Mensuration

Scenario

Estimating how many liters of water fit inside a society's overhead cylindrical tank.

Mental model

Perimeter = Walking the boundary. Area = Painting the surface. Volume = Filling with water.

Mensuration is about turning shapes into pure numbers. 2D deals with flat surfaces (floors, walls). 3D adds depth (boxes, cones, spheres). Every 3D volume formula is a 2D area formula scaled by a dimension.

Formula

2D: Rectangle Area = L x B, Perimeter = 2(L+B). Triangle Area = 0.5 x base x height. Circle Area = pi x r^2, Circumference = 2 x pi x r. Square Area = a^2, Diagonal = a x sqrt(2). 3D: Cuboid Volume = L x B x H, TSA = 2(LB + BH + LH). Cylinder Volume = pi x r^2 x h, CSA = 2 x pi x r x h, TSA = 2 x pi x r x (r + h). Cone Volume = (1/3) x pi x r^2 x h. Sphere Volume = (4/3) x pi x r^3, SA = 4 x pi x r^2.

Cylinder volume is just a circle's area (pi r^2) stacked up to height h. Cone is exactly 1/3 of a cylinder with the same base and height. CSA covers only curved sides; TSA includes flat lids. Always convert all measurements to the same unit before computing.

Worked example

Problem: A cylindrical water tank has radius 3.5 m and height 8 m. What is its volume? How many liters does it hold? (Use pi = 22/7, 1 cu m = 1000 L)

1. Formula: Volume = pi x r^2 x h = (22/7) x (3.5)^2 x 8.
2. (3.5)^2 = 12.25. (22/7) x 12.25 = (22 x 12.25) / 7 = 269.5 / 7 = 38.5.
3. Volume = 38.5 x 8 = 308 cubic meters.
4. Capacity in liters = 308 x 1000 = 3,08,000 liters.
5. Sanity-check: 22/7 x 3.5^2 x 8 = 22 x 1.75 x 8 = 22 x 14 = 308 cu m. Matches.

Answer: Volume = 308 cu m = 3,08,000 liters.

Problem: The length and breadth of a rectangular plot are in ratio 5:3. If the perimeter is 160 m, find the area.

1. Let L = 5x, B = 3x. Perimeter = 2(L + B) = 2(5x + 3x) = 16x = 160. So x = 10.
2. L = 50 m, B = 30 m.
3. Area = L x B = 50 x 30 = 1500 sq m.
4. Sanity-check: Perimeter = 2(50 + 30) = 2 x 80 = 160 m. Correct.

Answer: Area = 1500 sq m.

Shortcuts

• The 11 Divisibility Check: Almost all formulas with circles (pi) use 22/7. Hence, the final answer for area or volume is usually divisible by 11. Use this to eliminate options. (Find area of circle: Options A) 314 B) 308 C) 320 D) 310. Only 308 is divisible by 11 (3 - 0 + 8 = 11). Answer is B.)
• Cone is 1/3 of Cylinder: A cone's volume is exactly one-third of a cylinder with the same base radius and height. (If a cylinder's volume is 300 cc, a cone with the same base and height is 100 cc.)

Common mistakes

• Confusing Total Surface Area (TSA) and Curved Surface Area (CSA): CSA only covers the curved sides. TSA includes the flat top and bottom lids. If painting a completely closed tin, use TSA.
• Unit Mismatches: Radius might be in cm but height in meters. Always convert everything to a single unit before applying the formula.

Glossary

perimeter

The total distance around the outside edge of a flat shape.

volume

The amount of 3D space inside an object, like how much water a tank can hold.

dimension

A measurement of length, width, or depth.

Recall questions

• What is the perimeter of a rectangle?
• What is the volume of a cylinder?
• Is a cone's volume greater or smaller than a cylinder of same dimensions?

Questions & answers

A cylindrical tank has a base radius of 7m and a height of 10m. What is its capacity in cubic meters?

1540

Approach: Volume = pi x r^2 x h = (22/7) x 7 x 7 x 10 = 22 x 7 x 10 = 1540 cubic meters.

The diagonal of a square is 10*sqrt(2) cm. What is its area?

100

Approach: Diagonal = a*sqrt(2) = 10*sqrt(2), so side a = 10. Area = a^2 = 100 sq cm.

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