A triangular pyramid has one triangular base and three triangular faces. Let's take a look at the given diagram.

We can use the net of this

3 -

dimensional shape to find its surface area.

The surface area of a pyramid is the sum of the area of the base and the areas of the lateral faces. Let's calculate the area of the base and the faces one at a time. Then we can add them together.

Base

Looking at the net of this solid, let's think about its base.

We can see that the base of this triangle is

7

centimeters and its height is

6.1

centimeters. Let's find its area by substituting these values into the formula for the area of a triangle.

A = \frac{1}{2} b h

SubstituteII

$b = 7$ , $h = 6.1$

A = \frac{1}{2} (7) (6.1)

▼

Evaluate right-hand side

Multiply

A = \frac{1}{2} (42.7)

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{4 2 . 7}{2}

UseCalc

Use a calculator

A = 21.35

The area of the base is

21.35

square centimeters. Also, notice that all three sides of this triangle are

7

centimeters. This means that it is an equilateral triangle.

Lateral Faces

Because the base of the pyramid is an equilateral triangle, we know that all three of the lateral faces are congruent.

Since they are all exactly the same size, we only need to calculate the area of one of the faces. We can see in the diagram that the base of the lateral sides is

7

centimeters and the height is

10

centimeters. We can substitute these values into the formula for the area of a triangle.

A = \frac{1}{2} b h

SubstituteII

$b = 7$ , $h = 10$

A = \frac{1}{2} (7) (10)

▼

Evaluate right-hand side

Multiply

A = \frac{1}{2} (70)

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{7 0}{2}

CalcQuot

Calculate quotient

A = 35

The area of one lateral face is

35

square centimeters. Let's add this value

3

times to get the area of all three lateral faces combined.

Surface Area of the Three Lateral Faces 35 + 35 + 35 = 105 square centimeters

Surface Area of the Pyramid

Finally, to find the surface area of the triangular pyramid, we will add the area of the base and the area of the three lateral faces.

Surface Area of the Triangular Pyramid 21.35 + 105 = 126.35 square centimeters

Exercise