Mensuration, scale and the Earth as a sphere

Identify what is being measured

Length measures a boundary or distance, area measures a surface, and volume measures enclosed three-dimensional space. Their units differ: centimetres, square centimetres and cubic centimetres. A question about fencing asks for a perimeter, while covering a floor asks for area. A tank's capacity uses volume, but the material for its walls uses surface area. Sketch the shape and underline the requested quantity before selecting a formula. An accurately calculated volume cannot answer a surface-area question.

Convert units before combining measurements. Since and 1 m³ = 1,000,000 cm³. Also 1 litre. A tank holding 0.6 m³ contains 600 litres. Conversion factors must be squared or cubed with the dimension. Multiplying a square-centimetre value by 100 to obtain square metres is wrong in both factor and direction. A quick physical estimate can expose a conversion that makes a small container hold an implausibly large quantity.

Plane figures and composite areas

A rectangle has area lw and perimeter 2(l + w). A triangle has area bh/2, using perpendicular height h. A trapezium with parallel sides a and b and perpendicular separation h has area (a + b)h/2. A circle of radius r has area πr² and circumference 2πr. If a composite figure combines simple shapes, split it along known boundaries and add areas; subtract cut-outs. Do not add a shared interior line to an outside perimeter.

For an L-shaped floor, dividing it into two nonoverlapping rectangles is often easiest. Another approach is to take a large enclosing rectangle and subtract the missing corner. Both methods should agree. When dimensions are missing, derive them from total lengths before using a formula. If a rectangular border surrounds a garden, compute outside area minus inside area. A uniform border width adds twice that width to each full dimension, because the border occurs on both opposite sides.

Arcs, sectors, chords and segments

A sector with central angle θ degrees is θ/360 of a full circle. Its arc length is (θ/360)2πr and its area is (θ/360)πr². Its perimeter includes both radii as well as the arc. With r = 6 cm and length is 2π cm, sector area is 6π cm², and sector perimeter is 12 + 2π cm. The fraction of the circle is the same, but the measured quantities and units remain different.

The chord subtending central angle θ has length 2r sin(θ/2). For a minor segment, subtract the isosceles triangle area from sector area: segment area = (θ/360)πr² − r²sin θ/2 when using degree θ. For r = 6 and θ = 60°, the triangle area is 9√3 cm² and the minor segment area is 6π − 9√3 cm². If the angle is supplied in radians, arc length becomes rθ and sector area r²θ/2. Do not mix degree and radian versions.

Prisms, cylinders, cones and pyramids

A prism has constant cross-section, so volume is cross-sectional area × length. A cylinder is a circular prism with volume πr²h. A cone or pyramid has one third the volume of a prism with the same base area and perpendicular height. For a right circular cone, curved surface area is πrl, where l is slant height, and. The vertical height used for volume differs from the slant height used for curved area.

A sphere has surface area 4πr² and volume 4πr³/3. A hemisphere has half the sphere's curved surface and volume, but its total surface area includes the circular base. Thus a closed hemisphere has total surface area 3πr², while an open curved bowl has only 2πr² of curved surface. Read whether a base or lid is present. Composite solids may share a face that is no longer exposed; subtract such hidden faces when calculating the outer surface.

Use scale factors consistently

For similar figures with corresponding length ratio k, area ratio is k² and volume ratio is k³. If a model is at scale 1:50, a model length of 8 cm represents 400 cm = 4 m. A model area of 8 cm² represents. The numerical multiplier depends on dimension, not just on the stated scale. A map scale 1:100,000 means 1 cm represents 1 km on the ground, using consistent original units.

If two similar solids have volumes in ratio 27:64, their corresponding length ratio is 3:4 and surface-area ratio is 9:16. Taking a square root of the volume ratio would use the wrong dimension. In a density problem, mass = density × volume, with compatible units. A metal solid's mass can change after a cavity is cut out because material volume changes. The bounding box volume is not automatically the material volume. Diagram the removed region and subtract it before multiplying by density.

Model the Earth using latitude and longitude

In a simplified spherical-Earth model of radius R, a meridian is part of a great circle. Two points on the same meridian separated by latitude angle θ have surface distance (θ/360)2πR. Along a parallel of latitude φ, the circle's radius is R cos φ, so an angular longitude difference λ gives distance (λ/360)2πR cos φ along that parallel. This parallel route is generally not the shortest great-circle route. State which route is intended and use the radius supplied by the problem.

Self-check: find the volume of a cone of radius 3 cm and height 8 cm, the arc length of a 90° sector of radius 4 cm, and the length ratio of similar solids with volume ratio 8:125. Answers are 24π cm³, 2π cm and 2:5. Keep π exact when requested, and use a stated approximation such as 22/7 only when appropriate. Finish by checking dimensions: every term added to an area must have area units, and a negative physical length or volume signals an error in setup.

Calculate a frustum by subtraction

A cone cut by a plane parallel to its base produces a smaller similar cone and a frustum. One route to the frustum's volume is the large cone's volume minus the removed small cone's volume. Similarity determines missing heights from the radius ratio. A direct formula for perpendicular frustum height h and parallel circular radii R and r is. For R = 4 cm, r = 2 cm and h = 3 cm, the volume is 28π cm³. The height in this formula is the frustum's own perpendicular height, not its slant height or the height of the original full cone. A subtraction method provides a check when enough dimensions are known.