Mensuration

Perimeter measures the boundary of a shape. Area measures the flat space inside it. Volume measures the space inside a solid.

Write separate expressions for volume and exposed surface. Shared internal faces are not external surface.

A rectangle is 5 metres long and 3 metres wide. Find its area and perimeter. 1. Area: 5 × 3 = 15 square metres. 2. Perimeter: 5 + 3 + 5 + 3 = 16 metres. 3. Notice the different units: area uses square metres.

The shaded interior represents the area, which is 5 × 3 = 15 square metres. The outer line is the boundary, whose four lengths total 16 metres. The two answers have different units. If all lengths double, perimeter doubles while area becomes four times as large.

A cylinder of radius 3 and height 10 is melted into spheres of radius 1, with no material lost. Cylinder volume=pi x 9 x 10=90pi. One sphere volume=4pi/3. The quotient is 67.5, so at most 67 complete spheres can be formed; material remains. If the question assumes all material forms identical whole spheres, these dimensions cannot yield an integer count.

Area and perimeter measure different things and need different formulas. Do not round up a count when material is limited. Units must be cubic for volume and square for surface area.

1. Identify the shape and the quantity asked for. 2. Write the correct formula. 3. Convert all lengths to the same unit. 4. Substitute the values. 5. For a combined shape, add the required parts and subtract holes or overlaps.

A rectangle is 8 cm long and 3 cm wide. Its area is 8 x 3 = 24 square cm. Its perimeter is 2 x (8 + 3) = 22 cm. A box with those base dimensions and height 2 cm has volume 8 x 3 x 2 = 48 cubic cm.

Doubling every length makes area four times as large and volume eight times as large. Length, area, and volume do not grow at the same rate.

Write separate expressions for volume and exposed surface. Shared internal faces are not external surface.

A cylinder of radius 3 and height 10 is melted into spheres of radius 1, with no material lost. Cylinder volume=pi x 9 x 10=90pi. One sphere volume=4pi/3. The quotient is 67.5, so at most 67 complete spheres can be formed; material remains. If the question assumes all material forms identical whole spheres, these dimensions cannot yield an integer count.

Do not round up a count when material is limited. Units must be cubic for volume and square for surface area.

Cuboid A cuboid is a solid bounded by six rectangular faces. Opposite faces are parallel and equal. Its three perpendicular dimensions are length l, breadth b and height h. Volume = l × b × h. Total surface area = 2(lb + bh + hl).

For length 6 cm, breadth 4 cm and height 3 cm, volume = 6 × 4 × 3 = 72 cm³. Total surface area = 2(24 + 12 + 18) = 108 cm². Use cubic units for volume and square units for surface area.

Perimeter uses length units, area square units, and volume cubic units. Identify which faces or boundaries are included before calculating. All lengths must use the same unit. Key formulas Rectangle: area lb, perimeter 2(l+b). Triangle: area bh/2. Disk: area πr². Cuboid: volume lbh, total surface area 2(lb+bh+hl), space diagonal √(l²+b²+h²). Cube of side a: volume a³ and surface area 6a². Cylinder of radius r and height h: volume πr²h, curved surface area 2πrh, closed total surface area 2πr(h+r). Cone: slant height s=√(r²+h²), volume πr²h/3, curved area πrs, closed total area πr(s+r). Sphere: volume 4πr³/3 and surface area 4πr². Hemisphere: volume 2πr³/3, curved area 2πr² and closed total area 3πr².

A closed cylinder has radius 3 cm and height 5 cm. Volume = 45π cm³; curved area = 30π cm²; the two ends add 18π cm², giving total surface area 48π cm². If the top is open, the required area is instead 39π cm².

An open container does not include a missing face. A composite solid's internal joining faces do not count as exposed surface area. Scaling all lengths by k multiplies area by k² and volume by k³.