Surface area and volume of 3D solids
Geometry and measures • Mensuration and calculation
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Key concepts
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Volume of cuboids and right prisms
A cuboid is a right rectangular prism. Volume equals base area multiplied by height. For a cuboid, volume = length × width × height, so volume increases proportionally when any linear dimension increases. Right prisms share the same rule: volume = area of cross-sectional base × prism height. Prisms require parallel congruent faces and straight lateral edges. If the base area is known, the prism height directly scales the volume.
Volume of cylinders
A cylinder is a right prism with a circular base. Volume equals area of the circle base multiplied by its height. Formula: volume = πr²h, where r is the base radius and h is the height. Using a larger radius increases volume by the square of that change, because base area depends on r². Units for volume are cubic; unit consistency is required. An approximate value for π (for example 3.14 or 22/7) may appear in final answers depending on instructions.
Surface area of prisms and cylinders
Surface area of a prism equals the sum of the areas of all faces: two congruent bases plus the lateral faces. For right prisms, lateral faces are rectangles whose combined area equals perimeter of base × height. Therefore total surface area = 2 × area(base) + perimeter(base) × height. For cylinders, the curved surface area equals 2πrh and the total surface area equals 2πr² + 2πrh. Omitting end caps converts total surface area to curved surface area only, which occurs in open containers.
Spheres: surface area and volume
A sphere is perfectly symmetrical about its centre. Surface area formula: 4πr². Volume formula: (4/3)πr³. Increasing radius causes surface area to increase with r² and volume to increase with r³, so volume grows faster than surface area as size increases. Formulas assume a complete sphere. Partial spheres (spherical segments) require adjusted formulas and are outside basic formula application.
Pyramids and cones: volume and surface area
A pyramid has a polygonal base and triangular faces meeting at an apex. Volume formula for any pyramid: (1/3) × area(base) × height, where height is perpendicular from apex to base. A cone is a pyramid with a circular base, so cone volume = (1/3)πr²h. Surface area of a pyramid or cone includes base area plus lateral area. Lateral area for a cone equals πrl, where l is the slant height. Lateral areas for pyramids require calculating each triangular face; slant height must be perpendicular to the base edge for each face.
Composite solids: decomposition and recomposition
Composite solids combine basic solids such as prisms, cylinders, cones, spheres and pyramids. Surface area and volume calculations proceed by decomposing the shape into known parts, calculating each part, then summing volumes or summing exposed surface areas. Surface area requires care: internal faces that become shared when parts join are not included. Visual decomposition and labelling of shared and exposed faces prevents double counting or omission.
Units, significant figures and limitations
Volume units are cubic (for example cm³, m³). Surface area units are square (for example cm², m²). Dimensions must use consistent units before substitution; otherwise conversion is necessary. Approximation of π and rounding rules affect final answers. Exact answers using π are acceptable unless the problem specifies a numerical approximation. Formulas assume straight edges for prisms, perpendicular heights where stated, and full closed surfaces unless specified as open.
Key notes
Important points to keep in mind