# Volume of a Triangular Pyramid Lesson

The volume of a triangular pyramid is the measure of the space enclosed by a triangular pyramid. A triangular pyramid is a three dimensional figure where all faces are triangles. It has a triangular base and three triangular faces which meet at one vertex. If it is a regular triangular pyramid, that indicates that the base is equilateral and the three faces are congruent.

We calculate the **volume of a triangular pyramid** by multiplying the area of the base by the height and dividing the product by 3. The volume of a triangular pyramid is measured in cubic units.

V = ⅓ Bh

Where B represents the area of the base triangle and h is the height of the pyramid. In order to find B, the area of the base, you will need to use the formula for the area of a triangle.

A = ½ Bh

Where b is the length of the base and h is the height of the base.

Use the following process for finding the **volume of a triangular pyramid.**

- Find the area of the base of the pyramid.
- Determine the height of the pyramid.
- Use the formula V = ⅓ x Area of the Base x Height
- Label your final answer with the appropriate cubic units.

**Examples**

**Example 1:** Find the volume of a triangular pyramid with a height of 4 cm and an area of the base of 27 cm.

**Solution:** Start by recognizing the information that is given. We have the area of the base and the height of the pyramid. Therefore, we will substitute those values into the formula.

V = ⅓ Bh

V = ⅓ 27(4)

V = 9(4)

V = 36

The volume of the triangular pyramid is 36 cm^{3}.

**Example 2:** Find the volume of the triangular pyramid, with a base height of 15 in, the length of the base is 12 ft, and a pyramid height of 18 in.

**Solution:** Start by finding the area of the base of the pyramid. To do this, we need to use the formula

V = ⅓ Bh, where h is the height of the base of the triangle, in this case 15 in, and the length of the base of the triangle is 12 in. Therefore, the area of the base of the triangle is

A = ½ 12(15)

A = 6(15)

A = 90

We will then use the area of the base in the formula for the volume of a triangular pyramid.

V = ⅓ Bh

V = ⅓ 90(18)

V = 30(18)

V = 540

Therefore, the volume of the pyramid is 540 in^{3}.

**Example 3:** The base of a pyramid is a triangle with a base of 5 ft and a height of 7 ft. FInd the volume of the pyramid, if its height is 16 ft. Round your answer to the nearest tenth, if necessary.

**Solution: **Start by finding the area of the base of the pyramid, to do this we will multiply the base of the triangle and the height of the triangle, then divide the product by 2. A^{base} = ½ x 5 x 7.

A = ½ 5(7)

A = 2.5(7)

A = 17.5

Now that we know that the area of the base is 17.5, we will use that and the height of the pyramid to find the volume.

V = ⅓ 17.5(16)

V = 5.833(16)

V = 93.3

The volume of the triangular pyramid is 93.3 ft^{3}.

**FAQs on Volume of a Triangular Pyramid**

**1) What is a triangular pyramid?**

A triangular pyramid is a three-dimensional geometric shape characterized by a triangular base and triangular faces that meet at a single point called the apex or vertex.

**2) How do you calculate the volume of a triangular pyramid?**

The volume V of atriangular pyramid can be calculated using the formula:

V = ⅓ × Area of the Base × Height of the Pyramid

where the base area is based on the area of the triangular base, and the height is the perpendicular distance between the base and the vertex.

**3) What is the formula for the area of a triangular base?**

The area A of a triangular base is found by using the standard formula for area of a triangle, which is A = ½ × base length × height of base. If the base is an equilateral, you can use the following formula to find the area of an equilateral triangle:

**4) Can you explain how to find the height of a triangular pyramid?**

To find the height of a triangular pyramid, draw a perpendicular line from the apex to the base.

The length of this line is the height of the pyramid.

**5) What units should I use when calculating the volume of a triangular pyramid?**

The units used for calculating the volume of a triangular pyramid depend on the units used for the measurements of the base length, base height, and height of the pyramid. Ensure that all measurements are in the same units for accurate results.

**6) Are there any practical applications of calculating the volume of a triangular pyramid?**

Yes, the volume of a triangular pyramid is often used in architecture and engineering for calculating the volume of structures with triangular pyramid shapes, such as roofs, tents, and certain types of buildings.

**7) Can the triangular pyramid have a slanted height instead of a perpendicular height?**

Yes, in some cases, the height of the triangular pyramid may be slanted instead of perpendicular to the base. In such cases, the perpendicular height is typically used in the volume formula for accurate calculations.

**8) Can I use the volume formula for other types of pyramids?**

Yes, the volume formula for a triangular pyramid is part of a general formula that can be applied to other types of pyramids with different base shapes, such as square pyramids or pentagonal pyramids.