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- 51016
- Amy Lagusker
- College of the Canyons
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Area formulas
Let \(b\) = base
Let \(h\) = height
Let \(s\) = side
Let \(r\) = radius
| Shape Name | Shape | Area Formula |
|---|---|---|
| Rectangle |  | \(A=bh\) |
| Square |  | \(\begin{array}{l} |
| Parallelogram |  | \(A=bh\) |
| Triangle |  | \(A=\dfrac{1}{2} b h\) |
| Circle |  | \(A=\pi r^{2}\) |
| Trapezoid |  | \(A=\dfrac{1}{2} h\left(b_{1}+b_{2}\right)\) |
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
| Geometric Figure | Surface Area Formula | Surface Area Meaning |
|---|---|---|
|
| \(S A=2 B+P h\) | Find the area of each face. Add up all areas. |
|
| \(S A=B+\dfrac{1}{2} s P\) | Find the area of each face. Add up all areas. |
|
| \(S A=2 B+2 \pi r h\) | 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. |
|
| \(S A=4 \pi r^{2}\) | Find the area of the great circle and multiply it by 4. |
|
| \(S A=B+\pi r S\) | 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
| Geometric Figure | VolumeFormula | VolumeMeaning |
|---|---|---|
|
| \(V=B h\) | Find the area of the base and multiply it by the height |
|
| \(V=\dfrac{1}{3} B h\) | Find the area of the base and multiply it by 1/3 of the height. |
|
| \(V=B h\) | Find the area of the base and multiply it by the height. |
|
| \(V=\dfrac{4}{3} \pi r^{3}\) | Find the area of the great circle and multiply it by the radius and then multiply it by 4/3. |
|
| \(V=\dfrac{1}{3} B h\) | 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.
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.
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.
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.
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.
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
- Find the area of a triangle with a base of 40 inches and a height of 60 inches.
- Find the area of a square with a side of 15 feet.
- Find the surface area of Earth, which has a diameter of 7917.5 miles. Use 3.14 for PI.
- 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.
- Radius = 9 cm
- Diameter = 6 miles
- Radius = 8.6 cm
- Diameter = 14 meters
(Problems 5 – 8) Find the area of each polygon. Round answers to the nearest tenth.
(Problems 9 – 12) Name each figure.
(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.
(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.
Extension: Methods of Teaching Mathematics
Part 1
Assessments:
- What is the Difference between Formative and Summative Assessments? Which One is More Important?
- Formative Assessment Examples and When to Use Them
- 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.
