equal areas, and a median splits a triangle into two triangles of equal area.

baseh½ × base × h

In the figure: The dashed perpendicular is the height; it is not the slanting side.

Rule 1: Find the area — Multiply base by height, then take half.

Live figure

Area of a triangle

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In the figure: Drag the vertex along a parallel line: the area does not change.

  • Easy: Base 16, height 5: ½ × 80 = 40 cm².
  • Medium: Base 9, height 8: 36 cm².
  • Exam-type: Right triangle with short sides 6 and 8: ½ × 6 × 8 = 24 cm².

Rule 2: Find a base or height — Double the area, then divide by the known side.

  • Easy: Area 45, height 9: base 90 ÷ 9 = 10.
  • Medium: Area 28, base 8: height 56 ÷ 8 = 7.
  • Exam-type: Area 60, base 15: height 120 ÷ 15 = 8.

Rule 3: Same base, same height — Equal bases and equal heights give equal areas, even if the shapes look different.

  • Easy: Two triangles on a base of 12 between lines 5 apart: both ½ × 12 × 5 = 30.
  • Medium: A median of a triangle of area 70: each part is 35.
  • Exam-type: Isosceles triangle 10, 10, 12 with height 8 on the side 12: ½ × 12 × 8 = 48.
Remember

Any of the three sides can be the base; the matching height changes but the area stays the same.

Common mistake

Using the slanting side as the height. The height must be perpendicular to the base.

Shortcut

A triangle is half the box. Memory clue: "Half base-times-height: cut the rectangle on its diagonal".

Spot the difference

PointRectangleTriangle (same base, height)
Areabase × height½ × base × height
Base 12, height 56030
Comparedwholehalf

Practice questions

Easy
  1. 1Find the area of a triangle with base 16 cm and height 5 cm.
  2. 2Find the area with base 9 cm and height 8 cm.
  3. 3A triangle of area 45 cm² has base 10 cm. Find the height.
Medium
  1. 4A right triangle has short sides 6 cm and 8 cm. Find its area.
  2. 5A median splits a triangle of area 70 cm². Find the area of each part.
  3. 6A triangle of area 60 cm² has base 15 cm. Find the height.
Challenging
  1. 7The base of a triangle becomes 3 times while the height stays the same. The area becomes how many times?
  2. 8An isosceles triangle has sides 10, 10 and 12 cm and height 8 cm on the 12 cm side. Find its area.
Show answers (try them yourself first)
  1. 140
  2. 236
  3. 39
  4. 424
  5. 535
  6. 68
  7. 73 times
  8. 848

Area of polygons by splitting

From daily life

A farmer has an odd-shaped field. He walks along one long line across it, measures how far each corner is from that line, and finds the area without a formula for the whole shape.

Simple definition

To find the area of a polygon, split it into triangles, rectangles or trapeziums, find each area and add (or subtract the part cut out). For a quadrilateral with diagonal d and perpendiculars h1 and h2 from the other two vertices: area = ½ × d × (h1 + h2). If the diagonals are perpendicular: area = ½ × d1 × d2.

ABCDh1h2d

In the figure: Two triangles on the same diagonal: ½ × d × h1 + ½ × d × h2.

Rule 1: Diagonal and perpendiculars — Add the two perpendiculars, multiply by the diagonal, take half.

  • Easy: d = 30, h1 = 6, h2 = 4: ½ × 30 × 10 = 150.
  • Medium: d = 16, h1 = 5, h2 = 7: ½ × 16 × 12 = 96.
  • Exam-type: d = 24, h1 = 9, h2 = 3: ½ × 24 × 12 = 144.

Rule 2: Add or subtract shapes — Cut the figure into simple shapes and add, or take a big rectangle and subtract the missing corner.

  • Easy: Rectangle 9 by 4 plus a triangle (base 9, height 5): 36 + 22.5 = 58.5.
  • Medium: L-shape: 14 by 8 minus 6 by 3 = 112 − 18 = 94.
  • Exam-type: House: rectangle 8 by 5 plus a roof triangle (base 8, height 3): 40 + 12 = 52.

