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 √a2 + b2 because the triangle is formed with the sides a, b and √a2 + b2 is a right triangle whose hypotenuse is √a2 + b2.
In figure, the diagonal of cuboid is d.
Again, the triangle is formed with the sides c, d and √a2 + b2 is a right triangle.
Applying Pythagorean theorem,
d2 = c2 + (√a2 + b2)2
or, d2 = c2 + a2 + b2
or, d2 = a2 + b2 + c2
∴ d = √a2 + b2 + c2
If the edges of a cubboid are a, b, c and d is the diagonal,
d = √a2 + b2 + c2 unit
