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Mathematics LibreTexts

7.12: Area, Surface Area and Volume Formulas

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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

clipboard_efe15c30b1007547b342fc746a7efcf20.png

A=bh

Square

clipboard_ec24caea3d5296d300e6f1f98eccae4d4.png

A=bhA=s2

Parallelogram

clipboard_eb485ee8fb60488e5bde456d41c01adeb.png

A=bh

Triangle

clipboard_e2212c051d12fc4634a8ee3f1eda28490.png

A=12bh

Circle

clipboard_e062a8ba2f01f7dbdd5050dc75b4afcfb.png

A=πr2

Trapezoid

clipboard_e4f0ce691e022af3e89a66a7c366a6375.png

A=12h(b1+b2)

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 

clipboard_e81f9b4881086fecb9335c925c494c4e6.png 

SA=2B+Ph 

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

clipboard_e917a9764b931cea8719511a37f526a89.png 

SA=B+12sP 

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

clipboard_eaf05e34afb977462650d27c4706fb637.png 

SA=2B+2πrh 

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.  

clipboard_ea5d578ab0347abea37383a22a2492701.png

SA=4πr2

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

clipboard_e9c547851612b12562b5185fe0605b0ad.png

SA=B+πrS 

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 

Volume Formula 

Volume Meaning 

clipboard_e9a0de49a1375afc8775962a50a30a81b.png

V=Bh 

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

clipboard_e3a949ff5ec42fae21bd6c9a6773a69dc.png

V=13Bh 

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

clipboard_e88f8e5771fdb263d07c69b2ddf933536.png 

V=Bh 

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

clipboard_e24e7962afe42ee50f4fc0758451cd560.png 

V=43πr3

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

clipboard_ed295e65c6d76699d3776472f1c4b2c17.png 

V=13Bh

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

Example 7.12.1

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

clipboard_e41b579d2e55c37d8145b71fc46929023.png
Figure 6.5.1

Solution

A=πr2=π(7)2=49πfeet2=153.86feet2

Example 7.12.2

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

clipboard_eaa167458276c034c1497fbee20c48d82.png
Figure 6.5.2

Solution

A=12h(b1+b2)=12(12)(24+10)=6(34)=204 inches2

Example 7.12.3

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

clipboard_e7f0e0d855883d455b9c17bd32f5fbc15.png
Figure 6.5.3

Solution

SA=B+πrS=(πr2)+πrs=(π(32))+π(3)(8)=9π+24π=33πcm2=103.62cm2

Example 7.12.4

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.

clipboard_e1966ee3a811a2998d43c8e1a01f65351.png
Figure 6.5.4

Solution

SA=B+12sP=(bh)+12s(2b+2h)=(8)(12)+12(10)(2(8)+2(12))=96+12(10)(16+24)=96+5(40)=296 yards2

Example 7.12.5

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

clipboard_e8f88daf0b42e79373a2f60aef45758dc.png
Figure 6.5.5

Solution

V=43πr3=43π(3)3=43(27π)=36π meters 3=113.04 meters 3

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.

  1. clipboard_e7c11d2a06b604c10f855daeef8996806.png
  2.  clipboard_e71f2d376b673e2eeedce58fde84d357b.png
  3. clipboard_e2f13d1085ad717ca038c1450b452d3d1.png
  4. clipboard_e1abd666b733c9eb5f7c9c6b081cdd56a.png

(Problems 9 – 12) Name each figure.

  1.  clipboard_e9e2cc2664937ad831fbb674ad9999f14.png
  2. clipboard_e0434f0d23c6a5e226b248f81e884b497.png
  3. clipboard_e73a8e4539aa29805f2121f8d54380633.png
  4. clipboard_e1b38ca5b795760abf9ea3f4ed234a5f3.png

(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. clipboard_e2209ea68bc005aa5e461f595277d41d9.png
  2. clipboard_e6cc4dcb798c0d02022f26a2f8f7598c0.png
  3. clipboard_e3a982ab4b0487b87494a4fc61271e4e0.png
  4. clipboard_e4b6f66b4963ff0e17e58b75a7b6405ba.png
  5. clipboard_e584c2ff25992a84eaa7e40796c297b23.png

(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. clipboard_e0446db1e9293874d2e55a0a205226560.png
  2. clipboard_eac0f896eb03b29a9f16544a127879fbd.png
  3. clipboard_ee5baa989cb3bd5eb303b8cf115df882b.png
  4. clipboard_e5b24dffcb97d63b1510c5f204e131818.png
  5. clipboard_ebc8278c77a6141f1298d394b314e229e.png
  6. clipboard_e6599552f2317bca580dc5b815ff3baf1.png
  7. clipboard_ef7f0cda8e3c1436d52ef454dbff99d00.png
  8. clipboard_ef61bb9b9104ce69d8becd339515a44e1.png

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.


This page titled 7.12: Area, Surface Area and Volume Formulas is shared under a not declared license and was authored, remixed, and/or curated by Amy Lagusker.

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