Rule 3: Perpendicular diagonals and the triangle count — Half the product of the diagonals. A polygon with n sides splits into n − 2 triangles from one vertex.

  • Easy: Diagonals 18 and 10: ½ × 18 × 10 = 90.
  • Medium: A hexagon splits into 6 − 2 = 4 triangles.
  • Exam-type: A heptagon splits into 7 − 2 = 5 triangles.
Remember

Choose a split that uses lengths you already know: a diagonal with two perpendiculars is usually the quickest.

Common mistake

Forgetting to subtract a cut-out part, or using a sloping side where a perpendicular height is needed.

Shortcut

Split, Solve, Sum (or Subtract). Memory clue: "Split, Solve, Sum".

Spot the difference

PointAdd methodSubtract method
Use whenthe shape splits into piecesa corner is missing from a rectangle
Examplehouse shape: 40 + 12L-shape: 112 − 18
Result5294

Practice questions

Easy
  1. 1A quadrilateral has diagonal 20 m and perpendiculars 3 m and 5 m. Find the area.
  2. 2Into how many triangles does a hexagon split from one vertex?
  3. 3The diagonals of a quadrilateral are perpendicular, 12 cm and 7 cm. Find its area.
Medium
  1. 4An L-shape is a 16 m by 9 m rectangle with a 7 m by 4 m corner removed. Find its area.
  2. 5A house shape is a 10 m by 4 m rectangle with a roof triangle of base 10 m and height 3 m. Find its area.
  3. 6A plot has diagonal 36 m and perpendiculars 7 m and 11 m. Find the area.
Challenging
  1. 7A square of side 14 cm has a right triangle with short sides 3 cm cut from each of its 4 corners. Find the area left.
  2. 8A plot has diagonal 30 m and perpendiculars 8 m and 12 m. It is sold at ₹25 per m². Find the price.
Show answers (try them yourself first)
  1. 180
  2. 24 triangles
  3. 342
  4. 4116
  5. 555
  6. 6324
  7. 7178
  8. 8₹7500

Parallelogram and rhombus

From daily life

Cut a triangle off the slanted end of a parallelogram-shaped paper and move it to the other end: you get a rectangle with the same base and height. The area did not change.

Simple definition

Area of a parallelogram = base × height (the perpendicular height, not the slant side). A rhombus is a parallelogram, so base × height works for it too. Using its diagonals: area of a rhombus = ½ × d1 × d2. A diagonal splits a parallelogram into two equal triangles.

basehbase × h

In the figure: Slide the slanted piece across: base × height.

Rule 1: Base times height — Multiply the base by the perpendicular height.

Live figure

Parallelogram and rhombus

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In the figure: Change the slant but keep the height: the area stays the same.

  • Easy: Base 18, height 5: 90 cm².
  • Medium: Base 11, height 8: 88 cm².
  • Exam-type: Base 25, height 4, slant side 6: 100 cm² (the slant side is not used).

Rule 2: Find a missing base or height — Divide the area by the known one.

  • Easy: Area 72, base 9: height 8.
  • Medium: Area 90, height 6: base 15.
  • Exam-type: Sides 15 and 9, height 6 on the side 15: area 90, so the height on the side 9 is 10.

Rule 3: Rhombus with diagonals — Half the product of the two diagonals. The diagonals cut each other at right angles into halves.

  • Easy: Diagonals 14 and 10: ½ × 14 × 10 = 70.
  • Medium: Area 96, one diagonal 16: the other is 2 × 96 ÷ 16 = 12.
  • Exam-type: Side 17, one diagonal 30: half-diagonals 15 and 8, so the other diagonal is 16 and the area is ½ × 30 × 16 = 240.
Remember

A triangle on the same base and between the same parallels as a parallelogram has half its area.

Common mistake

Multiplying the two neighbouring sides (base × slant side) to get the area. Use the perpendicular height.

Shortcut

Slide the slant, use the straight height. Memory clue: "Parallelogram: slide it; rhombus: half of the cross".

