Question

Use the following information. The surface area of a cylinder equals the lateral surface area \((2 \pi r \cdot h)\) plus the area of the two bases \(\left(2 \cdot \pi r^{2}\right)\). Write the expression for the surface area of a cylinder.

Short answer

The surface area of a cylinder can be expressed with the formula \(2 \pi r \cdot h + 2 \cdot \pi r^{2}\).

Step-by-step solution

Understand the Concept

A cylinder has three parts - two bases which are circles and a curved lateral surface. The surface area of a cylinder is the sum of the areas of the lateral surface and the two bases.

Formulate the Lateral Surface Area

The lateral surface area of a cylinder is given by the formula: \(2 \pi r \cdot h\), where \(r\) is the radius of the base and \(h\) is the height of the cylinder.

Formulate the Area of the Bases

The area of each base (a circle) is \(\pi r^{2}\), so the total area of the two bases together is \(2 \cdot \pi r^{2}\).

Combine the Formulas

The total surface area of a cylinder is the sum of the lateral surface area and the area of the two bases. The combined formula for this is therefore: \(2 \pi r \cdot h + 2 \cdot \pi r^{2}\).

Key concepts

Lateral Surface Area

When we talk about the lateral surface area of a cylinder, we are referring to the area of the side that wraps around the cylinder's height. Picture it as the label on a soup can. To calculate this, we use the formula for the circumference of a circle \( 2 \times \pi \times r \) — which is the distance around the base — and multiply it by the height \( h \) of the cylinder. This gives us \( 2 \times \text\theta \times \pi \times r \times h \).

Understanding the lateral surface area is crucial because it represents the majority of the material that would be used to construct the cylinder, especially in practical applications like manufacturing.

Area of a Circle

For a cylinder, the top and bottom shapes are circles. The area of a circle is essential knowledge in geometry, and it's calculated using the formula \( \pi r^2 \), where \( r \) denotes the radius of the circle, or the distance from the center to the edge.

Here's a simple way to visualize why this formula makes sense: imagine slicing up a circle into tiny pieces and rearranging them to form a shape that closely resembles a rectangle. The sides of this 'rectangle' would be half the circumference of the original circle \( \pi \times r \), and the 'height' would be the radius \( r \). When you multiply these dimensions, you get \( \pi \times r^2 \). This is intuitive and helps understand why we square the radius and multiply by \( \pi \) to find the circle's area.

Algebraic Formulas

Algebraic formulas are the recipes for solving mathematical problems. They're like a blueprint that tells us how to find unknown quantities or how to express relationships between different quantities. When dealing with the area of shapes, especially for geometric shapes like cylinders, algebra helps translate the dimensions and shapes into a solvable equation.

Take the surface area of a cylinder as an example: by using algebra, we combined the lateral area and the area of the two bases into the succinct formula \( 2 \pi r \times h + 2 \pi r^2 \). This algebraic formula is practical, allowing us to plug in the known measurements (radius and height) to obtain the surface area of any cylindrical object.

Geometric Shapes

Geometry is packed with various shapes, each having unique properties and formulas to calculate measures like area and volume. Cylinders are 3-dimensional shapes with two congruent circular bases linked by a curved surface.

Understanding the components of geometric shapes like cylinders allows us to break down their complexity. Knowing that a cylinder is essentially two circles and a rectangle bent into a circle (if we unfold the lateral surface) helps conceptualize how its surface area is calculated. Awareness of these geometrical nuances is critical to mastering how to compute areas and volumes, which is crucial for fields ranging from architecture to engineering and beyond.