Volume Of Cube Formula and Area Of Isosceles Triangle Formula

Volume of Cube Formula

A cube is a three-dimensional shape with equal width, height, and length measurements. A cube has six square faces, and each face will have a side of equal length. Finding the volume of a cube is sometimes very much essential. Generally to find the volume of a cuboid we needed to multiply its length × width × height. In this article, we will explore the formula used to calculate the volume of a cube. The student will also gain a conceptual understanding for volume of cube formula and appropriate units to use. Let us start learning!

Volume of a cube formula

What is Cube?

What do children’s building blocks, a milk crate, and dice have all in common? Also, what is unique about the shape of their sides? All such objects are the example of a perfect cube. A cube is a unique three-dimensional shape that has squares for all six of its sides.

The volume of a cube will define the number of cubic units, that cube will occupy. A cube has a solid three-dimensional figure, which has 6 square faces or sides. Many objects like water tank, ludo dice etc are found in the shape of a cube.

The formula for Volume of Cube:

We can easily find the volume of the cube by knowing the length of its one edge. Suppose, the length of the edges of the cube is ‘a’. Then the V will be the product of length, height, and width. Its derivation is given below.

Derivation for Volume of a Cube:

The volume of an object is defined as the amount of space that a solid occupies. As we know that a cube is a 3-dimensional object with all equal sides i.e. length, breadth, and height. Cube volume formula derivation is as follows:

• Consider a square piece of paper.
• The area that square sheet will take up will be its surface area i.e. its length X breadth.
• As the square will have equal length and breadth, so the surface area will be a².
• Now, a simple cube can be made by stacking multiple square sheets on top of each other to get height as a unit.
• Hence it can be concluded that the overall area covered by the cube will be the area of the base multiplied by the height.
• So, Volume of Cube,

V = a² X a

i.e. V = a³

Where,

V	Volume of cube
a	Length of one side

Solved Examples

Q.1: Find the volume of a cube with sides of length 7 cm.

Solution:

Given, the length of the sides of the cube is 7cm.

We know, V =  a³

Therefore, Volume, V = (7 cm)³

So, Volume V  = 343 cubic cm.

Q.2: Find the length of the edges of the cube, if its volume is equal to 125 cubic cm.

Solution:

Given, Volume of the cube = 125 cubic cm.

Let the length of the edges is a.

We know,

The volume V =  a³

Substituting the value, we get,

V = a³

Or

a = V−−√3

= 125−−−√3

= 5 cm

Therefore, the length of the cube is 5 cm.

Area of Isosceles Triangle Formula

The word isosceles triangle is a type of triangle, it is the triangle that has two sides the same length. If all three sides are equal in length then it is called an equilateral triangle. Obviously all equilateral triangles also have all the properties of an isosceles triangle. In this article, we will discuss the isosceles triangle and area of isosceles triangle formula. Let us begin learning!

Area of Isosceles Triangle Formula

Definition of Isosceles Triangle:

An isosceles triangle is a triangle with two sides of equal length and two equal internal angles adjacent to each equal sides. It is unlike an equilateral triangle where we can use any vertex to find out the altitude. Thus in an isosceles triangle, we have to draw a perpendicular from the vertex which is common to the equal sides.

Therefore, in an isosceles triangle, two equal sides join at the same angle to the base i.e. the third side. These special properties of the isosceles triangle will help us to calculate its area from just a couple of pieces of information.

Let us learn the methods to find out the area, altitude, and perimeter of such an isosceles triangle.

Properties:

• The unequal side of an isosceles triangle is normally referred to as the ‘base’ of the triangle.
• The base angles of the isosceles triangle are always equal.
• If the 3rd angle is a right angle, it is called a “right isosceles triangle”.
• The altitude of a triangle is a perpendicular distance from the base to the topmost

Procedure to compute the area of an isosceles triangle:

Step-1: Find the isosceles triangle’s base. The base is the easy part, and just use the third unequal side as the base. The length of base will be b.

Step-2: Draw a perpendicular line between the base to the opposite vertex.  The length of this line will be the height of the triangle, so label it as h. After computing h we can find the area. In the isosceles triangle, this line will always hit the base at its exact midpoint.

Step-3: The perpendicular line will divide the triangle into two equal right-angled triangles. The hypotenuse s of the right triangle is one of the two equal sides of the isosceles. Its base will be half of the base i.e. b/2. Thus using the Pythagorean Theorem we will determine the length of perpendicular i.e. h using,

h= s2−(b2)2−−−−−−−−√

Where,

S	The hypotenuse of a right-angled triangle
B	Length of base
H	Height of triangle

Step-4: Put the base and height into the area formula. The formula is as follows:

A = bXh2

A	Area of the isosceles triangle
B	Length of base
H	Height of triangle

Remember to write the answer in terms of square units.

Solved Examples

Q.1: Find the area, altitude, and perimeter of an isosceles triangle given a = 5 cm, b = 9 cm, c =5 cm?

Solution:

Given,

a = 5 cm

b = 9 cm

C= 5 cm.

Perimeter of an isosceles triangle

= a + b + c cm

= 5 + 9 + 5 cm

= 19 cm

Height of an isosceles triangle

h = s2−(b2)2−−−−−−−−√

= 52−(92)2−−−−−−−−√

= 25–814−−−−−√

= 2.18 cm

Area of an isosceles triangle,

A = bXh2

= 9X2.182  cm²

= 9.81 cm²