9.5: Area and Volume of Geometric Figures and Objects (2024)

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    Learning Objectives
    • know the meaning and notation for area
    • know the area formulas for some common geometric figures
    • be able to find the areas of some common geometric figures
    • know the meaning and notation for volume
    • know the volume formulas for some common geometric objects
    • be able to find the volume of some common geometric objects

    Quite often it is necessary to multiply one denominate number by another. To do so, we multiply the number parts together and the unit parts together. For example,

    \(\begin{array} {rcl} {\text{8 in.} \cdot \text{8 in.}} & = & {8 \cdot 8 \cdot \text{in.} \cdot \text{in.}} \\ {} & = & {64 \text{ in.}^2} \end{array}\)

    \(\begin{array} {rcl} {\text{4 mm} \cdot \text{4 mm} \cdot \text{4 mm}} & = & {4 \cdot 4 \cdot 4 \cdot \text{mm} \cdot \text{mm} \cdot \text{mm}} \\ {} & = & {64 \text{ mm}^3} \end{array}\)

    Sometimes the product of units has a physical meaning. In this section, we will examine the meaning of the products \(\text{(length unit)}^2\) and \(\text{(length unit)}^3\)

    The Meaning and Notation for Area

    The product \(\text{(length unit)} \cdot \text{(length unit)} = \text{(length unit)}^2\), or, square length unit (sq length unit), can be interpreted physically as the area of a surface.

    Area
    The area of a surface is the amount of square length units contained in the surface.

    For example, 3 sq in. means that 3 squares, 1 inch on each side, can be placed precisely on some surface. (The squares may have to be cut and rearranged so they match the shape of the surface.)

    We will examine the area of the following geometric figures.

    9.5: Area and Volume of Geometric Figures and Objects (4) 9.5: Area and Volume of Geometric Figures and Objects (5)

    9.5: Area and Volume of Geometric Figures and Objects (6) 9.5: Area and Volume of Geometric Figures and Objects (7)

    9.5: Area and Volume of Geometric Figures and Objects (8)

    Area Formulas

    We can determine the areas of these geometric figures using the following formulas.

    Figure Area Formula Statement
    9.5: Area and Volume of Geometric Figures and Objects (9) Triangle \(A_T = \dfrac{1}{2} \cdot b \cdot h\) Area of a triangle is one half the base times the height.
    9.5: Area and Volume of Geometric Figures and Objects (10) Rectangle \(A_R = l \cdot w\) Area of a rectangle is the length times the width.
    9.5: Area and Volume of Geometric Figures and Objects (11) Parallelogram \(A_P = b \cdot h\) Area of a parallelogram is base times the height.
    9.5: Area and Volume of Geometric Figures and Objects (12) Trapezoid \(A_{Trap} = \dfrac{1}{2} \cdot (b_1 + b_2) \cdot h\) Area of a trapezoid is one half the sum of the two bases times the height.
    9.5: Area and Volume of Geometric Figures and Objects (13) Circle \(A_c = \pi r^2\) Area of a circle is \(\pi\) times the square of the radius.

    Finding Areas of Some Common Geometric Figures

    Sample Set A

    Find the area of the triangle.

    9.5: Area and Volume of Geometric Figures and Objects (14)

    Solution

    \(\begin{array} {rcl} {A_T} & = & {\dfrac{1}{2} \cdot b \cdot h} \\ {} & = & {\dfrac{1}{2} \cdot 20 \cdot 5 \text{ sq ft}} \\ {} & = & {10 \cdot 6 \text{ sq ft}} \\ {} & = & {60 \text{ sq ft}} \\ {} & = & {60 \text{ ft}^2} \end{array}\)

    The area of this triangle is 60 sq ft, which is often written as 60 \(\text{ft}^2\).

    Sample Set A

    Find the area of the rectangle.

    9.5: Area and Volume of Geometric Figures and Objects (15)

    Solution

    Let's first convert 4 ft 2 in. to inches. Since we wish to convert to inches, we'll use the unit fraction \(\dfrac{\text{12 in.}}{\text{1 ft}}\) since it has inches in the numerator. Then,

    \(\begin{array} {rcl} {\text{4 ft}} & = & {\dfrac{\text{4 ft}}{1} \cdot \dfrac{\text{12 in.}}{\text{1 ft}}} \\ {} & = & {\dfrac{4 \cancel{\text{ ft}}}{1} \cdot \dfrac{\text{12 in.}}{1 \cancel{\text{ ft}}}} \\ {} & = & {\text{48 in.}} \end{array}\)

    Thus, \(\text{4 ft 2 in. = 48 in. + 2 in. = 50 in.}\)

    \(\begin{array} {rcl} {A_R} & = & {l \cdot w} \\ {} & = & {\text{50 in.} \cdot \text{8 in.}} \\ {} & = & {400 \text{ sq in.}} \end{array}\)

    The area of this rectangle is 400 sq in.

