SURFACE AREA, VOLUME AND CAPACITY (Cuboid, Cube and Cylinder)

200. The Cube's Secrets · Volume, Surface Area, and Diagonals

Volume = a³, TSA = 6a², LSA = 4a², Diagonal = a√3 always hold for any cube.

aaaDiagonal = a√3V = a³TSA = 6a²LSA = 4a²a = 150 cma = 150 cmV = 3375000 cm³V = 3375000 cm³TSA = 135000 cm²TSA = 135000 cm²LSA = 90000 cm²LSA = 90000 cm²D = 259.8 cmD = 259.8 cm
Cube formulas (side a): Volume = a³, Total Surface Area = 6a² (6 squares), Lateral Surface Area = 4a² (4 vertical faces, excluding top and bottom), Diagonal = a√3 (space diagonal corner to opposite corner).

Stuck? Ask Guru

Selina ICSE: SURFACE AREA, VOLUME AND CAPACITY (Cuboid, Cube and Cylinder)

What this lesson covers

Try to break it

Drag A to move the cube around the canvas. The edge length a stays fixed, so volume = a³, total surface = 6a², lateral surface = 4a², and the body diagonal = a√3 don't change. Try to find a position where one of those readouts disagrees with its formula — impossible.

How you build it

Construct a cube.

  • Draw the front face — square ABCD (side a), the face closest to you.
  • Draw the back face — square EFGH, the same size as ABCD, offset up and to the right to show depth.
  • Mark point A — the front bottom-left corner.
  • Mark point B — the front bottom-right corner.
  • Mark point C — the front top-right corner.
  • Mark point D — the front top-left corner.
  • Mark point E — the back bottom-left corner (behind A).
  • Mark point F — the back bottom-right corner (behind B).
  • Mark point G — the back top-right corner (behind C).
  • Mark point H — the back top-left corner (behind D).
  • Connect A to E — the first depth edge.
  • Connect B to F.
  • Connect C to G.
  • Connect D to H to complete the cube.
  • Draw the space diagonal BH — from front-bottom-right B to back-top-left H. Its length is a√3.

The proof, step by step

Prove that a cube has volume a³, total surface area 6a², and diagonal a√3.

  • A cube has 6 identical square faces, each with area a².
  • Total Surface Area (TSA) is the sum of areas of all 6 faces: 6 × a² = 6a².
  • Lateral Surface Area (LSA) covers the 4 vertical faces: 4 × a² = 4a².
  • Volume is the space occupied: length × breadth × height = a × a × a = a³.
  • The face diagonal splits a square face into two right triangles, giving a√2 by Pythagoras.
  • The space diagonal forms a right triangle with a face diagonal and an edge, giving a√3.

Worked example

The edge of a cube is 7 cm. Find its total surface area.

TSA = 6a² = 6 × 7² = 6 × 49 = 294 cm².

  • 147 cm²
  • 294 cm² — correct
  • 343 cm²
  • 196 cm²
Hold to talk

Subscription Status