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# Area and Volume

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Suppose you are redecorating your room and need to know the amount of floor space available. So, you measure the floor which is, in fact, finding the area of that room. Now, suppose you plan to fill up your pool with water. You measure how much water is needed to fill the pool by measuring its volume.

In geometry, it is important to know the space covered by certain 2-dimensional and 3-dimensional figures. We can determine this space by calculating area and volume based on the type of figure. Here we will understand the basic concept of area and volume as related to various types of figures.

## Meaning of area in geometry

The total space occupied by the two-dimensional surface of an object or a flat shape is called an area.

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

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

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

Area of Triangle - StudySmarter Originals

### Area of circle

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

Area of circle

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

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.

The total area occupied by the outer surfaces of any 3-dimensional object is called the surface area of that 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

The surface area of the entire 3D shape excluding the surface area of the base and top is known as the lateral surface area.

The surface area of all curved surfaces of the 3D shape is known as the 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.

Lateral surface area - StudySmarter Originals

Here in the figure of the cylinder, the gray surface is the lateral surface, which is the surface that surrounds the cylinder. When calculating the lateral surface area, we will consider the gray color surface shown but not the white top surface or white bottom surface of the cylinder.

### Total surface area

The total area of all surfaces and sides of an object is called the 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.

Total surface area - StudySmarter Originals

Here, we consider the surface area of both the lateral surface (light gray color) and top & base (dark gray color) when we calculate the total surface area of this cylinder.

## 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

Lateral surface area of cube

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

Lateral surface area of cuboid

### 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

Lateral surface area of cylinder

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

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.

The space enclosed within the object or the space occupied by any three-dimensional solid object is known as 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

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

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

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

where r is the radius of the sphere

Volume of sphere - StudySmarter Originals

## Area and volume - Key takeaways

• The total space occupied by the 2-dimensional surface of an object or a flat shape is called area. Area uses square units, like feet squared or meters squared.
• The area of two congruent (identical) figures will be the same, but sometimes two non-congruent figures might have the same area.
• The total area occupied by the outer surfaces of any 3-dimensional object is called the surface area of that object. Surface area uses square units.
• Surface can be classified into three types: lateral surface area, curved surface area, and total surface area.
• The space enclosed within the object or the space occupied by any three-dimensional solid object is known as volume.

Surface area measures the area of the outer surface of any 3D object. Whereas, volume measures the space occupied by the 3D object.

To find volume, we measure the capacity of an object by cubic units, and to find the surface area we calculate the lateral and total surface of the object.

The surface area and volume of a cube can be calculated as

Total Surface area of cube = 6a2

Volume of Cube = a3

We can find area of all shapes by multiplying its length and width.

## Final Area and Volume Quiz

Question

What is the shape of a wheel?

Circle.

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Question

Is the diameter of a circle twice its radius?

Yes.

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Question

What is a distance from one endpoint to another on a circle that does not necessarily have to pass through the origin?

Chord.

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Question

What is a line that cuts the circle at exactly one point?

Tangent.

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Question

What is the arc length of the circle referred as?

Circumference.

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Question

A mathematical constant that is defined as the ratio of the circumference to the diameter of a circle is known as:

Pi.

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Question

What is the line that intersects a circle in two points that does not pass through the origin?

Secant line.

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Question

What is the value of pi rounded of to 3 decimal places?

3.141

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Question

What does the surface area of a cone mean?

The surface area of a cone is the total surface area covered by both its circular base and its curved surface.

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Question

What is the difference between internal height and slant height?

The internal height is the perpendicular distance between the top of the cone and the base.

The slant height is the height of the cone from the top to the outer edge of the base.

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Question

How do you calculate the surface area of a cone?

You plug the correct values into the equation of:

a = πr+ πrl, where r is the radius of the base and l is the slant height.

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Question

Given a cone of internal height of 48 feet and base diameter of 28 feet, what is the surface area?

