1. Perimeter, Area, Surface Area, and Volume Examples
Geometry
and
Measurement

2. Geometry Polyhedron: V + F – E = 2
Vertices
Edges
Faces
Should be able to draw ALL of the following:
Sphere
Prisms – Cube, Rectangular, Triangular
Cylinder
Cone
Pyramids – Triangular, Square

3. Measurement Rectangle Perimeter
3 ft
5 ft
Rectangle
Perimeter
P = 2l + 2w, where l = length and w = width
Example: l = 5 ft and w = 3 ft
P rectangle = 2l + 2w
P = 2(5 ft) + 2(3 ft)
P = 10 ft + 6 ft
P = 16 ft

4. Measurement Rectangle Area A = lw where l = length and w = width
3 ft
5 ft
Rectangle
Area
A = lw where l = length and w = width
Example: l = 5 ft and w = 3 ft
A rectangle = lw
A = (5 ft)(3 ft)
A = 15 ft2

5. Measurement Square Perimeter P = 4s, where s = length of a side
3 ft
Square
Perimeter
P = 4s, where s = length of a side
Example: s = 3 ft
P square = 4s
P = 4(3 ft)
P = 12 ft

6. Measurement Square Area A = s2 where s = length of a side
3 ft
Square
Area
A = s2 where s = length of a side
Example: s = 3 ft
A square = s2
A = (3 ft)2
A = 9 ft2

7. Measurement Triangle Perimeter
P = a + b + c, where a, b, and c are the lengths of the sides of the triangle
Example: a = 3 m; b = 4 m; c = 5 m
P triangle = a + b + c
P = 3 m + 4 m + 5 m
P = 12 m

8. Measurement Triangle Area
A = ½ bh, where b is the base and h is the height of the triangle
Example: b = 3 m; h = 4 m
A triangle = ½ bh
A = ½ (3 m) (4 m)
A = 6 m2

9. Measurement Circle Circumference
3 cm
Circle
Circumference
C circle = d or C = 2r, where d = diameter and r = radius
Example: r = 3 cm
C circle = 2r
C = 2(3 cm)
C = 6 cm

10. Measurement Circle Area A = r2, where r = radius Example: r = 3 cm
A circle = r2
A = (3 cm)2
A = 9 cm2

11. Measurement Rectangular Prism
7 cm
6 cm
5 cm
Measurement
Rectangular Prism
Surface Area: sum of the areas of all of the faces
Example: There are 4 lateral faces: 2 lateral faces are 6 cm by 7 cm (A1= wh) and 2 lateral faces are 5 cm by 7 cm (A2 = lh). There are 2 bases 6 cm by 5 cm (A3 = lw)
A1 = (6 cm)(7 cm) = 42 cm2
A2 = (5 cm)(7 cm) = 35 cm2
A3 = (6 cm)(5 cm) = 30 cm2
SA rectangular prism = 2wh + 2lh + 2lw
SA = 2(42 cm2) + 2(35 cm2) + 2(30 cm2)
SA = 84 cm cm cm2
SA = 214 cm2

12. Measurement Rectangular Prism Volume:
7 cm
6 cm
5 cm
Rectangular Prism
Volume:
V = lwh where l is length; w is width; and h is height
Example: l = 6 cm; w = 5 cm; h = 7 cm
V rectangular prism = Bh = lwh
V = (6 cm)(5 cm)(7 cm)
V = 210 cm3

13. Measurement
5 cm
Cube
Surface Area: sum of the areas of all 6 congruent faces
Example: There are 6 faces: 5 cm by 5 cm (A = s2)
SA cube = 6A = 6s2
SA = 6(5 cm)2
SA = 6(25 cm2)
SA = 150 cm2

14. Measurement Cube Volume: V = s3 where s is the length of a side
5 cm
Cube
Volume:
V = s3 where s is the length of a side
Example: s = 5 cm
V cube = Bh = s3
V = (5 cm)3
V = 125 cm3

15. Measurement Triangular Prism
Surface Area: sum of the areas of all of the faces
Example: There are 3 lateral faces: 6 m by 7 m (A1= bl). There are 2 bases: 6 m for the base and 5 m for the height (2A2 = bh).
A1 = (6 m)(7 m) = 42 m2
2A2 = (6 m)(5 m) = 30 m2
SA triangular prism = bh + 3bl
SA = 30 m2 + 3(42 m2)
SA = 30 m m2
SA = 156 m2

16. Measurement Triangular Prism Volume:
V = ½ bhl where b is the base; h is height of the triangle; and l is length of the prism
Example: b = 6 m; h = 5 m; l = 7 m
V triangular prism = Bh = ½ bhl
V = ½ (6 m)(5 m)(7 m)
V = 105 m3

17. Measurement
3 ft
12 ft
Cylinder
Surface Area: area of the circles plus the area of the lateral face
Example: r = 3 ft; h = 12 ft
SA cylinder= 2rh +2r2
SA = 2 (3 ft)(12 ft) + 2 (3 ft)2
SA = 72 ft2 + 2 (9 ft2)
SA = 72 ft2 + 18 ft2
SA = 90 ft2

18. Measurement
3 ft
12 ft
Cylinder
Volume of a Cylinder: V = r2h where r is the radius of the base (circle) and h is the height.
Example: r = 3 ft and h = 12 ft.
V cylinder = Bh = r2h
V = (3 ft)2  (12 ft)
V = (9 ft2)(12 ft)
V = 108 ft3

19. 5 ft
13 ft
12 ft
Measurement
Cone
Surface Area: area of the circle plus the area of the lateral face
Example: r = 5 ft; t = 13 ft
SA cone= rt +r2
SA = (5 ft)(13 ft) +  (5 ft)2
SA = 65 ft2 +  (25 ft2)
SA = 65 ft2 + 25 ft2
SA = 90 ft2

20. 5 ft
13 ft
12 ft
Measurement
Cone
Volume: V = r2h/3 where r is the radius of the base (circle) and h is the height.
Example: r = 5 ft; h = 12 ft
V cone= r2h/3
V = [(5 ft)2  12 ft ]/ 3
V = [(25 ft2)(12 ft)]/3
V = (25 ft2)(4 ft)
V = 100 ft3

21. Measurement Sphere Surface Area: 4r2 where r is the radius
8 mm
Sphere
Surface Area: 4r2 where r is the radius
Example: r = 8 mm
SA sphere = 4r2
SA = 4(8 mm)2
SA = 4(64 mm2)
SA = 256 mm2

22. Measurement
6 mm
Sphere
Volume of a Sphere: V = (4/3) r3 where r is the radius
Example: r = 6 mm
V sphere = 4r3/3
V = [4 x (6 mm)3]/3
V = [4 x 216 mm3]/3
V = [864 mm3]/3
V = 288 mm3

23. Measurement
Triangular Pyramid
Square Pyramid