1. Surface Area of Prisms, Cylinders, and Pyramids
Shape and Space
The aim of this unit is to teach pupils to:
Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning
Understand congruence and similarity
Identify and use the properties of circles
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
Surface Area of Prisms, Cylinders, and Pyramids

2. Surface Area of Rectangular Prisms
The surface area of a prism is the entire area of the outside of the object.
To calculate surface area, find the area of each side and add them together.
There are 6 faces to this rectangular prism.
Front and back are the same
Top and Bottom are the same
Two ends are the same.

3. Surface Area of Rectangular Prisms
To find the surface area, add the areas together.
Top and Bottom
A = bh
A = (90)(130)
A = cm2
Ends
A = bh
A = (90)(50)
A = 4500 cm2
Front and back
A = bh
A = (130)(50)
A = 6500 cm2
Total Surface Area = 2(top and Bottom) + 2(Ends) + 2(Front and Back)
= 2(11700) + 2(4500) + 2(6500)
= cm2

4. YOU TRY!
To find the surface area, add the areas together.

5. SOLUTION To find the surface area, add the areas together.
Top and Bottom
A = bh
A = (4)(10)
A = 40 m2
Ends
A = bh
A = (2)(4)
A = 8 m2
Front and back
A = bh
A = (2)(10)
A = 20 m2
Total Surface Area = 2(top and Bottom) + 2(Ends) + 2(Front and Back)
= 2(40) + 2(8) + 2(20)
= 136 m2

6. Surface Area of Triangular Prisms
The surface area of a triangular prism is the entire area of the outside of the object.
To calculate surface area, find the area of each side and add them together.
There are 5 faces to this triangular prism.
Two ends are the same.
Three sides depend on the type of triangle:
Equilateral
Isosceles
Scalene

7. Surface Area of Triangular Prisms
To find the surface area, add the areas together.
Bottom
A = bh
A = (1.3)(2.1)
A = 2.73 m2
Ends
A = bh  2
A = (1.3)(0.5)  2
A = m2
Front
A = bh
A = (2.1)(0.5)
A = 1.05 m2
Back
A = bh
A = (2.1)(1.2)
A = 2.52 m2
Total Surface Area = Bottom + 2(Ends) + Front + Back
= (0.325)
= 6.95 m2

8. YOU TRY!
To find the surface area, add the areas together.

9. SOLUTION To find the surface area, add the areas together. Sides
a2 = c2 - b2
a2 = (1)2 - (0.5)2
a2 =
a2 = 0.75
a = 0.866
Sides
A = bh
A = (1)(3)
A = 3 m2
Ends
A = bh  2
A = (1)(0.866)  2
A = m2
Using Pythagorean Theorem you can find the height of the triangle.
c2 = a2 + b2
Total Surface Area = 2(Ends) + 3(sides)
= 2(0.433) + 3(3)
= m2

10. Surface Area of Trapezoidal Prisms
Back
Front
Ends
Base
A = bh
A = ½ h (a + b)
A = (10)(7)
= 70 cm2
A = (4)(7)
= 28 cm2
A = (7)(3)
= 21 cm2
A = ½ (2)(4 + 10)
= 14 cm2
Total Surface Area = Back + Front + 2(Ends) + 2(Base)
= (21) (14) = 168 cm2

11. Finding Surface Area of Cylinders:
The total surface area of a cylinder is the sum of the lateral surface area and the areas of the bases. The lateral surface is the curved surface on a cylinder. You can think of the lateral surface as a wrapper. You can slice the wrapper and lay it flat to get a rectangular region.

12. Cylinder’s Lateral Surface
The height of the rectangle is the height of the cylinder.
The base of the rectangle is the circumference of the circular base of the cylinder.
The lateral surface area is the area of the rectangular region.
Area of a Rectangle:
A=bh H H H

13. Surface Area of a Cylinder
Surface Area of a Cylinder r r
SA = 2r² + dH

14. How many circles are there?
EXAMPLES CONTINUED…
How many circles are there? 2
SA= _______________ + _______________
SA = ____________________

15. Surface area of rectangular pyramids
Base
Lateral side #1
Lateral side #2
A = bh
A = (10)(6)
= 60 m2
A = ½ bh
A = ½ (6)(8)
= 24 m2
A = ½ (10)(12)
Total Surface Area = Base + 2 (Lateral Side #1) + 2 (Lateral Side #2)
= (24) + 2(60) =
= 228 m2

16. Surface Area of Pyramids
The surface area of a pyramid is the entire area of the outside of the object.
To calculate surface area, find the area of each side and add them together.
There are 5 faces to this triangular pyramid.
One square bottom
Four triangular sides are the same.

17. Surface Area of Pyramids
To find the surface area, add the areas together.
Bottom
A = s2
A = (4)(4)
A = 16 cm2
sides
A = bh  2
A = (4)(3)  2
A = 6 cm2
Total Surface Area = Bottom + 4(sides)
= (6)
= 40 cm2

18. YOU TRY!
To find the surface area, add the areas together.

19. SOLUTION To find the surface area, add the areas together. Bottom
A = 25 cm2
sides
A = bh  2
A = (5)(6)  2
A = 15 cm2
Total Surface Area = Bottom + 4(sides)
= (15)
= 85 cm2

20. Base Lateral side A = ½ bh A = ½ (4)(3.5) = 7 in2 A = ½ (4)(6)
Surface Area of Triangular Pyramids
Base
Lateral side
A = ½ bh
A = ½ (4)(3.5)
= 7 in2
A = ½ (4)(6)
= 12 in2
6 in
Total Surface Area = Base + 3 (Lateral Side)
= 7 + 3(12) =
= 43 in2
4 in
3.5 in
4 in
4 in

21. Surface Area of a Cone

22. The axis is also an altitude
LATERAL AREAS OF CONES
The axis is also an altitude
Slant height
Axis
Altitude l
Oblique Cone
Right Cone

23. LATERAL AREAS OF CONES l l
We can use the net for the cone to derive the formula for the lateral area. r l l r

24. LATERAL AREAS OF CONES l l
The lateral region of the cone is a sector of a circle with radius l
The arc length of the sector is the same as the circumference of the base, or 2r.
The circumference of the circle containing the sector is 2l
The area of the sector is proportional to the area of the circle. l r r l

25. LATERAL AREAS OF CONES l l
The area of the sector is proportional to the area of the circle. l
area of sector = measure of arc
area of circle = circumference of circle
area of sector = 2r
2l2 = 2l
(l2)(2r)
area of sector =
2l
area of sector = rl r r l

26. Surface Area of a Cone Total Surface Area = Base + Lateral side
= 22.09π
= cm2
A = πrl
A = π(4.7)(13.6)
= π
= cm2
13.6 cm
4.7 cm.
Total Surface Area = Base + Lateral side =
= cm2

27. Surface Area of Spheres
A sphere is the set of all points in space that are the same distance from a point, the center of the sphere.

28. Surface Area of a Sphere
Surface area = 4 π (radius)2
S = 4 π r2 C
Radius

29. Find the Surface Area of a Sphere
Find the surface area of the sphere. Round your answer to the nearest whole number.
= (4)(3.14)(64) = 804 in2
The radius is 8 in.
8 in.

30. Video Links https://www.youtube.com/watch?v=K3X0Y7CnV6U