    Sample Set A

    Find the area of the parallelogram.

    9.5: Area and Volume of Geometric Figures and Objects (16)

    Solution

    \(\begin{array} {rcl} {A_P} & = & {b \cdot h} \\ {} & = & {\text{10.3 cm} \cdot \text{6.2 cm}} \\ {} & = & {63.86 \text{ sq cm}} \end{array}\)

    The area of this parallelogram is 63.86 sq cm.

    Sample Set A

    Find the area of the trapezoid.

    9.5: Area and Volume of Geometric Figures and Objects (17)

    Solution

    \(\begin{array} {rcl} {A_{Trap}} & = & {\dfrac{1}{2} \cdot (b_1 + b_2) \cdot h} \\ {} & = & {\dfrac{1}{2} \cdot (\text{14.5 mm + 20.4 mm}) \cdot (4.1 \text{ mm})} \\ {} & = & {\dfrac{1}{2} \cdot (\text{34.9 mm}) \cdot (4.1 \text{ mm})} \\ {} & = & {\dfrac{1}{2} \cdot \text{(143.09 sq mm)}} \\ {} & = & {71.545 \text{ sq mm}} \end{array}\)

    The area of this trapezoid is 71.545 sq mm.

    Sample Set A

    Find the approximate area of the circle.

    9.5: Area and Volume of Geometric Figures and Objects (18)

    Solution

    \(\begin{array} {rcl} {A_c} & = & {\pi \cdot r^2} \\ {} & \approx & {(3.14) \cdot (16.8 \text{ ft})^2} \\ {} & \approx & {(3.14) \cdot (\text{282.24 sq ft})} \\ {} & \approx & {888.23 \text{ sq ft}} \end{array}\)

    The area of this circle is approximately 886.23 sq ft.

    Practice Set A

    Find the area of each of the following geometric figures.

    9.5: Area and Volume of Geometric Figures and Objects (19)

    Answer

    36 sq cm

    Practice Set A

    9.5: Area and Volume of Geometric Figures and Objects (20)

    Answer

    37.503 sq mm

    Practice Set A

    9.5: Area and Volume of Geometric Figures and Objects (21)

    Answer

    13.26 sq in.

    Practice Set A

    9.5: Area and Volume of Geometric Figures and Objects (22)

    Answer

    367.5 sq mi

    Practice Set A

    9.5: Area and Volume of Geometric Figures and Objects (23)

    Answer

    452.16 sq ft

    Practice Set A

    9.5: Area and Volume of Geometric Figures and Objects (24)

    Answer

    44.28 sq cm

    The Meaning and Notation for Volume

    The product \(\text{(length unit)}\text{(length unit)}\text{(length unit)} = \text{(length unit)}^3\), or cubic length unit (cu length unit), can be interpreted physically as the volume of a three-dimensional object.

    Volume
    The volume of an object is the amount of cubic length units contained in the object.

    For example, 4 cu mm means that 4 cubes, 1 mm on each side, would precisely fill some three-dimensional object. (The cubes may have to be cut and rearranged so they match the shape of the object.)

    9.5: Area and Volume of Geometric Figures and Objects (25) 9.5: Area and Volume of Geometric Figures and Objects (26)9.5: Area and Volume of Geometric Figures and Objects (27) 9.5: Area and Volume of Geometric Figures and Objects (28)

    Volume Formulas

    Figure Volume Formula Statement
    9.5: Area and Volume of Geometric Figures and Objects (29) Rectangular solid \(\begin{array} {rcl} {V_R} & = & {l \cdot w \cdot h} \\ {} & = & {\text{(area of base)} \cdot \text{(height)}} \end{array}\) The volume of a rectangular solid is the length times the width times the height.
    9.5: Area and Volume of Geometric Figures and Objects (30) Sphere \(V_s = \dfrac{4}{3} \cdot \pi \cdot r^3\) The volume of a sphere is \(\dfrac{4}{3}\) times \(\pi\) times the cube of the radius.
    9.5: Area and Volume of Geometric Figures and Objects (31) Cylinder \(\begin{array} {rcl} {V_{Cyl}} & = & {\pi \cdot r^2 \cdot h} \\ {} & = & {\text{(area of base)} \cdot \text{(height)}} \end{array}\)
    The volume of a cylinder is \(\pi\) times the square of the radius times the height.
    9.5: Area and Volume of Geometric Figures and Objects (32) Cone \(\begin{array} {rcl} {V_c} & = & {\dfrac{1}{3} \cdot \pi \cdot r^2 \cdot h} \\ {} & = & {\text{(area of base)} \cdot \text{(height)}} \end{array}\) The volume of a cone is \(\dfrac{1}{3}\) times \(\pi\) times the square of the radius times the height.

