📐 Class 8 Mathematics — Mensuration
Interactive Visual Revision Notes
Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
🌎 Real-World Connection: A farmer needs to know the area of his trapezoidal field to decide how much fertiliser to buy. A carpenter calculates the surface area of a wooden box to know how much paint is needed. A water tank company works out the volume of a cylindrical tank to know how many litres it holds. Mensuration — the mathematics of measuring shapes — is the foundation of every construction, packaging, and engineering project!
📐 Topic 1: What Is Mensuration?
Mensuration means measuring shapes.
We measure area of flat shapes.
We measure perimeter around shapes.
We also measure surface area of solids.
We find volume of 3D objects.
Area is measured in square units.
Volume is measured in cubic units.
📐 Topic 2: Area of Trapezium
A trapezium has two parallel sides.
The parallel sides are called a and b.
The distance between them is height h.
Add the two parallel sides first.
Multiply the sum by the height.
Then divide by 2.
Area = ½ × (a + b) × h
📐 Topic 3: Area of Rhombus
A rhombus has four equal sides.
Its diagonals cross at 90 degrees.
The diagonals are called d₁ and d₂.
Multiply the two diagonals together.
Then divide by 2.
The diagonals cut each other in half.
Area = ½ × d₁ × d₂
📐 Topic 4: Area of General Quadrilateral
A quadrilateral has four sides.
Draw one diagonal inside it.
This splits it into two triangles.
Find the area of each triangle.
Add both areas to get the total.
Use: Area = ½ × base × height.
📐 Topic 5: Area of Polygon
A polygon has many sides.
Split the polygon into simple shapes.
Use triangles and rectangles.
Find the area of each part.
Add all the parts together.
This gives the total area.
📐 Topic 6: Surface Area of Cube
A cube has 6 equal square faces.
Each face has side length a.
Area of one face = a².
Total surface area = 6 faces.
So we multiply a² by 6.
Surface Area of Cube = 6a²
📐 Topic 7: Surface Area of Cuboid
A cuboid has 6 rectangular faces.
It has length l, breadth b, height h.
Top and bottom = l × b each.
Front and back = b × h each.
Left and right = h × l each.
Add all three pairs together.
Surface Area of Cuboid = 2(lb + bh + hl)
📐 Topic 8: Surface Area of Cylinder
A cylinder has two circular ends.
It has a curved surface around it.
The radius of the circle is r.
The height of the cylinder is h.
Curved surface area = 2πrh.
Total = curved area + two circles.
Total Surface Area = 2πr(r + h)
📐 Topic 9: Volume of Cube and Cuboid
Volume = space inside a solid.
Volume of cube = a × a × a.
We write this as a³.
Volume of cuboid = l × b × h.
Volume is in cubic units.
Example: cm³, m³.
Cube = a³ | Cuboid = l × b × h
📐 Topic 10: Volume of Cylinder
A cylinder has a circular base.
Area of the base = πr².
Multiply the base area by height h.
This gives the volume.
Volume is in cubic units.
Volume of Cylinder = πr²h
🎨 Section 2 — Easy Illustrations
Picture 1: Trapezium with Parallel Sides
A trapezium has two parallel sides a and b. The height h is the distance between them.
Picture 2: Rhombus with Diagonals
The two diagonals of a rhombus cross at right angles. Each half is shaded differently.
Picture 3: Cube (3D View)
A cube has 6 equal square faces. All edges have the same length a.
Picture 4: Cuboid (3D View)
A cuboid has length l, breadth b, and height h. Opposite faces are equal.
Picture 5: Cylinder (3D View)
A cylinder has radius r and height h. The curved surface wraps around the body.
✏️ Section 3 — Worked Examples
Example 1: Area of a Trapezium
Question: Find the area of a trapezium. Parallel sides are 10 cm and 14 cm. Height is 8 cm.
1 Write the formula: Area = ½ × (a + b) × h.
2 Here a = 10 cm, b = 14 cm, h = 8 cm.
3 Add the parallel sides: 10 + 14 = 24 cm.
4 Multiply by height: 24 × 8 = 192.
5 Divide by 2: 192 ÷ 2 = 96.
Area of Trapezium = 96 cm²
Example 2: Surface Area and Volume of Cuboid
Question: A cuboid has length 12 cm, breadth 8 cm, height 5 cm. Find surface area and volume.
1 Here l = 12 cm, b = 8 cm, h = 5 cm.
2 Surface Area = 2(lb + bh + hl).
3 lb = 12 × 8 = 96.
4 bh = 8 × 5 = 40.
5 hl = 5 × 12 = 60.
6 Add: 96 + 40 + 60 = 196.
7 Surface Area = 2 × 196 = 392 cm².
8 Volume = l × b × h = 12 × 8 × 5.
9 Volume = 480 cm³.
Surface Area = 392 cm² | Volume = 480 cm³
Example 3: Curved Surface Area and Volume of Cylinder
Question: A cylinder has radius 7 cm and height 15 cm. Find curved surface area and volume. (Use π = 22/7)
1 Here r = 7 cm, h = 15 cm, π = 22/7.
2 Curved Surface Area = 2πrh.
3 = 2 × (22/7) × 7 × 15.
