GRE Geometry | Three Dimensional Figures (original) (raw)

Last Updated : 11 Jul, 2025

An object which has only height and length are called 2-dimensional figures while an abject which has height length and width are called 3-dimensional object.

Examples of 2-D figures:

Examples of 3-dimensional figures:

  1. Cube:

  1. In a cube length breadth and height are all equal.

length = breadth = height

  1. let's length, breadth and height as a.
    It has 6 square face each with side length a.

Volume of cube = a * a * a Volume of cube = a3

Surface area of cube = 6 * area of one face Surface area of cube = 6 * a2

  1. Cuboid:

  1. In a cuboid length breadth and height are not equal.
    It has 6 square face,

Volume of cuboid = length * breadth * height

Surface area of cuboid = 6 * area of one face Surface area of cuboid = length * breadth
+ length * breadth
+ breadth * height + breadth * height + length * height + length * height

Surface area of cuboid = 2(length * breadth
+ breadth * height + length * height )

  1. Sphere:

  1. A sphere has radius, volume and surface area:

Volume of sphere = (4 / 3) * π * radius * radius * radius Volume of sphere = (4 / 3) * π * r3

Surface area of sphere = 4 * &pi * radius * radius Surface area of sphere = 4 * &pi * radius2

Volume of hemisphere = (2 / 3) * π * radius * radius * radius Surface area of sphere = 2 * &pi * radius * radius

  1. Cylinder:

  1. A cylinder is a solid or hollow circular shape object with two circular base.

Volume of cylinder = π * radius * radius * h Curved surface area of cylinder = 2 * π * radius * h Total surface area = 2 * π * radius * h + 2 * &pi * radius * radius

  1. Cone:

  1. A cylinder is a solid or hollow with one circular base and which tappers from base to a point.

Volume of a cone = (1 / 3)π * radius * radius * height

Curved surface area of a cone = π * radius * slant height

Total surface area of cone = π * radius * slant height + π * radius * radius

  1. Pyramid:

  1. Pyramid is 3-d object with polygon base, this polygon base is connected to an apex in pyramid.

Volume of a pyramid = (area of base * height) / 3

Examples:

  1. What will be the curved surface area of a cylinder having radius 5cm and height 10cm?

Curved surface area of a cylinder = 2 * π * r * height = 2 * 22 / 7 * 5 * 10 = 314.285714286 cm2

  1. What will be the slant height of a cone which have curved surface area 990 cm2 and radius 5 cm?

Curved surface area of a cone = π * radius * slant height π * radius * slant height = (22 / 7) * 5 * l l = 63cm

  1. What is the volume of a square base pyramid having base side 4 cm and height 6cm?

Volume of a square = area of base * height / 3 = 42 * 6 / 3 = 16 * 6 / 3 = 32 cm3

  1. What will be the volume of a sphere having radius 2.1cm

Volume of a sphere = 4 * π * radius3 / 3 = (4 / 3) * (22 / 7) * 2.13 = 38.808 cm3

  1. What will be the surface area of a cube having side length breadth and height 8 cm?

Surface area of a cube = 6 * side2 = 6 * 82 = 6 * 64 = 384 cm2