    Finding Volumes of Some Common Geometric Objects

    Sample Set B

    Find the volume of the rectangular solid.

    9.5: Area and Volume of Geometric Figures and Objects (33)

    Solution

    \(\begin{array} {rcl} {V_R} & = & {l \cdot w \cdot h} \\ {} & = & {\text{9 in.} \cdot \text{10 in.} \cdot \text{3 in.}} \\ {} & = & {\text{270 cu in.}} \\ {} & = & {\text{270 in.}^3} \end{array}\)

    The volume of this rectangular solid is 270 cu in.

    Sample Set B

    Find the approximate volume of the sphere.

    9.5: Area and Volume of Geometric Figures and Objects (34)

    Solution

    \(\begin{array} {rcl} {V_S} & = & {\dfrac{4}{3} \cdot \pi \cdot r^3} \\ {} & \approx & {(\dfrac{4}{3}) \cdot (3.14) \cdot \text{(6 cm)}^3} \\ {} & \approx & {(\dfrac{4}{3}) \cdot (3.14) \cdot \text{(216 cu cm)}} \\ {} & \approx & {\text{904.32 cu cm}} \end{array}\)

    The approximate volume of this sphere is 904.32 cu cm, which is often written as 904.32 cm\(^3\).

    Sample Set B

    Find the approximate volume of the cylinder.

    9.5: Area and Volume of Geometric Figures and Objects (35)

    Solution

    \(\begin{array} {rcl} {V_{Cyl}} & = & {\pi \cdot r^2 \cdot h} \\ {} & \approx & {(3.14) \cdot (\text{4.9 ft})^2 \cdot \text{(7.8 ft)}} \\ {} & \approx & {(3.14) \cdot (\text{24.01 sq ft}) \cdot \text{(7.8 ft)}} \\ {} & \approx & {(3.14) \cdot \text{(187.278 cu ft)}} \\ {} & \approx & {\text{588.05292 cu ft}} \end{array}\)

    The volume of this cylinder is approximately 588.05292 cu ft. The volume is approximate because we approximated \(\pi\) with 3.14.

    Sample Set B

    Find the approximate volume of the cone. Round to two decimal places.

    9.5: Area and Volume of Geometric Figures and Objects (36)

    Solution

    \(\begin{array} {rcl} {V_{c}} & = & {\dfrac{1}{3} \cdot \pi \cdot r^2 \cdot h} \\ {} & \approx & {(\dfrac{1}{3}) \cdot (3.14) \cdot (\text{2 mm})^2 \cdot \text{(5 mm)}} \\ {} & \approx & {(\dfrac{1}{3}) \cdot (3.14) \cdot (\text{4 sq mm}) \cdot \text{(5 mm)}} \\ {} & \approx & {(\dfrac{1}{3}) \cdot (3.14) \cdot \text{(20 cu mm)}} \\ {} & \approx & {20.9\overline{3} \text{ cu mm}} \\ {} & \approx & {\text{20.93 cu mm}} \end{array}\)

    The volume of this cone is approximately 20.93 cu mm. The volume is approximate because we approximated \(\pi\) with 3.14.

    Practice Set B

    Find the volume of each geometric object. If \(\pi\) is required, approximate it with 3.14 and find the approximate volume.

    9.5: Area and Volume of Geometric Figures and Objects (37)

    Answer

    21 cu in.

    Practice Set B

    Sphere

    9.5: Area and Volume of Geometric Figures and Objects (38)

    Answer

    904.32 cu ft

    Practice Set B

    9.5: Area and Volume of Geometric Figures and Objects (39)

    Answer

    157 cu m

    Practice Set B

    9.5: Area and Volume of Geometric Figures and Objects (40)

    Answer

    0.00942 cu in.

    Exercises

    Find each indicated measurement.

