Area and Volume

Meaning of area in geometry

Many different area formulas are available to determine the area of various two-dimensional shapes. We can also determine a shape's area by counting the number of unit squares that cover the entire surface of the object. This method helps us to learn and understand the concept of area. See the image below which displays the concept of the square unit. You will find that area formulas are easier and faster to use, however. The unit of area is always measured in square units like square centimeters, square meters, square inches, and so on.

Area of 2D figure, splashlearn.com

Here are some facts about area to keep in mind:

• The area and perimeter of shapes are different calculations and can often be confused with one another.
• The area of any figure is mostly calculated based on the terms like side, length, base, height, and radius.
• The area of two congruent (identical) figures will be the same, but sometimes two non-congruent (not identical) figures might have the same area.

Area formulas for plain figures (2D)

Let's take a look at some of the different formulas for area of basic and common geometric figures, including:

• Area of rectangle
• Area of square
• Area of triangle
• Area of circle
• Area of parallelogram

Area of rectangle

We can calculate the area of a rectangle with width w and height h as follows:

Area of Rectangle$\mathbf{=}\mathit{w}\mathbf{\times }\mathit{h}$

Area of rectangle - StudySmarter Originals

Area of square

Squares have four sides which are all equal in length. So, we can calculate the area with the given formula:

Area of Square$\mathbf{=}{\mathit{a}}^{\mathbf{2}}$

where a is the length of side

Area of square - StudySmarter Originals

Area of triangle

We can calculate the area of the triangle with the help of its altitude h and its base b.

Area of Triangle$\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\times }\mathit{b}\mathbf{\times }\mathit{h}$

Area of Triangle - StudySmarter Originals

Area of circle

The space occupied by the circle can be calculated based on its radius.

Area of circle$\mathbf{=}\mathit{\pi }\mathbf{\times }{\mathit{r}}^{\mathbf{2}}$

where r is the radius of the circle

Area of circle - StudySmarter Originals

Area of parallelogram

A parallelogram is a figure with opposite sides as a pair of parallel lines.

Area of parallelogram$\mathbf{=}\mathit{b}\mathbf{\times }\mathit{h}$

where b is the base and h is the height of the parallelogram

Area of parallelogram - StudySmarter Originals

Meaning of surface area in geometry

The term surface area is associated with three-dimensional figures and objects. However, its concept is essentially the same as area in that it measures the size of the surface using square units like inches squared (in²). Formulas for surface areas differ based on the type of 3-dimensional object.

Surface Area - StudySmarter Originals

In other words, the area occupied by all the faces and sides of any 3D object is its surface area. For any figure, surface area can be classified into three types:

1. Lateral surface area
2. Curved surface area
3. Total surface area

Types of surface area

The three types of surface area (lateral, curved, and total) are explained here.

Lateral and curved surface area

Sometimes, the lateral surface area and the curved surface area are actually referring to the same part of the 3D shape. This is the case for shapes with curved surfaces like cylinders and cones, for example. Shapes like cubes, however, do not have curved surface area, and this term does not apply.

Total surface area

In other words, we include any and all surfaces of an object when calculating the total surface area, as opposed to the lateral surface area. Usually, we simply refer to the total surface area as surface area.

Surface area formulas

Let's take a look at some of the surface area formulas for common three-dimensional figures.

Surface area of cube

A cube is a three-dimensional figure consisting of six square faces. The surface area of a cube can be obtained by finding the area of all six faces and summing them up together. For the lateral surface area of the cube, we take only four faces into consideration, eliminating the top and bottom sides.

Surface area of cube - StudySmarter Originals

The formula for the surface area of a cube is as follows:

Total surface area of cube$\mathbf{=}\mathbf{6}{\mathit{a}}^{\mathbf{2}}$

Lateral surface area of cube$\mathbf{=}\mathbf{4}{\mathit{a}}^{\mathbf{2}}$

where a is the length of all sides

Surface area of cuboid

A cuboid is a three-dimensional figure with all six faces rectangular. The total surface area and lateral surface of the cuboid can be calculated in the same manner as a cube. However, for a cuboid, instead of using the same measurement for all six faces, we consider and measure the different faces separately. Hence, we calculate the surface area of the cuboid in terms of length l, width w, and height h.