2814.867 ft2

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Question

Given a cone with a radius of 3 feet and a slant height of 7.6 feet, calculate the surface area of the cone.

99.903 ft2

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Question

A cone has a slant height of 10 inches and a radius of 7 inches, what is the internal height?

7.141 in

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Question

What is the area of a circle with a radius of 6 inches?

36π

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Question

What is the area of the curved surface, with a slant height of 16 inches and a radius of 7.4 inches?

371.965 in2

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Question

What is a regular polygon?

A polygon that is equilateral and equiangular

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Question

What does equilateral mean?

All sides have equal length

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Question

What does equiangular mean?

All angles have equal size

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Question

What is an apothem?

The segment going from the center of the polygon to the midpoint of one of the sides.

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Question

How to find the perimeter of a regular polygon?

By multiplying the length of one side by the number of sides.

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Question

What is the apothem of a square also equal to?

Half of the side length

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Question

What is the area of a hexagon with side length of 6cm?

93.5cm2

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Question

What is the area of an octagon with side length of 9cm?

391 cm2

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Question

What is the area of a nonagon (9 sides) with side length of 4cm?

98.9 cm2

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Question

What is the apothem of a heptagon with a side length of 5cm?

5.19 cm

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Question

What is the apothem of a decagon (10 sides) with a perimeter of 1 m?

15.4 cm

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Question

What is the apothem of a dodecagon (10 sides) with a perimeter of 36 cm?

5.6 cm

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Question

What is the area of a decagon (10 sides) with a perimeter of 50 cm?

192 cm2

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Question

What is the area of a pentagon with a perimeter of 15 m?

15.5 m2

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Question

Given a cone with a slant height of 13 inches and a radius of 5 inches, what is the surface area?

282.743 in2

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Question

Given an internal height of 15 inches and a diameter of 16 inches, calculate the slant height.

17 inches

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Question

Given an internal height of 12 inches and a radius of 5 inches, calculate the surface area of the cone.

282.74 in2

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Question

What is the general rule for finding the surface area of solids?

When finding the surface area of a solid, you generally need to sum the area of all the faces and bases of the shape.

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Question

What is the formula for finding the surface area of a prism?

\begin{align} S&= 2B + Ph \\ &= aP+Ph.\end{align}

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Question

What is the formula for finding the surface area of a cylinder?

\begin{align} S &= 2B+Ch \\ &= 2\pi r^2 + 2\pi rh. \end{align}

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Question

What is the formula for finding the surface area of a cone?

\begin{align} S &= B + \frac{1}{2}Cl \\ &= \pi r^2 + \pi rl. \end{align}

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Question

What is the formula for finding the surface area of a sphere?

$S = 4\pi r^2$

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Question

What is the formula for finding the surface area of a pyramid?

$S = B +\frac{1}{2}Pl$

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Question

What is the volume of a solid?

The volume of a solid describes the cubic units that fit inside the 3D shape.

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Question

What is a rhombus?

A rhombus is a quadrilateral whose all 4 sides are equal.

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Question

State whether the following statement is true or false:

Every parallelogram is a rhombus.

False

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Question

State whether the following statement is true or false:

Every rhombus is a parallelogram.

True

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Question

State whether the following statement is true or false:

The opposite angles of a rhombus are supplementary.

False

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Question

A rhombus with an area of 96 has a diagonal of length 8. Find the length of the other diagonal.

24

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Question

A rhombus with an area of 63 has a diagonal of length 7. Find the length of the other diagonal.

18

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Question

A rhombus with an area of 25 has a diagonal of length 5. Find the length of the other diagonal.

10

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Question

A rhombus with an area of 18 has a diagonal of length 6. Find the length of the other diagonal.

6

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Question

State whether the following statement is true or false: Not every square is a rhombus.

False

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Question

Consider a rhombus ABCD with sides of length, 6, and internal angle ∠ABC = 90. Find the area of the rhombus.

36

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