    Exercise \(\PageIndex{1}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (41)

    Answer

    16 sq m

    Exercise \(\PageIndex{2}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (42)

    Exercise \(\PageIndex{3}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (43)

    Answer

    1.21 sq mm

    Exercise \(\PageIndex{4}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (44)

    Exercise \(\PageIndex{5}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (45)

    Answer

    18 sq in.

    Exercise \(\PageIndex{6}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (46)

    Exercise \(\PageIndex{7}\)

    Exact area

    9.5: Area and Volume of Geometric Figures and Objects (47)

    Answer

    \((60.5 \pi + 132) \text{ sq ft}\)

    Exercise \(\PageIndex{8}\)

    Approximate area

    9.5: Area and Volume of Geometric Figures and Objects (48)

    Exercise \(\PageIndex{9}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (49)

    Answer

    40.8 sq in.

    Exercise \(\PageIndex{10}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (50)

    Exercise \(\PageIndex{11}\)

    Approximate area

    9.5: Area and Volume of Geometric Figures and Objects (51)

    Answer

    31.0132 sq in.

    Exercise \(\PageIndex{12}\)

    Exact area

    9.5: Area and Volume of Geometric Figures and Objects (52)

    Exercise \(\PageIndex{13}\)

    Approximate area

    9.5: Area and Volume of Geometric Figures and Objects (53)

    Answer

    158.2874 sq mm

    Exercise \(\PageIndex{14}\)

    Exact area

    9.5: Area and Volume of Geometric Figures and Objects (54)

    Exercise \(\PageIndex{15}\)

    Approximate area

    9.5: Area and Volume of Geometric Figures and Objects (55)

    Answer

    64.2668 sq in.

    Exercise \(\PageIndex{16}\)

    Area

    9.5: Area and Volume of Geometric Figures and Objects (56)

    Exercise \(\PageIndex{17}\)

    Approximate area

    9.5: Area and Volume of Geometric Figures and Objects (57)

    Answer

    43.96 sq ft

    Exercise \(\PageIndex{18}\)

    Volume

    9.5: Area and Volume of Geometric Figures and Objects (58)

    Exercise \(\PageIndex{19}\)

    Volume

    9.5: Area and Volume of Geometric Figures and Objects (59)

    Answer

    512 cu cm

    Exercise \(\PageIndex{20}\)

    Exact volume

    9.5: Area and Volume of Geometric Figures and Objects (60)

    Exercise \(\PageIndex{21}\)

    Approximate volume

    9.5: Area and Volume of Geometric Figures and Objects (61)

    Answer

    11.49 cu cm

    Exercise \(\PageIndex{22}\)

    Approximate volume

    9.5: Area and Volume of Geometric Figures and Objects (62)

    Exercise \(\PageIndex{23}\)

    Exact volume

    9.5: Area and Volume of Geometric Figures and Objects (63)

    Answer

    \(\dfrac{1024}{3} \pi \text{ cu ft}\)

    Exercise \(\PageIndex{24}\)

    Approximate volume

    9.5: Area and Volume of Geometric Figures and Objects (64)

    Exercise \(\PageIndex{25}\)

    Approximate volume

    9.5: Area and Volume of Geometric Figures and Objects (65)

    Answer

    22.08 cu in.

    Exercise \(\PageIndex{26}\)

    Approximate volume

    9.5: Area and Volume of Geometric Figures and Objects (66)

    Exercises for Review

    Exercise \(\PageIndex{27}\)

    In the number 23,426, how many hundreds are there?

    Answer

    4

    Exercise \(\PageIndex{28}\)

    List all the factors of 32.

    Exercise \(\PageIndex{29}\)

    Find the value of \(4 \dfrac{3}{4} - 3 \dfrac{5}{6} + 1 \dfrac{2}{3}\).

    Answer

    \(\dfrac{31}{12} = 2 \dfrac{7}{12} = 2.58\)

    Exercise \(\PageIndex{30}\)

    Find the value of \(\dfrac{5 + \dfrac{1}{3}}{2 + \dfrac{2}{15}}\).

    Exercise \(\PageIndex{31}\)

    Find the perimeter.

