Right Prisms and Cylinders Revision Notes for Grade 10 NSC Matric Mathematics

Right Prisms and Cylinders

What is a right prism?

A right prism is a three-dimensional geometric solid that has a polygon as its base and vertical faces that are perpendicular to the base. The base and top surface are identical in shape and size. It is called a "right" prism because the angles between the base and the side faces are all right angles (90°).

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The key feature that makes a prism "right" is that all the angles between the base and the side faces are exactly 90 degrees. This creates perfectly vertical side faces that are perpendicular to both the top and bottom surfaces.

Types of right prisms

Different types of right prisms are named according to the shape of their base:

Triangular prism: has a triangle as its base
Rectangular prism: has a rectangle as its base
Cube: a special rectangular prism where all sides are equal length
Cylinder: has a circle as its base (technically not a prism but follows the same principles)

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While a cylinder is technically not a prism because it has a curved surface rather than flat polygonal faces, it follows the same mathematical principles for calculating surface area and volume as right prisms.

Surface area of prisms and cylinders

Surface area is the total area of all the exposed or outer surfaces of a three-dimensional shape. To find the surface area, we need to calculate the area of each face and add them all together.

Understanding nets

The easiest way to understand surface area is to imagine unfolding the prism into a flat pattern called a net. When a prism is unfolded, you can clearly see each face and calculate the area of each one separately.

Net patterns for different shapes:

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Common Net Patterns:

Rectangular prism net: Made up of six rectangles
Cube net: Made up of six identical squares
Triangular prism net: Made up of two triangles and three rectangles
Cylinder net: Made up of two identical circles and one rectangle

Calculating surface area

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Standard Process for Finding Surface Area:

Sketch and label the net of the prism
Find the area of each different shape in the net
Add up all the areas

This three-step process works for all types of prisms and cylinders.

Worked example 1: Surface area of a rectangular prism

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Worked Example: Surface Area of a Rectangular Prism

Question: Find the surface area of a rectangular prism with dimensions 5 cm × 2 cm × 10 cm.

Solution:

Step 1: Sketch the net of the prism The net shows one large rectangle and two small rectangles.

Step 2: Calculate the areas

Large rectangle = perimeter of small rectangle × length
Large rectangle = (2 + 5 + 2 + 5) × 10 = 14 × 10 = 140 cm²
Small rectangles = 2 × (5 × 2) = 2 × 10 = 20 cm²

Step 3: Add the areas together Surface area = 140 + 20 = 160 cm²

Worked example 2: Surface area of a triangular prism

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Worked Example: Surface Area of a Triangular Prism

Question: Find the surface area of a triangular prism with base 8 cm, height 3 cm, and length 12 cm.

Solution:

Step 1: Sketch the net The net consists of a large rectangle and two triangular ends.

Step 2: Find missing measurements using Pythagoras' theorem For the triangle with base 8 cm and height 3 cm:

$x^2 = 3^2 + \left(\frac{8}{2}\right)^2 = 9 + 16 = 25$

Therefore $x = 5$ cm

Perimeter of triangle = 5 + 5 + 8 = 18 cm

Step 3: Calculate areas

Large rectangle = perimeter of triangle × length = 18 × 12 = 216 cm²
Triangle area = $\frac{1}{2} × 8 × 3 = 12$ cm²
Two triangles = 2 × 12 = 24 cm²

Step 4: Total surface area Surface area = 216 + 24 = 240 cm²

Worked example 3: Surface area of a cylinder

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Worked Example: Surface Area of a Cylinder

Question: Find the surface area of a cylinder with radius 10 cm and height 30 cm.

Solution:

Step 1: Sketch the net

The net consists of two circles and one rectangle.

Step 2: Calculate the areas

Rectangle area = circumference of circle × height = $2πr × h = 2π(10) × 30 = 1884.96$ cm²
Circle area = $πr^2 = π(10)^2 = 314.16$ cm²
Two circles = $2 × 314.16 = 628.32$ cm²

Step 3: Total surface area Surface area = 1884.96 + 628.32 = 2513.3 cm²

Volume of prisms and cylinders

Volume is the three-dimensional space occupied by an object or the contents of an object. Volume is always measured in cubic units (cm³, m³, etc.).

Universal volume formula

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Universal Volume Formula for All Right Prisms and Cylinders:

$\text{Volume} = \text{Area of base} × \text{Height}$

This fundamental formula applies to every type of right prism and cylinder, regardless of the shape of the base.

Volume formulas for specific shapes

Shape	Base	Volume Formula
Rectangular prism	Rectangle	$V = l × b × h$
Triangular prism	Triangle	$V = \frac{1}{2} × b × h × H$
Cylinder	Circle	$V = πr^2 × h$

Worked example 4: Volume of a cube

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Worked Example: Volume of a Cube

Question: Find the volume of a cube with side length 3 cm.

Solution:

Step 1: Find the area of the base Base area = $s^2 = 3^2 = 9$ cm²

Step 2: Apply the volume formula Volume = base area × height = $9 × 3 =$ 27 cm³

Worked example 5: Volume of a triangular prism

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Worked Example: Volume of a Triangular Prism

Question: Find the volume of a triangular prism with base 8 cm, height 10 cm, and length 20 cm.

Solution:

Step 1: Find the area of the triangular base Base area = $\frac{1}{2} × b × h = \frac{1}{2} × 8 × 10 = 40$ cm²

Step 2: Apply the volume formula Volume = base area × length = $40 × 20 =$ 800 cm³

Worked example 6: Volume of a cylinder

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Worked Example: Volume of a Cylinder

Question: Find the volume of a cylinder with radius 4 cm and height 15 cm.

Solution:

Step 1: Find the area of the circular base Base area = $πr^2 = π(4)^2 = 16π$ cm²

Step 2: Apply the volume formula Volume = base area × height = $16π × 15 =$ 754.0 cm³

Key formulas to remember

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Essential Formulas and Methods:

Surface area:

Find the net of the shape
Calculate area of each face
Add all areas together

Volume:

Universal formula: Volume = Area of base × Height
Rectangular prism: $V = l × b × h$
Triangular prism: $V = \frac{1}{2} × \text{base} × \text{height} × \text{length}$
Cylinder: $V = πr^2 × h$

Exam tips

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Critical Exam Strategies:

Always draw and label the net when calculating surface area
For triangular prisms, you may need to use Pythagoras' theorem to find missing side lengths
Remember that cylinders follow the same volume principle as prisms
Check your units - surface area uses square units (cm²) and volume uses cubic units (cm³)
Show all working clearly, step by step

Remember!

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Key Points to Remember:

Right prisms have perpendicular faces and identical top and bottom surfaces
Surface area is found by unfolding into nets and adding all face areas
Volume always equals base area multiplied by height for right prisms and cylinders
Nets help visualise the individual faces of complex 3D shapes
Always check units - surface area in square units, volume in cubic units

Right Prisms and Cylinders (Grade 10 NSC Matric Mathematics): Revision Notes