# Cuboid Shape

By the end of this lesson

You wiil be introduced to cuboid faces, edges and vertices.

You will be familiar to a set of formulas for surface area, diagonal and volume of a cubiod.

You will have learned how to solve cuboid problems with examples.

## Cuboid

A cuboid is a polyhedron formed by six rectangles that are placed at right angles.

A cuboid showing length, width and height.

So, a cuboid is a polyhedron of rectangular faces. All rectangles are also known as faces of a cuboid. There are three pairs of opposite faces. The opposite faces or rectangles of each pair are congruent. All adjacent faces or rectangles are at right angles and as a result, several edges and vertices are made.

A cuboid must have

- 6 faces
- 12 edges
- 8 vertices.

## Area of Cuboid

A cuboid is showing its length, width and height.

The area of a cuboid means the sum of the area its total faces. A cuboid is made of 3 pairs of rectangles that are placed opposite. And each rectangle is congruent to its opposite.

Let the length, width and height of a cuboid be a, b and c respectively.

∴ Area of top face = length × width = ab square unit

∴ Area of top and bottom faces = (ab + ab) square unit

Again, area of right face = width × height = bc square unit

∴ area of right and faces = (bc + bc) square unit

At last, area of back face = length × height = ca square unit

∴ area of back and front faces = (ca + ca) square unit

So, total surface area = sum of the three pair of congruent faces

∴ If the area of total surface area is A,

A = (ab+ab+bc+bc+ca+ca) square unit

or, A = (2ab+2bc+2ca) square unit

∴ A = 2(ab+bc+ca) square unit

If the lengths of edges of a cuboid are a, b, c and the total surface area is A,

A = 2(ab+bc+ca) square unit

A cuboid is showing its edges and diagonal.

Let the length, width and height of a cubiod are a, b and c respectively.

∴ Volume of cubiod = (area of base) × height

or, Volume of cubiod = ab × c cubic unit

∴ Volume of cubiod = abc cubic unit

∴ Volume of cubiod = (length × width × height) cubic unit

If a, b, c are the three edges and V is the volume of a cubboid,

V = abc cubic unit

## Diagonal of Cuboid

First, try to determine the diagonal of bottom face.

Tf length and width of a rectangle are a and b, its diagonal is √a^{2} + b^{2} because the triangle is formed with the sides a, b and √a^{2} + b^{2} is a right triangle whose hypotenuse is √a^{2} + b^{2}.

In figure, the diagonal of cuboid is d.

Again, the triangle is formed with the sides c, d and √a^{2} + b^{2} is a right triangle.

Applying Pythagorean theorem,

d^{2} = c^{2} + (√a^{2} + b^{2})^{2}

or, d^{2} = c^{2} + a^{2} + b^{2}

or, d^{2} = a^{2} + b^{2} + c^{2}

∴ d = √a^{2} + b^{2} + c^{2}

If the edges of a cubboid are a, b, c and d is the diagonal,

d = √a^{2} + b^{2} + c^{2} unit