    9.5: Area and Volume of Geometric Figures and Objects (67)

    Answer

    27.9m

    9.5: Area and Volume of Geometric Figures and Objects (2024)

    FAQs

    How do you find the area of a geometric figure? ›

    Here are the most important and useful area formulas for sixteen geometric shapes:
    1. Square area formula: A = a²
    2. Rectangle area formula: A = a × b.
    3. Triangle area formulas: ...
    4. Circle area formula: A = πr²
    5. Circle sector area formula: A = r² × angle / 2.
    6. Ellipse area formula: A = a × b × π
    Jul 29, 2024

    How to calculate volume of geometric shapes? ›

    The volume of a rectangular solid is the length times the width times the height. The volume of a sphere is 43 times π times the cube of the radius. The volume of a cylinder is π times the square of the radius times the height. The volume of a cone is 13 times π times the square of the radius times the height.

    What is the surface area and volume of a 3D shape Grade 9? ›

    The volume of a 3D shape is the space inside it. The surface area of a 3D shape is the total area of all its faces. To find the volume of a cuboid we use the formula, Volume = length × width × height .

    What is the area surface area and volume in geometry? ›

    Surface area and volume are calculated for any three-dimensional geometrical shape. The surface area of any given object is the area or region occupied by the surface of the object. Whereas volume is the amount of space available in an object.

    How do you solve area geometry problems? ›

    To find the area of a square or rectangle, multiply the length times the width. To find the area of a circle, multiply pi times the radius squared. To find the area of a triangle, multiply one-half the base times the height.

    What is an example of area in geometry? ›

    For example, if you want to know the area of a square box with side 40 cm, you will use the formula: Area of Square = a2, where a is the side of the square. Similarly, the area of a triangle can also be found using its Area formula (1/2 × b ×h).

    How do I calculate surface area and volume? ›

    For a cube, the surface area and volume formulas are SA = 6s^2 and V = s^3, where s is the length of one side. Therefore, the surface area to volume ratio is SA/V = 6/s.

    What is the volume of a 3D shape and area? ›

    Unit 9 Section 4 : Surface Area and Volume of 3-D Shapes
    CubeVolume = x³ Surface area = 6x²
    CylinderVolume = π r²h Area of curved surface = 2π rh Area of each end = π r² Total surface area = 2π rh + 2π r²
    PrismA prism has a uniform cross-section Volume = area of cross section × length = A l
    1 more row

    What is area and volume surface? ›

    Surface area and volume are different attributes of three-dimensional figures. Surface area is a two-dimensional measure, while volume is a three-dimensional measure. Two figures can have the same volume but different surface areas.

    Is area and volume part of geometry? ›

    Volume and area are both used to measure figures in geometry. They both calculate the amount of space a figure takes up.

    What is the formula for surface area in geometry? ›

    Variables:
    Surface Area FormulaSurface Area Meaning
    SA=B+12sPFind the area of each face. Add up all areas.
    SA=2B+2πrhFind the area of the base, times 2, then add the areas to the areas of the rectangle, which is the circumference times the height.
    SA=4πr2Find the area of the great circle and multiply it by 4.
    2 more rows
    Sep 17, 2020

    What is area and volume formula? ›

    What is the formula for surface area volume Class 10?
    Name of ShapeCurved Surface AreaVolume
    Cube4a2a3
    Cylinder2πrhπr2h
    Sphere4πr24/3π r3
    Coneπrl1/3π r2h
    2 more rows
    Aug 2, 2024

    How do you find the geometric formula? ›

    What is the rule for the geometric sequence? Each term of a geometric sequence is formed by multiplying the previous term by a constant number r, starting from the first term a1. Therefore, the rule for the terms of a geometric sequence is an=a1(r)^(n-1).

    What is the area method in geometry? ›

    The area method is a decision procedure for a fragment of Euclidean plane geometry. The method deals with problems stated in terms of sequences of specific geometric construction steps.

    What is the formula for the area of this figure? ›

    What is the formula for area?
    ShapeArea Formula
    Square/RectangleA = l x w
    CircleA = πr^2
    TriangleA = ½ (b x h)
    ParallelogramA = b x h
    1 more row
    Nov 8, 2023

    What is the formula for finding area? ›

    Area Formulas
    FiguresFormulaVariables
    RectangleArea = l × wl = length w = width
    SquareArea = a 2a = sides of square
    TriangleArea = 1 2 bhb = base h = height
    CircleArea = π r 2r = radius of circle
    2 more rows

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    Phone: +5958753152963

    Job: National Specialist

    Hobby: Kayaking, Photography, Skydiving, Embroidery, Leather crafting, Orienteering, Cooking

    Introduction: My name is Twana Towne Ret, I am a famous, talented, joyous, perfect, powerful, inquisitive, lovely person who loves writing and wants to share my knowledge and understanding with you.