SURFACE AREA OF A TRIANGULAR PRISM

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SURFACE AREA OF A TRIANGULAR PRISM

Objective: Find the surface area of a triangular prism

Find the area of the following plane figures. Write your answer on your notebook.

1) 2) 3)

Area = ____ Area = ____ Area = ____

4) 5)

Area = ____ Area = ____ Area = ____

The surface area of a solid figure is the sum of the areas of all its faces.

The surface area of a triangular prism is the sum of the areas of all the faces.

Review

Study and Learn

4 cm

10 cm 6 cm

7 cm

12 cm

2 cm

8 cm

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2

 How many faces does the figure have?

5

 What are the shapes of the faces?

triangles and squares

To visualize clearly, let us unfold the triangular prism.

 The faces may vary depending on how the prism stands. The important thing you must remember is the shapes that each face is made of.

How many faces does the prism have? 5

Yes, 5 faces and these faces are triangles and squares. How many triangular faces does it have?

How many rectangular faces does it have? Are the triangles the same in size? Yes

What kind of triangles are these? equilateral triangle Are the rectangles the same?

Since we are after the surface area, we need to recall the formula for finding the area of a rectangle and a triangle.

Do you still remember the formulas?

Area of a Rectangle = length x width or A = l x w

If the length of a rectangle is 5 cm and its width is 4 cm, what is the area?

20 cm2 because 5 cm x 4 cm = 20 cm2

For a triangle, the area is one-half of the product of base and height.

In symbol, A =

2 1

bh or

2 bh

.

If the base of a triangle is 10 cm and its height is 8 cm, what is the area? 40 cm2 because 10 cm x 8 cm = 80 cm2 2 = 40 cm2

base height

B o t t o m S

i d e

S i d e Front

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3

Now, let us try to find the surface area of a triangular prism.

 What is the area of one rectangle?

A = l x w

= 7 cm x 6 cm = 42 cm2

 How many rectangles does the figure has? 3

 Are they all equal? Yes

So we will multiply the area of 1 rectangle to 3. 42 cm2 x 3 = 126 cm2

 What is the area of one triangle?

A = bh  6 cm x 4 cm = 24 cm2 = 12 cm2 2 2 2

 How many triangles do the figure has? 2

 Are they all equal? Yes, because it is an equilateral triangle

 So, we will multiply the area of 1 triangle to 2.

12 cm2 x 2 = 24 cm2

We can also solve the given figure by area of each face then add all the faces.

Area of Face A Area of Face B Area of Face C

A = l x w

= 7 cm x 6 cm = 42 cm2

A = l x w

= 7 cm x 6 cm = 42 cm2

A = l x w

= 7 cm x 6 cm = 42 cm2

Area of Face D Area of Face E

A =

2 bh

A =

2 bh

6 cm 4 cm

7 cm A B C

D

E 6 cm

6 cm

4 cm 6 cm 6 cm

(4)

4 = 6 cm x 4 cm

2 = 24 cm2 2

= 12 cm2

= 6 cm x 4 cm 2

= 24 cm2 2

= 12 cm2

Area of Face A - 42 cm2 - 42 cm2 - 42 cm2 - 12 cm2 - 12 cm2 Total SA = 150 cm2

Let us have another one.

Area of Triangle =

2 bh

A = 6 cm x 5 cm  30 cm2 15 cm2

2 2

2 x 15 cm2 = 30 cm2 Area of 2 triangles

Area of Rectangle = l x w A = 6 cm x 12 cm  72 cm2

3 x 72 = 216 cm2 Area of 3 rectangles

 So, 30 cm2 + 216 cm2 = 246 cm2 the area of the triangular prism.

 It is easy, isn’t it?

 But what if the given triangle is not equilateral? What if the given one is an isosceles triangle or an scalene?

 Can you still use the formula? No

 If the given triangle is not an equilateral triangle, then it’s best to use the flattened method of finding the surface area. It is also important that you can visualize the figure correctly.

Let’s have one example.

 Isosceles Triangle

12 cm

5 cm

6 cm

4 cm

5 cm 8 cm

5 cm

D

A B C

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5

 Which faces are the same in size and shape (congruent)?

Face A and C and Face D and E

 Which faces is different to all faces? Face B

Area of Face A  A = 8 cm x 5 cm = 40 cm2

Area of Face B  A = 8 cm x 4 cm = 32 cm2

Area of Face C  A = 8 cm x 5 cm = 40 cm2

Area of Face D  A = bh  4 cm x 5 cm 2 2 = 20 cm2 10 cm2

2

Area of Face E  A = bh  4 cm x 5 cm 2 2 = 20 cm2 10 cm2

2

Face A - 40 cm2 B - 32 cm2 C - 40 cm2 D - 10 cm2 E - 10 cm2 Total SA = 132 cm2

 Did you find a shorter way of solving the surface of a triangular prism with an isosceles triangle?

Yes, just multiply Face A by two and Face D will be the sum of Face D and E if you will not divide it by 2.

Let’s have an scalene triangle.

