Trapezoidal Prism

Trapezoid Prism or Trapezoidal Prism is a geometric solid characterized by its unique trapezoidal base and parallel rectangular faces. A trapezoid prism has this special characteristic of having trapezoidal bases, unlike other prisms that may have rectangular or square bases.

In this article, we will understand the concepts of trapezoidal prism, including its definition, properties, construction, and formulas, providing a comprehensive understanding of this three-dimensional shape.

What is a Trapezoid Prism?

Trapezoid Prism is a 3D geometric shape characterized by two parallel trapezoidal bases and rectangular or parallelogram faces connecting the corresponding sides. It belongs to the category of prisms, which are polyhedra with two identical parallel bases and rectangular faces.

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Trapezoidal Prism Definition

Trapezoidal Prism is a 3D shape with two trapezoidail face joined by 4 rectangular surfaces.

Trapezoidal Prism Faces, Edges And Vertices

A trapezoidal prism is a three-dimensional geometric shape characterized by specific features:

Number of Faces, Edges and Vertices in Trapezoidal Prism
Property	Numbers
Faces	6 (2 trapezoidal faces, 4 rectangular faces)
Edges	12
Vertices	8

Properties of Trapezoidal Prism

Below listed are the properties of Trapezoidal Prism:

• Trapezoid Prisms have parallel trapezoidal bases
• They have rectangular faces
• Trapezoid Prisms have equal edge lengths connecting corresponding vertices.

Cross-Section of Trapezoid Prism

Cross-section of a Trapezoidal Prism exhibits a trapezoidal shape, providing a visual representation of the prism’s unique base structure. Trapezoid Prism structure is made of a polyhedron with two parallel trapezoidal bases and rectangular faces connecting the corresponding sides of the bases. The cross-section of a Trapezoid Prism reveals a trapezoidal shape showcasing the distinct features of its base.

Trapezoidal Prism Net

The net of a Trapezoid Prism unfolds into trapezoidal and rectangular sections showcasing how the prism can be flattened into a 2D representation. The three-dimensional geometric figure of Trapezoidal Prism is flattened into 2D highlighting the two parallel trapezoidal bases and rectangular faces connecting the corresponding sides.

Shape of a Trapezoid Prism

The trapezoidal shape of the base influences the appearance of the Trapezoid Prism distinguishing it from other geometric solids. The shape of a trapezoid prism includes two parallel trapezoidal bases. Connecting the corresponding sides of the trapezoids are rectangular or parallelogram faces. Also, the connecting faces are perpendicular to the bases forming a solid and closed structure. Thus, it will give Trapezoidal Prism a three-dimensional geometric figure with two parallel trapezoidal bases and rectangular faces connecting the corresponding sides.

Construction of Trapezoid Prism

To construct a Trapezoidal Prism, draw the trapezoidal base on a plane, extend its sides to create the prism’s height and connect corresponding vertices to form rectangular faces. Below are the construction steps of trapezoid prism :

Step 1. Draw the Trapezoidal Base: Sketch a trapezoid on a plane so that its sides and angles align with the desired dimensions.

Step 2. Extend the Sides: Extend lines from each trapezoidal side to create the prism’s height.

Step 3. Connect Corresponding Vertices: Join the corresponding vertices of the trapezoidal bases to form rectangular faces.

Applications of Trapezoid Prism

Trapezoid prisms find application in various fields, including: Architecture, Engineering, Geometry Education, Optics etc. Below listed are the applications of Trapezoid Prism:

• Architecture: Used in the design of buildings and structures.
• Engineering: Employed for creating components in machinery.
• Geometry Education: A useful example for teaching 3D geometry concepts.
• Optics: Applied in the design of optical devices.

Trapezoid Prism Volume

Trapezoid Volume Formula is given as:

V = 1/2 × (a+b) × h × l

where

• a and b are the lengths of the trapezoidal bases,
• h is the height,
• l is the slant height.

Example: Find the Volume of Trapezoid when the bases are 4 cm and 3 cm, height is 6 cm and slant height is 8 cm then calculate the volume of trapezoid prism.

Solution:

h=6cm (height)

a = 4cm (length of the longer base)

b = 3cm (length of the shorter base)

l = 8cm (slant height).

Volume (V) of trapezoid prism is calculated by formula:

V = 1/2 × (a+b) × h × l

Substitute values in the above formula, we get:

V = 1/2 × (4+3)cm × 6cm × 8cm

V =168 cm³

Trapezoidal Prism Surface Area

Like every 3D shape, Trapezoid Prism has two types of surface areas:

• Lateral Surface Area of Trapezoid Prism
• Total Surface Area of Trapezoid Prism

Lateral Surface Area of Trapezoid Prism

LSA = (a + b + c + d) × H

where a, b, c, d are the lengths of the trapezoidal bases and h is the height.

Example: Given H = 6 meter and the sides of trapezoidal are a = 4m, b = 3m, c = 5m and d = 8m then find the LSA of the trapezoid prism.

Solution:

Lateral Surface Area (LSA) of trapezoid prism is calculated by formula:

LSA = (a + b + c + d) × H

Substitute values in the above formula, we get:

LSA = (4 + 3 + 5 + 8) m × H

LSA = (4 + 3 + 5 + 8) m × H

= 20 m × (6)m

= 120 m²

Total Surface Area of Trapezoid Prism

TSA = h (b + d) + l (a + b + c + d)

where a, b, c, d are the lengths of the trapezoidal bases, h is the height and l is the slant height.

Example: Given height = 5m, slant height = 10m and the sides of trapezoidal are a = 4m, b = 6m, c = 5m and d = 8m then find the TSA of the trapezoidal prism. Then Calculate the Total Surface Area (TSA) of trapezoid prism.

