1. Cones, Pyramids and Spheres
Volume & Surface Area
Volume & Surface Area
Cones, Pyramids and Spheres 1

2. Identical isosceles triangles
Pyramids
A Pyramid is a three dimensional figure with a regular polygon as its base and lateral faces are identical isosceles triangles meeting at a point.
Identical isosceles triangles
base = quadrilateral
base = heptagon
base = pentagon

3. Pyramids

4. Surface Area of Pyramids
Find the surface area of the pyramid.
height h = 8 m side s = 6 m apothem a = 4 m
Surface Area
= area of base + (#of sides of base) (area of one lateral face)
What shape is the base?
Area of a pentagon h
= ½ Pa (P = perimeter)
= ½ (5)(6)(4)
= 60 m2 l a s

5. Surface Area of Pyramids
What shape are the lateral sides?
Find the surface area of the pyramid height h = 8 m apothem a = 4 m side s = 6 m
Area of a triangle = ½ base (height)
= ½ (6)(8.9)
= 26.7 m2
Attention! the height of the triangle is the slant height ”l ” h l
l 2 = h2 + a2 =
= 80 m2
l = 8.9 m a s

6. Surface Area of Pyramids
Surface Area of the Pyramid
= 60 m2 + 5(26.7) m2
= 60 m m2
= m2
Find the surface area of the pyramid height h = 8 m apothem a = 4 m side s = 6 m h l a s

7. Pyramids
Creamed
Coconut
1/3 of the
calories 7

8. = + + Volume of Pyramids Volume of a Pyramid:
V = (1/3) Area of the base x height
V = (1/3) Ah
Volume of a Pyramid = 1/3 x Volume of a Prism = + +

9. Exercise #2
Find the volume of the pyramid height h = 8 m apothem a = 4 m side s = 6 m
Volume = 1/3 (area of base) (height)
= 1/3 ( 60m2)(8m)
= 160 m3 h
Area of base = ½ Pa a
= ½ (5)(6)(4)
= 60 m2 s

10. Pyramids Volume = 1/3 x base area x height
Find the volume of this pyramid 10

11. Cones Pringles A third of the calories A fact:
If Pringles came in a cone, which was the same height and diameter as the tall tube, it would contain one third of the calories!!!
Why??
Pringles
A third
of the
calories 11

12. Cones

13. The Cone + + = Volume of a Cone =
A Cone is a three dimensional solid with a circular base and a curved surface that gradually narrows to a vertex. + + =
Volume of a Cone =

14. Exercise #1 = (1/3)(3.14)(1)2(2) = 3.14(1)2(2) = 2.09 m3 = 6.28 m3
Find the volume of a cylinder with a radius r=1 m and height h=2 m. Find the volume of a cone with a radius r=1 m and height h=1 m
Volume of a Cylinder = base x height = pr2h
= 3.14(1)2(2)
= 6.28 m3
Volume of a Cone = (1/3) pr2h
= (1/3)(3.14)(1)2(2)
= 2.09 m3

15. Find the area of a cone with a radius r=3 m and height h=4 m.
Surface Area of a Cone
Find the area of a cone with a radius r=3 m and height h=4 m.
r = the radius h = the height l = the slant height
Use the Pythagorean Theorem to find l
l 2 = r2 + h2
l 2= (3)2 + (4)2
l 2= 25
l = 5
Surface Area of a Cone
= pr2 + prl
= 3.14(3) (3)(5)
= m2

16. Cones Slant height (l) Volume = Example: find the volume of this cone16

17. A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres

18. Spheres

19. Example 1A: Finding Volumes of Spheres
Find the volume of the sphere. Give your answer in terms of .
Volume of a sphere. __
= 2304 in3
Simplify.

20. Example 1B: Finding Volumes of Spheres
Find the diameter of a sphere with volume 36,000 cm3.

21. Example 1C: Finding Volumes of Spheres
Find the volume of the hemisphere.

22. Example 3A: Finding Surface Area of Spheres
Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of .
SA = 4r2
Surface area of a sphere
SA = 4(38)2
SA = 5776 cm2
Simplify.

23. Example SA = 4r2 SA = 4(1)2 SA = 4 m2 V = 4 (1)3 3 V = 4 m3 3
Find the surface area and volume of the sphere.
SA = 4r2
SA = 4(1)2
SA = 4 m2
V = 4 (1)3 3
V = 4 m3 3

24. Example Find the surface area and volume of the hemisphere.
SA = 2r2 + r2
= 2(11)2 +(11)2
= 363 in2