Surface area of cuboid - StudySmarter Originals

The formula for the surface area of a cuboid is as follows:

Total surface area of cuboid$\mathbf{=}\mathbf{2}\mathbf{(}\mathit{l}\mathit{w}\mathbf{+}\mathit{w}\mathit{h}\mathbf{+}\mathit{l}\mathit{h}\mathbf{)}$

Lateral surface area of cuboid$\mathbf{=}\mathbf{2}\mathit{h}\mathbf{(}\mathit{l}\mathbf{+}\mathit{w}\mathbf{)}$

Surface area of cylinder

A cylinder is a 3D shape with two flat circular faces, one on the top and one on the bottom, attached at the opposite ends of a curved face. A simple example is a pipe with both of the ends sealed or closed. The surface area of a cylinder is the sum of the surface areas of both circular bases and the curved surface area. When determining only the lateral or curved surface area, just the curved surface of the cylinder is considered.

Surface area of cylinder - StudySmarter Originals

The formula for the surface area of a cylinder is as follows:

Total surface area of cylinder$\mathbf{=}\mathbf{2}\mathit{\pi }\mathit{r}\mathbf{(}\mathit{r}\mathbf{+}\mathit{h}\mathbf{)}$

Lateral surface area of cylinder$\mathbf{=}\mathbf{2}\mathit{\pi }\mathit{r}\mathit{h}$

where r is the radius of circular base and h is the height of a cylinder

Surface area of sphere

A sphere is the three-dimensional representation of a circle. As spheres have only one overall surface and no other faces, discussing its lateral surface area doesn't make sense, as it would be the same as its total surface area. Therefore, we only calculate the total surface area for spheres.

Surface area of sphere - StudySmarter Originals

The formula for the surface area of a cuboid is as follows:

Total surface area of sphere$\mathbf{=}\mathbf{4}\mathit{\pi }{\mathit{r}}^{\mathbf{2}}$

where r is the radius of the sphere

Meaning of volume in geometry

Any three-dimensional object in nature occupies space. So, the space occupied by that object is called its volume.

To measure the volume we divide the occupied space into equal cubical units and calculate the total occupied cubical units, The unit of volume can be in a cubic centimeter, cubic meter, cubic inches, and so on.

Volume of 3D object visualization, tasks.illustrativemathematics.org

Volume formulas

Let's understand some of the formulas of volume for familiar 3D objects and shapes.

Volume of cube

As all the sides in the cube are of equal length, we can calculate the volume of a cube using the formula given below:

Volume of cube$\mathbf{=}{\mathit{a}}^{\mathbf{3}}$

where a is the length of sides for all faces

Volume of cube - StudySmarter Originals

Volume of cuboid

Unlike a cube, a cuboid has sides of different measures. So, we calculate the volume in the terms of length l, width w, and height h. The volume formula for a cuboid is as follows:

Volume of cuboid$\mathbf{=}\mathit{l}\mathbf{\times }\mathit{b}\mathbf{\times }\mathit{h}$

Volume of cuboid - StudySmarter Originals

Volume of cylinder

As the cylinder is formed from its circular bases and the distance between them, we consider the radius r of the circular base and height h of the cylinder when calculating its volume. The formula is as follows:

Volume of cylinder$\mathbf{=}\mathit{\pi }{\mathit{r}}^{\mathbf{2}}\mathit{h}$

Volume of Cylinder - StudySmarter Originals

Volume of sphere

As it can be used in some applications to approximate the shape of Earth, we have a special interest in the sphere, the three-dimensional representation of a circle. The volume of a sphere can be derived using multiple methods, including the Archimedes formula and Cavalieri's principle, for example. For now, we will just take a brief glance at its commonly accepted volume formula:

Volume of sphere$\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathit{\pi }{\mathit{r}}^{\mathbf{3}}$

where r is the radius of the sphere

Volume of sphere - StudySmarter Originals