Solution:

Total Surface Area (SA) of trapezoid prism is calculated by formula:

TSA = h (b + d) + l (a + b + c + d)

Substitute values in the above formula, we get:

TSA = h (6m + 4m) + l (4m + 6m + 5m + 8m)

= h×(10m) + l×(23m)

= 5m × (10m) + 10m × (23m)

= 50m² + 230m²

= 280m²

Trapezoid Prism Formulas

The Trapezoid Prism formula includes

Formula Name	Formula Expression
Volume of Trapezoid Prism	V = 1/2 × (a+b) × h × l
LSA of Trapezoid Prism	(a + b + c + d) × H
TSA of Trapezoid Prism	h (b + d) + l (a + b + c + d)

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Trapezoid Prism – Solved Examples

Example 1: Given a trapezoid prism with bases of lengths 6 cm and 8 cm, a height of 10 cm, and a slant height of 5 cm, calculate its volume.

Solution:

Given

• Bases of Prism = 6 cm and 8 cm
• h= 10 cm
• l= 5cm

Volume = 1/2 × (a+b) × h × l

V = 1/2 × (6+8) × 10 × 5

V = 70 cm³ 

Example 2: Given height = 4m, slant height = 10m and the sides of trapezoidal are a = 5m, b = 7m, c = 4m and d = 8m then find the TSA of the trapezoidal prism. Then Calculate the Total Surface Area (TSA) of trapezoid prism.

Solution:

Total Surface Area (SA) of trapezoid prism is calculated by formula:

TSA = h (b + d) + l (a + b + c + d)

Substitute values in the above formula, we get:

TSA = h (7 m + 8 m) + l (5 m + 7 m + 4 m + 8 m)

= h×(15 m) + l×(24 m)

= 4 m × (15 m) + 10 m × (24 m)

= 60 m² + 240 m²

= 300 m²

Example 3: A trapezoid prism has a volume of 48 cm³, height= 2 cm and its bases have lengths of 8 cm and 4 cm. Find the slant height.

Solution:

Given,

Volume = 48 cm³

Bases Lengths of Prism = 8 cm and 4 cm

h= 2 cm

Volume = 1/2 × (a+b) × h × l

48 = 1/2 × (8+4) × 2 cm × l

l = 48/12

l = 4 cm

Example 4: Find the lateral surface area of a trapezoid prism with bases of lengths 4cm, 5 cm, 4 cm and 7 cm and a height of 9 cm.

Solution:

Given,

• Trapezoid Prism with bases of lengths = 5 cm and 7 cm
• Height = 9 cm

Lateral SurfaceArea= (Perimeter of Trapezoidal Base) × Height

Lateral SurfaceArea = (4cm + 5 cm + 4cm + 7cm) × 9cm

LSA = (20cm) × 9 cm

LSA = 180 cm²

Example 5: A trapezoid prism has a volume of 420 cm³, height= 2 cm and its bases have lengths of 18 cm and 22 cm. Find the slant height.

Solution:

Given,

• Volume = 420 cubic cm³
• Bases Lengths of Prism = 18 cm and 22 cm
• h= 2 cm
• Volume = 1/2 × (a+b) × h × l

420 = 1/2 × (18+22) × 2 cm × l

l = 420/60

l = 7 cm

Trapezoid Prism – Practice Problems

Q1. Given a trapezoid prism with bases of lengths 7 cm, 5 cm, 4cm, 8cm, height of 12 cm and a slant height of 13 cm, calculate its LSA.

Q2. A trapezoid prism has a volume of 400 cubic units, and its bases have lengths of 30m and 10m. Find the height.

Q3. Given a trapezoid prism with bases of lengths 5cm, 3 cm, 6cm, 8cm, height of 10 cm and a slant height of 12 cm, calculate its LSA.

Q4. Given a trapezoid prism with bases of lengths 5 cm, 4 cm, 3cm, 6cm, height of 8 cm and a slant height of 12 cm, calculate its TSA.

Q5. Given a trapezoid prism with bases of lengths 3 cm and 4 cm, a height of 5 cm and a depth of 2 cm, calculate its volume.

Trapezoid Prism – Frequently Asked Questions

1. How is a Trapezoid Prism Different from a Rectangular Prism?

A trapezoid prism has trapezoidal bases, while a rectangular prism has rectangular bases.

2. Can a Trapezoid Prism have more than Two Trapezoidal Faces?

No, a trapezoid prism has two trapezoidal bases, but the connecting faces are typically rectangles or parallelograms.

3. How is Trapezoid Prism different from other Prism Shapes?

A trapezoid prism is characterized by having trapezoidal bases, unlike other prisms that may have rectangular or square bases.

4. Can a Trapezoid prism have Equal Length bases?

Yes, a trapezoid prism can have equal-length bases, which would essentially make it a rectangular prism.

5. Is it Possible for a Trapezoid Prism to have more than Six Faces?

No, a trapezoid prism inherently has six faces: two trapezoidal bases and four rectangular or parallelogram faces connecting corresponding sides of the bases.

6. Does a Trapezoidal Prism Have Volume?

Yes, a trapezoidal prism does have volume.

7. What is the Formula To Find the Volume of a Trapezoidal Prism?

The formula to find the volume of a trapezoidal prism is V = (1/2) × (a + b) × h × l, where ‘a’ and ‘b’ are the lengths of the two parallel bases (the trapezoid’s sides), ‘h’ is the height of the trapezoid, and ‘l’ is the length of the prism.

8. How Can You Find the Volume of a Trapezoidal Prism when the Height is given?

If the height of the trapezoidal prism is given, you can still use the same formula: V = (1/2) × (a + b) × h × l. Simply substitute the known values for ‘a,’ ‘b,’ ‘h,’ and ‘l’ into the formula and solve for the volume.