Geometry Formulas - Area, Volume, Perimeter

Definition of the perimeter

The perimeter of a shape is defined as the total distance around the shape. It is the length of the outline or boundary of any two-dimensional geometric shape. The perimeter of different figures can be equal in measure depending upon the dimensions. For example, imagine a triangle made of a wire of length L. The same wire can be reused to make a square, considering that all the sides are equal in length.

Definition of area 

In geometry, the area can be defined as the space occupied by a flat shape or the surface of an object.  The area of a figure is the number of unit squares that cover the surface of a closed figure. Area is measured in square units such as square centimetres, square feet, square inches, etc.

The area of the squares below, with unit squares of sides 1 centimetre each, will be measured in square centimetres (cm²).

Definition of volume

Volume is a mathematical quantity that shows the amount of three-dimensional space occupied by an object or a closed surface. The unit of volume is in cubic units such as m3, cm3, in3 etc. Volume is also termed as capacity. A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder or a sphere. Volume of a solid is measured in cubic units.

How do you find the perimeter of a geometric shape?

Perimeter of Rectangle

A rectangle has right angles like a square does, but it has two longer sides that are the same (length) and two shorter sides that are the same (width). If we know the length of one side and the width of another, we can add them together and multiply by 2.

Formula:

P = 2(l + w)

To find the perimeter of the rectangle, we need to have the length of one of the longer sides and the width of one of the shorter sides. We see from the labels that the length is 6 and the width is 3.

P = 2(l + w)

we substitute 6 for the l and 3 for the w: P = 2(6 + 3). Adding 6 and 3 equals 9, so our equation now looks like this: P = 2(9). Multiplying 2 times 9 gives us 18, which is the perimeter of the rectangle.

Example 1

The two sides of the rectangle are given. What will be the perimeter of the rectangle?

 

One side of the rectangle is 2 cm and the other side is 5 cm.

We know that, the perimeter of a rectangle = 2 × (sum of adjacent sides)

Therefore, the perimeter of the rectangle = 2 × (5 + 2) = 2 × (7) = 14 cm

Example 2

If the perimeter of the given rectangle is 10 cm and the length of one of its sides is 2 cm. What will be the other side?

 

The perimeter of the rectangle, with one of the sides equal to 2 cm, is 10 cm.

Let the missing side be ‘a’.

We know that, the perimeter of a rectangle = 2 × (sum of adjacent sides)

10 = 2 × (2 + a) 5 = (2 + a)

 a = 5 - 2 = 3 cm

 How do you find the area of a geometric shape? 

Area of Square

Formula

Area of square = Side² square units

Example 1

Area of a square with a side of 7 cm.

Solution:

Area of a square = side × side. Here, side = 7 cm

Substituting the values, 7 × 7= 49.

Therefore, the area of the square = 49 square cm.

Example 2

Square clipboard whose side measures 120 cm.

Side of the clipboard = 120 cm = 1.2 m

Area of the clipboard = side × side

= 120 cm ×120 cm

= 14400 sq. cm

= 1.44 sq. m

Example 3

The side of a square wall is 75 m. What is the cost of painting it at the rate of Rs. 3 per sq. m?

Side of the wall = 75 m

Area of the wall = side × side = 75 m × 75 m = 5625 sq. m

For 1 sq. m, the cost of painting = Rs. 3

Thus, for 5625 sq. m, the cost of painting = Rs. 3 × 5625 = Rs 16875

How do you find the volume of a geometric shape? 

Volume of Sphere or Circle

The volume V of a sphere is four-thirds times pi times the radius cubed.

Formula:

V=4/3πr

Example 1

Find the volume of the sphere whose radius is 8 m. Round to the nearest cubic meter.

Solution

The formula for the volume of a sphere is

V=43πr3

The radius of the sphere is 8 m.

Substitute 8 for r in the formula.

V=4/3π(8)³

Simplify.

V=4/3π(512)

≈2145

Therefore, the volume of the sphere is about 2145 m³

Example 2

Find the volume of a sphere whose circumference is 144 units.

Solution: Given that the circumference of a sphere is 144 units.

We know the circumference of a circle is given by 2���r, where r is the radius.

Therefore, C = 2���r = 144

Solving this, we get r = 22.92 units.

The formula gives the volume of a sphere, V = 4/3 ��� r3

Putting the value of r, we get, V = 4/3 ��� (22.92)3

V = 50453.197unit³.

Example 3

Find the volume of a sphere if its surface area is 100 square meters.

Solution: We know that the surface area of a sphere is given by S = 4���r, where r is the radius of a sphere.

Therefore, S = 4���r = 100

Finding the value of r, we get, r = 7.96 m

The volume of a sphere is given by, V = 4/3 ��� r3

Putting the value of r, we get,

V = 4/3 ��� (7.96)3 = 2111.58 m3.