Surface Area of a Square Pyramid Calculator | Free Online Surface Area of a Square Pyramid Calculator with Steps - BYJUS

# Surface Area of a Square Pyramid Calculator

The surface area of a square pyramid calculator is a free online tool that can be used to calculate the surface area of the pyramid as well as its base length and slant height. Let us familiarize ourselves with the calculator....Read MoreRead Less

## Online Surface Area of a Square Pyramid Calculator

### How to use the ‘Surface Area of a Square Pyramid Calculator’?

Follow these steps to use the surface area of a square pyramid calculator:

Step 1: Enter any two known measures into the respective input boxes and the unknown measure will be calculated.

Step 2: Select the appropriate units for the input and output.

Step 3: Click on the ‘Solve’ button to obtain the result.

Step 4: Click on the ‘Show Steps’ button to view the stepwise solution applied to find the missing measure.

Step 5: Click on the   button to enter new inputs and start again.

Step 6: Click on the ‘Example’ button to play with different random input values.

Step 7: Click on the ‘Explore’ button to visualize the square pyramid and understand the formulas used in the calculation of surface area.

Step 8: When on the ‘Explore’ page, click the ‘Calculate’ button, if you want to go back to the calculator.

### What is a square pyramid?

A square pyramid is a three dimensional figure that has a square base and four lateral faces that are triangles. These triangles meet at a vertex called the apex. A perpendicular dropped from the apex to the center of the square base is the height of the pyramid. The altitude of the triangular face is the slant height, that is, the perpendicular from the apex to the midpoint of the side of the base.

### Formulas used in the ‘surface area of a square pyramid calculator’

The surface area of a solid is the sum of the areas of all its faces. Hence, the surface area of a square pyramid is the sum of the area of its square base and the area of its four triangular lateral faces.

Consider a square pyramid of side of base b and slant height h.

Surface area,  S = Area of base + Area of four lateral faces

Area of base = $$b^2$$

Area of one lateral face = $$\frac{1}{2}\times b \times h$$

When the surface area S and side of base b is known, slant height h can be calculated as,

Area of one lateral face = $$\frac{S-\text{Area of base}}{4}$$

Therefore, slant height h = $$\frac{\text{Area of one lateral face}\times 2}{b}$$

When the surface area S and slant height h is known, side of base b can be calculated as,

Surface area, S = $$b^2+4\times \frac{1}{2}\times b \times h$$

S = $$b^2+2\times b \times h$$

$$b^2+2bh-S$$ = 0       (1)

Therefore, this is a quadratic equation, where b is the variable.

A quadratic equation $$ax^2+bx+c$$ = 0, can be solved for x as:

x = $$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

On applying the above expression to equation (1):

a = 1, b = 2h and c = – S

The side of the base is a measure of length, hence, the value will always be positive. Therefore, we consider only the positive root of the equation.

Therefore, side of base, b = $$\frac{-2h+\sqrt{(-2h)^2-4\times 1 \times (-S)}}{2\times 1}$$

### Solved Examples

Example 1: Find the surface area of a square pyramid with a base of side length 4 centimeters and with a slant height of 3 centimeters.

Solution:

b = 4 cm

h = 3 cm

S = Area of base + Area of four lateral faces

S = $$b^2+4 \times \left(\frac{1}{2}\times b \times h\right)$$

= $$4^2+4 \times \left(\frac{1}{2}\times 4 \times 3\right)$$

= 40

Therefore, the surface area of the square pyramid is 40 square centimeters.

Example 2: Find the side of the base of a square pyramid whose surface area is 64 square centimeters and slant height is 6 centimeters.

Solution:

S = 64 $$cm^2$$

h = 6 cm

S = $$b^2+2\times b \times h$$

Rearrange the formula of surface area to find the side of base as in:

Side of base, b = $$\frac{-2h+\sqrt{(-2h)^2-4 \times 1 \times (-S)}}{2 \times 1}$$

b = $$\frac{-12+\sqrt{(-12)^2-4(-64)}}{2}$$

b = 4 cm

So, the side of the base is 4 centimeters.

Example 3: Annie traveled to Egypt to see the pyramids. Her tour guide claimed that the surface area of one of the pyramids is 25,620 square meters and has a square base with an approximate side length of 100 meters. Annie wondered what the slant height of the pyramid could be. Help Annie find the answer.

Solution:

S = 25,620 $$m^2$$

b = 100 m

Area of the square base = $$b^2$$

= $$100^2$$

= 10,000 $$m^2$$

Area of one lateral face = $$\frac{S-\text{Area of base}}{4}$$

= $$\frac{25,620-10,000}{4}$$

= 3905 $$m^2$$

Therefore, slant height h = $$\frac{\text{Area of one lateral surface} \times 2}{b}$$

h = $$\frac{3905 \times 2}{100}$$

= 78.1  m

The slant height of the square pyramid is 78.1 meters.