Mensuration word problems, solved step by step

Mensuration questions are stories about fences, tiles, tanks and tents. The skill is to see which shape and which measure the story is about. Here are ten solved problems of the kinds set in classes 6 to 10. Each begins by naming the shape and the measure. Where π is needed, 22/7 is used.

1. Fencing a field

A rectangular field is 80 m long and 60 m wide. Find the cost of fencing it at ₹50 a metre.

Fencing goes round the edge: perimeter. 2 × (80 + 60) = 280 m. Cost = 280 × 50 = ₹14,000.

2. Tiling a floor

A room is 6 m by 4.5 m. How many square tiles of side 30 cm are needed?

Tiles cover the surface: area. Work in centimetres: 600 × 450 = 2,70,000 cm². One tile is 30 × 30 = 900 cm². Number of tiles = 2,70,000 ÷ 900 = 300.

3. A path round a circular park

A circular park of radius 35 m has a path 7 m wide round it on the outside. Find the area of the path.

The path is a ring. Outer radius 42 m, inner radius 35 m. Area = 22/7 × (42² − 35²) = 22/7 × 539 = 1694 m².

4. Turns of a wheel

A wheel has diameter 70 cm. How many turns does it make in going 1.1 km?

One turn covers the circumference: 22/7 × 70 = 220 cm. The distance is 1,10,000 cm. Turns = 1,10,000 ÷ 220 = 500.

5. Painting the walls

A room is 5 m long, 4 m wide and 3 m high. Its door and windows take up 4 m². Find the area to be painted on the four walls.

Four walls: lateral surface area of a cuboid. 2 × 3 × (5 + 4) = 54 m². Less the openings: 54 − 4 = 50 m².

6. A water tank

How many litres does a tank 2 m long, 1.5 m wide and 1 m deep hold?

Holding means volume. 2 × 1.5 × 1 = 3 m³. Since 1 m³ = 1000 litres, the tank holds 3000 litres.

7. A round tank

A cylindrical tank has radius 0.7 m and height 2 m. Find its capacity in litres.

Volume of a cylinder: 22/7 × 0.7 × 0.7 × 2 = 3.08 m³, which is 3080 litres.

8. Canvas for a tent

A conical tent is 24 m high and its base has radius 7 m. How much canvas is needed?

A tent has no floor, so only the curved surface. Slant height = √(7² + 24²) = √625 = 25 m. Canvas = 22/7 × 7 × 25 = 550 m².

9. Melting a ball

A metal sphere of radius 6 cm is melted and made into a cylinder of radius 4 cm. Find the height of the cylinder.

Melting keeps the volume the same. (4/3) × π × 6³ = π × 4² × h. The π cancels: 288 = 16h, so h = 18 cm.

10. A triangular plot

A plot has sides 13 m, 14 m and 15 m. Find its area.

Three sides and no height: Heron's formula. Half the perimeter is s = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 m².

A method for any mensuration problem

1. Name the shape. Draw it and write the measures on the drawing.
2. Name the measure. Is the story about an edge (perimeter), a covering (area or surface area) or a filling (volume)?
3. Make the units agree before calculating.
4. Write the formula, put in the numbers, and calculate.
5. Give the unit, and ask whether the size of the answer is sensible.

Clue words

Use the calculator to check your own answers, and see units and conversions for the unit steps.

More