6.5: Area, Surface Area and Volume Formulas (2024)

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    Area formulas

    Let \(b\) = base

    Let \(h\) = height

    Let \(s\) = side

    Let \(r\) = radius

    Table 6.5.1: Area formulas

    Shape Name

    Shape

    Area Formula

    Rectangle

    6.5: Area, Surface Area and Volume Formulas (2)

    \(A=bh\)

    Square

    6.5: Area, Surface Area and Volume Formulas (3)

    \(\begin{array}{l}
    A=b h \\
    A=s^{2}
    \end{array}\)

    Parallelogram

    6.5: Area, Surface Area and Volume Formulas (4)

    \(A=bh\)

    Triangle

    6.5: Area, Surface Area and Volume Formulas (5)

    \(A=\dfrac{1}{2} b h\)

    Circle

    6.5: Area, Surface Area and Volume Formulas (6)

    \(A=\pi r^{2}\)

    Trapezoid

    6.5: Area, Surface Area and Volume Formulas (7)

    \(A=\dfrac{1}{2} h\left(b_{1}+b_{2}\right)\)

    Surface Area Formulas

    Variables:

    \(SA\) = Surface Area

    \(B\) = area of the base of the figure

    \(P\) = perimeter of the base of the figure

    \(h\) = height

    \(s\) = slant height

    \(r\) = radius

    Table 6.5.2: Surface Area formulas

    Geometric Figure

    Surface Area Formula

    Surface Area Meaning

    6.5: Area, Surface Area and Volume Formulas (8)

    \(S A=2 B+P h\)

    Find the area of each face. Add up all areas.

    6.5: Area, Surface Area and Volume Formulas (9)

    \(S A=B+\dfrac{1}{2} s P\)

    Find the area of each face. Add up all areas.

    6.5: Area, Surface Area and Volume Formulas (10)

    \(S A=2 B+2 \pi r h\)

    Find the area of the base, times 2, then add the areas to the areas of the rectangle, which is the circumference times the height.

    6.5: Area, Surface Area and Volume Formulas (11)

    \(S A=4 \pi r^{2}\)

    Find the area of the great circle and multiply it by 4.

    6.5: Area, Surface Area and Volume Formulas (12)

    \(S A=B+\pi r S\)

    Find the area of the base and add the product of the radius times the slant height times PI.

    Volume Formulas

    Variables:

    \(SA\) = Surface Area

    \(B\) = area of the base of the figure

    \(P\) = perimeter of the base of the figure

    \(h\) = height

    \(s\) = slant height

    \(r\) = radius

    Table 6.5.3: Volume formulas

    Geometric Figure

    VolumeFormula

    VolumeMeaning

    6.5: Area, Surface Area and Volume Formulas (13)

    \(V=B h\)

    Find the area of the base and multiply it by the height

    6.5: Area, Surface Area and Volume Formulas (14)

    \(V=\dfrac{1}{3} B h\)

    Find the area of the base and multiply it by 1/3 of the height.

    6.5: Area, Surface Area and Volume Formulas (15)

    \(V=B h\)

    Find the area of the base and multiply it by the height.

    6.5: Area, Surface Area and Volume Formulas (16)

    \(V=\dfrac{4}{3} \pi r^{3}\)

    Find the area of the great circle and multiply it by the radius and then multiply it by 4/3.

    6.5: Area, Surface Area and Volume Formulas (17)

    \(V=\dfrac{1}{3} B h\)

    Find the area of the base and multiply it by 1/3of the height.

    Example \(\PageIndex{1}\)

    Find the area of a circle with diameter of 14 feet.

    6.5: Area, Surface Area and Volume Formulas (18)

    Solution

    \[\begin{aligned}A&=\pi r^{2}\\&=\pi(7)^{2}\\&=49 \pi \text {feet}^{2}\\&=153.86 \text {feet}^{2} \end{aligned} \nonumber \]

    Example \(\PageIndex{2}\)

    Find the area of a trapezoid with a height of 12 inches, and bases of 24 and 10 inches.