4 The 7s cancel: 2 × 22 × 15 = 660 cm².
5 Volume = πr²h.
6 = (22/7) × 7 × 7 × 15.
7 = 22 × 7 × 15 = 2310 cm³.
Curved SA = 660 cm² | Volume = 2310 cm³
Tip: When r = 7 and π = 22/7, the 7s cancel!
📊 Section 4 — Cheat Sheet
Shape
Formula
Key Fact
Trapezium Area
½ × (a + b) × h
Two parallel sides
Rhombus Area
½ × d₁ × d₂
Diagonals at 90°
Quadrilateral
Split into 2 triangles
Draw a diagonal
Polygon Area
Split into simple shapes
Add all parts
Cube SA
6a²
6 equal faces
Cuboid SA
2(lb + bh + hl)
3 pairs of faces
Cylinder SA
2πr(r + h)
Curved + 2 circles
Cube Volume
a³
All sides equal
Cuboid Volume
l × b × h
3 different sides
Cylinder Volume
πr²h
Base area × height
📄 Section 5 — Take-Home Handout
📝 A. Fill in the Blanks
1. Area of trapezium = ½ × (a + b) × _________.
2. A rhombus has _________ equal sides.
3. Surface area of a cube = _________.
4. Volume of a cuboid = l × b × _________.
5. Curved surface area of cylinder = _________.
✓ B. True or False
1. A cube has 8 faces. _______
2. Volume is measured in square units. _______
3. The diagonals of a rhombus cross at 90 degrees. _______
4. A cylinder has two circular ends. _______
5. Surface area of cuboid = 2(lb + bh + hl). _______
✏️ C. Solve These
1. A trapezium has parallel sides 8 cm and 12 cm. Height is 6 cm. Find the area.
Show your work here
2. A cube has side 5 cm. Find its surface area and volume.
Show your work here
🔗 D. Match the Columns
Column A
Column B
1. Volume of cube
a. 2πrh
2. Area of rhombus
b. a³
3. Curved SA of cylinder
c. 6a²
4. SA of cube
d. ½ × d₁ × d₂
🎨 E. Draw and Label
Draw a cuboid and label its length, breadth, and height. Write the formulas for surface area and volume next to it.
Draw your cuboid here
✅ Detailed Solutions — Section 5 Handout
A. Fill in the Blanks — Answers
1. Area of trapezium = ½ × (a + b) × height (h).
2. A rhombus has 4 equal sides.
3. Surface area of a cube = 6a². (6 faces, each of area a².)
4. Volume of a cuboid = l × b × h.
5. Curved surface area of cylinder = 2πrh.
B. True or False — Answers
1. A cube has 8 faces. — False. A cube has 6 faces (top, bottom, front, back, left, right).
2. Volume is measured in square units. — False. Volume is measured in cubic units (cm³, m³). Square units are for area.
3. The diagonals of a rhombus cross at 90 degrees. — True. Diagonals of a rhombus are perpendicular bisectors of each other. ✓
4. A cylinder has two circular ends. — True. Total SA = 2πr² + 2πrh. ✓
5. Surface area of cuboid = 2(lb + bh + hl). — True. Standard formula. ✓
C1. Trapezium with parallel sides 8 cm and 12 cm, height 6 cm. Find area.
1 Formula: Area = ½ × (a + b) × h.
2 a = 8 cm, b = 12 cm, h = 6 cm.
3 Area = ½ × (8 + 12) × 6 = ½ × 20 × 6 = ½ × 120 = 60 cm².
Area of trapezium = 60 cm²
C2. Cube with side 5 cm. Find surface area and volume.
1 Side a = 5 cm.
2Surface area = 6a² = 6 × 5² = 6 × 25 = 150 cm².
3Volume = a³ = 5³ = 5 × 5 × 5 = 125 cm³.
Surface area = 150 cm², Volume = 125 cm³
D. Match the Columns — Answers
1. Volume of cube → b. a³
2. Area of rhombus → d. ½ × d₁ × d₂ (product of diagonals ÷ 2).
3. Curved SA of cylinder → a. 2πrh
4. SA of cube → c. 6a²
🃏 Section 6 — Revision Flashcards
👉
📐
What is mensuration?
Mensuration is the study of measuring area, perimeter, surface area, and volume of shapes.
▲
What is the area of a trapezium?
½ × (a + b) × h
a and b are parallel sides. h is the height.
◆
What is the area of a rhombus?
½ × d₁ × d₂
d₁ and d₂ are the two diagonals.
📦
What is the surface area of a cube?
6a²
A cube has 6 equal square faces.
📦
What is the surface area of a cuboid?
2(lb + bh + hl)
Three pairs of rectangular faces.
🔂
What is the volume of a cylinder?
πr²h
Base area (πr²) multiplied by height.
📏
What is the volume of a cuboid?
l × b × h
Length times breadth times height.
🎯
What is the curved surface area of a cylinder?
2πrh
This does not include the two circular ends.
🧠 Section 7 — Mind Map
Hover on any branch to focus it. All key ideas connected together.
Class 8 Mathematics — Mensuration
Original Visual Revision Notes — Different Learner Friendly
Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
💜 Created with love by a father, to make learning easier for his daughter 💜