Curved-Surface Solids

Aan Triono included in category Grade 9

2021-01-05 2026-08-08 717 words 4 minutes

A lesson on cylinders as part of curved-surface solids, including elements, surface area, and volume.

Contents

Introduction

Curved-surface solids include cylinders, cones, and spheres. This article focuses on the cylinder, a solid formed by two parallel congruent circles and one curved lateral surface.

A. Basic Competencies

3.7 Generalize the surface area and volume of curved-surface solids such as cylinders, cones, and spheres.
4.7 Solve contextual problems related to the surface area and volume of curved-surface solids and their combinations.

B. Learning Objectives

After studying this lesson, students are expected to be able to:

identify the elements of a cylinder
calculate the base area, lateral area, and top area
calculate the volume of a cylinder
determine cylinder elements when the volume is known
compare cylinder volumes after a change in radius
calculate the change in cylinder volume when the radius changes

C. Lesson Material

1. Elements of a Cylinder

A cylinder is a curved-surface solid formed by two identical parallel circles and a rectangle wrapped around them.

Calculating the area and volume of a cylinder

From the picture we see:

a cylinder has three surfaces: the base, the top, and the curved side
the base and top are circles
the curved side is called the lateral surface
the distance between base and top is the height, denoted by $t$
the radius is denoted by $r$ and the diameter by $d$

2. Cylinder Net

A cylinder net consists of:

two congruent circles
one rectangle

Cylinder net

The net is useful because the surface area of the cylinder is equal to the total area of this unfolded figure.

3. Surface Area of a Cylinder

The surface area is the sum of:

the area of the base circle
the area of the lateral surface
the area of the top circle

Details:

base area = $\pi r^2$
lateral area = circumference of base $\times$ height = $2\pi rt$
top area = $\pi r^2$

So:

\begin{aligned} S &= \pi r^2 + 2\pi rt + \pi r^2\\ &= 2\pi r^2 + 2\pi rt\\ &= 2\pi r(r+t) \end{aligned}

Example

A cylinder has height 13 cm and base radius 7 cm. Find its surface area.

\begin{aligned} S &= 2\pi r(r+t)\\ &= 2 \times \frac{22}{7} \times 7 \times (7+13)\\ &= 44 \times 20\\ &= 880 \end{aligned}

Therefore:

880\ \text{cm}^2

4. Volume of a Cylinder

Since the base is a circle, the volume of a cylinder is the base area times the height.

Formula:

V = \pi r^2 t

or, in terms of diameter:

V = \frac{1}{4}\pi d^2 t

Example 1

A cylinder has radius 14 cm and height 20 cm. Find its volume.

\begin{aligned} V &= \pi r^2 t\\ &= \frac{22}{7}\times 14^2 \times 20\\ &= 12{,}320 \end{aligned}

So the volume is:

12{,}320\ \text{cm}^3

Example 2

A cylindrical drinking container has volume 693 ml. If its height is 18 cm and the container is full, determine:

the diameter
the total surface area

Since 1 ml = 1 cm^3, we have:

V = 693\ \text{cm}^3

a. Diameter

\begin{aligned} V &= \pi r^2 t\\ 693 &= 3.14 \times r^2 \times 18\\ 693 &= 56.52r^2\\ r^2 &= \frac{693}{56.52}\\ r^2 &\approx 12.26\\ r &\approx 3.5 \end{aligned}

So:

d = 2r = 7\ \text{cm}

b. Surface area

\begin{aligned} S &= 2\pi r(r+t)\\ &= 2 \times \frac{22}{7} \times 3.5 \times (3.5+18)\\ &= 473 \end{aligned}

Therefore:

473\ \text{cm}^2

Important Formulas

base area = $\pi r^2$
lateral area = $2\pi rt$
top area = $\pi r^2$
total surface area = $2\pi r(r+t)$
surface area without top = $\pi r(r+2t)$
volume = $\pi r^2t$

GeoGebra Simulation

Use the following simulation to explore how changes in radius and height affect the surface area and volume of a cylinder.

D. Evaluation

Try the following problems:

A cylinder has height 14 cm and base radius 3 cm. Find its volume.
A cylinder is made of sheet metal with radius 14 cm and height 20 cm. Find the sheet area needed to make it.
A cylindrical water tank has height 2 m and diameter 7 dm. The bottom leaks and water flows out at an average rate of 5 liters per minute. If the tank is full, after how many minutes will it be empty?

Closing

The cylinder is one of the most common curved-surface solids in daily life, appearing in cans, pipes, glasses, and water tanks. Understanding its elements, surface area, and volume makes many applied geometry problems much easier to solve.