    6.5: Area, Surface Area and Volume Formulas (19)

    Solution

    \[\begin{aligned} A&=\dfrac{1}{2} h\left(b_{1}+b_{2}\right)\\ &=\dfrac{1}{2}(12)(24+10)\\ &=6(34)\\ &=204 \text { inches}^2 \end{aligned}\nonumber \]

    Example \(\PageIndex{3}\)

    Find the surface area of a cone with a slant height of 8 cm and a radius of 3 cm.

    6.5: Area, Surface Area and Volume Formulas (20)

    Solution

    \[\begin{aligned}
    SA&= B+\pi rS\\ &=\left(\pi r^{2}\right)+\pi rs\\ &=\left(\pi\left(3^{2}\right)\right)+\pi(3)(8) \\
    &=9 \pi+24 \pi\\ &=33 \pi \text {cm}^{2}\\ &=103.62 \text {cm}^{2}
    \end{aligned} \nonumber \]

    Example \(\PageIndex{4}\)

    Find the surface area of a rectangular pyramid with a slant height of 10 yards, a base width (b) of 8 yards and a base length (h) of 12 yards.

    6.5: Area, Surface Area and Volume Formulas (21)

    Solution

    \[\begin{aligned}
    SA&=B+\dfrac{1}{2} s P\\
    &=(b h)+\dfrac{1}{2} s(2 b+2 h) \\
    &=(8)(12)+\dfrac{1}{2}(10)(2(8)+2(12)) \\
    &=96+\dfrac{1}{2}(10)(16+24) \\
    &=96+5(40) \\
    &=296 \text { yards}^{2}
    \end{aligned} \nonumber \]

    Example \(\PageIndex{5}\)

    Find the volume of a sphere with a diameter of 6 meters.

    6.5: Area, Surface Area and Volume Formulas (22)

    Solution

    \[\begin{aligned} V&=\dfrac{4}{3} \pi r^{3}\\ &=\dfrac{4}{3} \pi(3)^{3}\\ &=\dfrac{4}{3}(27 \pi)\\ &=36 \pi \text { meters }^{3}\\ &=113.04 \text { meters }^{3} \end{aligned} \nonumber \]

    Partner Activity 1

    1. Find the area of a triangle with a base of 40 inches and a height of 60 inches.
    2. Find the area of a square with a side of 15 feet.
    3. Find the surface area of Earth, which has a diameter of 7917.5 miles. Use 3.14 for PI.
    4. Find the volume of a can a soup, which has a radius of 2 inches and a height of 3 inches. Use 3.14 for PI.

    Practice Problems

    (Problems 1 – 4) Find the area of each circle with the given parameters. Use 3.14 for PI. Round your answer to the nearest tenth.

    1. Radius = 9 cm
    2. Diameter = 6 miles
    3. Radius = 8.6 cm
    4. Diameter = 14 meters

    (Problems 5 – 8) Find the area of each polygon. Round answers to the nearest tenth.

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    (Problems 9 – 12) Name each figure.

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    4. 6.5: Area, Surface Area and Volume Formulas (30)

    (Problems 13 – 17) Find the surface area of each figure. Leave your answers in terms of PI, if the answer contains PI. Round all other answers to the nearest hundredth.

    1. 6.5: Area, Surface Area and Volume Formulas (31)
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    5. 6.5: Area, Surface Area and Volume Formulas (35)

    (Problems 18 – 25) Find the volume of each figure. Leave your answers in terms of PI, for answers that contain PI. Round all other answers to the nearest hundredth.

    1. 6.5: Area, Surface Area and Volume Formulas (36)
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    Extension: Methods of Teaching Mathematics

    Part 1

    Assessments:

    1. What is the Difference between Formative and Summative Assessments? Which One is More Important?
    2. Formative Assessment Examples and When to Use Them
    3. Summative Assessment Examples and When to Use Them

    Part 2

    Write a Formative and Summative Assessment for Your Lesson Plan

    Part 3

    Make sure you are working on Khan Academy throughout the semester.

    6.5: Area, Surface Area and Volume Formulas (2024)
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