Spot the difference

PointParallelogramRhombus (with diagonals)
Areabase × height½ × d1 × d2
Example18 × 5½ × 14 × 10
Result9070

Practice questions

Easy
  1. 1A parallelogram has base 12 cm and height 9 cm. Find its area.
  2. 2A rhombus has diagonals 8 cm and 6 cm. Find its area.
  3. 3A parallelogram of area 60 cm² has base 12 cm. Find its height.
Medium
  1. 4A diagonal splits a parallelogram of area 84 cm² into two triangles. Find the area of each.
  2. 5A rhombus has area 120 cm² and one diagonal 24 cm. Find the other diagonal.
  3. 6A parallelogram has sides 15 cm and 9 cm and height 6 cm on the 15 cm side. Find the height on the 9 cm side.
Challenging
  1. 7A rhombus has side 10 cm and one diagonal 16 cm. Find its area.
  2. 8A parallelogram has area 90 cm². A triangle on the same base and between the same parallels has what area?
Show answers (try them yourself first)
  1. 1108
  2. 224
  3. 35
  4. 442
  5. 510
  6. 610
  7. 796
  8. 845

Trapezium and areas in real life

From daily life

A field has two parallel edges of 50 m and 30 m, 20 m apart. If you picture its average width, (50 + 30) ÷ 2 = 40 m, the field is like a 40 m by 20 m rectangle: 800 m².

Simple definition

A trapezium has one pair of parallel sides. Area = ½ × (sum of the parallel sides) × height. Two copies of a trapezium (one turned upside down) make a parallelogram with base a + b, which gives the formula. Useful units: 1 m² = 10000 cm², 1 square foot = 144 square inches, 1 acre = 43560 square feet.

baabh

In the figure: The parallelogram has area (a + b) × h, so one trapezium is half of it.

Rule 1: Find the area — Add the parallel sides, multiply by the height, take half.

  • Easy: Sides 13 and 7, height 6: ½ × 20 × 6 = 60.
  • Medium: Sides 18 and 10, height 5: ½ × 28 × 5 = 70.
  • Exam-type: Sides 9 and 3, height 8: ½ × 12 × 8 = 48.

Rule 2: Find a missing part — Double the area, divide by the height to get the sum of the parallel sides.

  • Easy: Area 80, height 8, one side 12: sum 20, other side 8.
  • Medium: Area 75, sides 12 and 18: height 150 ÷ 30 = 5.
  • Exam-type: Area 54, height 6: the sum of the parallel sides is 18.

Rule 3: Change units — Square the length factor: 1 foot = 12 inches gives 1 square foot = 12 × 12 = 144 square inches.

  • Easy: 2 square feet = 2 × 144 = 288 square inches.
  • Medium: 3 m² = 3 × 10000 = 30000 cm².
  • Exam-type: 3 acres = 3 × 43560 = 130680 square feet.
Remember

If the two parallel sides are equal, the trapezium becomes a parallelogram. If one of them shrinks to 0, it becomes a triangle: the same formula covers both.

Common mistake

Using a slanting side as the height, or mixing units (feet with inches) in one calculation.

Shortcut

Average width times height. Memory clue: "Mean of the parallels, times the gap".

Spot the difference

PointTrapeziumParallelogram
Area½ × (a + b) × hbase × h
Parallel sidestwo differentboth pairs equal
Sides 12 and 8, height 55060

Practice questions

Easy
  1. 1A trapezium has parallel sides 15 cm and 9 cm and height 6 cm. Find its area.
  2. 2Parallel sides 10 cm and 6 cm, height 3 cm. Find the area.
  3. 3How many square inches are there in 1 square foot?
Medium
  1. 4A trapezium of area 105 cm² has height 7 cm. Find the sum of its parallel sides.
  2. 5A trapezium of area 150 cm², height 10 cm has one parallel side 18 cm. Find the other.
  3. 6Convert 5 m² into cm².
Challenging
  1. 7A trapezium-shaped field has parallel sides 50 m and 30 m, 20 m apart. Ploughing costs ₹10 per m². Find the cost.
  2. 8If one parallel side of a trapezium becomes 0, the formula ½ × (a + b) × h becomes the area of which shape?
Show answers (try them yourself first)
  1. 172
  2. 224
  3. 3144 square inches
  4. 430
  5. 512
  6. 650000 cm²
  7. 7₹8000
  8. 8a triangle