6 cm 3 cm 7 cm 4 cm 5 cm 5 cm 5 cm 4 cm

8 cm 8 cm 8 cm

8 cm 4 cm 5 cm 5 cm 5 cm 5 cm

Face A Face C

5 cm

Face E Face D

Face B

A B C

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6 Solution:

Area of Face A  A = 7 cm x 5 cm = 35 cm2 Area of Face B  A = 7 cm x 6 cm = 42 cm2 Area of Face C  A = 7 cm x 4 cm = 28 cm2

Area of Face D  A = 6 cm x 3 cm = 18 cm2 = 9 cm2

2 2

Area of Face E  A = 6 cm x 3 cm = 18 cm2 = 9 cm2

2 2

A. Find the surface area of the flattened triangular prism. Copy and answer this on your answer on your notebook.

1)

Area of Face A = _____ Area of Face B = _____ Area of Face C = _____ Area of Face D = _____ Area of Face E = _____

Total Surface Area = _____

Try These

D

E

C

A B

8 cm

10 cm h = 7 cm

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7 2)

Area of Face F = _____ Area of Face G = _____ Area of Face H = _____ Area of Face I = _____ Area of Face J = _____

3)

Area of Face K = _____ Area of Face L = _____ Area of Face M = _____ Area of Face N= _____ Area of Face O = _____

10 cm h = 8 cm

15 cm 6 cm

I

J

H

F G

h = 4 cm 9 cm 7 cm

5 cm

8 cm

K

L

M

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B. What is the surface area of the following figure? Write your answer on your notebook.

1) 3)

SA = _____

2) SA = _____

SA = _____

 The surface area of a solid figure is the sum of the areas of all its faces.

 The surface area of a triangular prism is the sum of the areas of all the faces.

 A triangular prism has 5 faces compose of two triangles and 3 rectangles.

 The best way to solve for the surface area of a triangular prism is to unfold or flatten the figure so that you can visualize clearly the kind of face it has and its measurement.

 The unit used for surface area is square unit.

Wrap Up

h = 4 cm

5 cm

5 cm 10 cm

8 dm

12 dm

10 dm 13 dm

10 dm

h = 9 dm 15 dm

10 dm

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9

A. Below are triangular prisms that have a letter on it. To complete the message below, you need to find the surface area of each figure. Then write the code letter beside your answer on the correct lines at the bottom of the page. The first one has been done for you. Copy and answer the message on your notebook.

1) 2)

3) 4)

5) 6)

N !

306 112 580 306 174 174 306 304 180

On Your Own

N

10 cm _{h = 4 cm}

12 cm 5 cm 7 cm

10 dm

20 dm 8 dm

8 dm

C

h = 6 dm

4 m

8 m

X

5 cm h = 6 cm

10 cm

T

5 cm

6 m 9 m

7 m

12 m

E

L

9 cm

5 cm 5 cm

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B. Find the surface area of the following. Write your answer on your notebook.

1)

SA = _____

2)

SA = _____

3)

SA = _____

4)

SA = _____

5)

h = 5 cm

6 cm

6 cm 8 cm

10 cm

6 cm 5 cm

6 cm

h = 5 cm

8 dm

6 dm

7 dm

12 dm h = 7 dm

8 cm h = 7 cm

10 cm

9 cm

9 m

12 m

h = 8 cm

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SA = _____

Check your answer with the answer key. If you get…

8-10 Excellent! You may now proceed to the next lesson.

5-7 You need to review the processes you missed.

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Key to Correction

COMPREHENSION OF SURFACE AREA

REVIEW

1) 21 cm2 2) 40 cm2 3) 28 cm2 4) 24 cm2 5) 32 cm2

TRY THESE

A.

1) Face A – 8 x 10 = 80 cm2 Face B – 8 x 10 = 80 cm2

Face C – 8 x 10 = 80 cm2

Face D – (8 x 7)  2 = 28 cm2 Face E – (8 x 7)  10 = 28 cm2

Total Surface Area = 296 cm2

2) Face F – 10 x 15 = 150 cm2 Face G – 6 x 15 = 90 cm2

Face H – 10 x 15 = 150 cm2

Face I – (8 x 6)  2 = 24 cm2 Face J – (8 x 6)  2 = 24 cm2

3) Face K – 7 x 8 = 56 cm2 Face L – 9 x 8 = 72 cm2

Face M – 5 x 8 = 40 cm2

Face N – (9 x 4)  2 = 18 cm2 Face O – (9 x 4)  2 = 18 cm2

B.

1) Front Triangle – A = 5 x 4 = 20 = 10 cm2 2 2

Back Triangle – A = 5 x 4 = 20 = 10 cm2

2 2 1 Rectangle – A = 5 x 10 = 50 cm2

3 x 50 cm2 = 150 cm2 total area of 3 rectangles

Surface Area = 10 cm2 + 10 cm2 + 150 cm2 = 170 cm2

ON YOUR OWN

A. B.

1) V = 400 cm3 2) V = 720 cm3 3) V = 324 cm3

1) V = 1126.67 m3 2) V = 625 m3