4. [Suface Aea] Skill 4.1 Calculating the total suface aea (TSA)) of ectangula pisms and cubes using nets (1). Find any unknown side lengths. Calculate the aea of each face as shown on the net. Hint: Rectangula pisms have 6 faces of 3 diffeent sizes: base and top () font and back () othe faces () Add togethe the aea of all faces. Hints: Sides maked with a dash ( ) ae of equal length. Sides maked with two dashes ( ) ae of equal length etc. Q. Find the total suface aea of the cube by finding the aea of its net. 5 mm A. Aea of squae face 5 mm 5 mm 5 mm 5 m 6 150 m A cube has 6 identical faces a) Find the total suface aea of the ectangula pism by finding the aea of its net. b) Find the total suface aea of the cube by finding the aea of its net. 30 cm 0 cm F o n t Top B a c k 3 cm F o n t Top B a c k Aea: base & top 0 3 10 Aea: font & back 30 3 180 Aea: othe faces 30 0 100 10 + 180 + 100 Aea of 1 face page 77 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.1 Calculating the total suface aea (TSA)) of ectangula pisms and cubes using nets (). c) Find the total suface aea of the squae pism by finding the aea of its net. d) Find the total suface aea of the ectangula pism by finding the aea of its net. Top 3 m Back Top L a te f a ce Top a l Font 1 cm 0 cm 5 m Aea: base & top Aea: 4 lateal faces Aea: base & top Aea: font & back Aea: othe faces cm e) Find the total suface aea of the squae pism by finding the aea of its net. f) Find the total suface aea of the ectangula pism by finding the aea of its net. 4 mm 1 4 cm 7 mm 30 cm m cm page 78 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4. Calculating the total suface aea (TSA)) of ectangula pisms. ectangula pism (length width) + (length height) + (width height) lw + lh + wh (lw + lh + wh) l w h cube 6(length length) 6l l Q. Lewis wants to make a box, with a lid, fo his cad collection. The box needs a base of 11 cm by 0 cm and must be 1 cm high. How much wood does Lewis need? a) The locke block needs to be esufaced. What is the suface aea of this ectangula pism disegading its base? A. (11 0 + 11 1 + 0 1) (0 + 13 + 40) 59 1184 b) Zoe s mattess was ton in emoval. What is the minimum amount of mattess ticking needed to e-cove the mattess? 1.5 m 190 cm 0.5 m 5 1 Subtact 1 base aea m lw + lh + wh 110 55 + 110 190 + 55 190 6050 + 0 900 + 10 450 6050 + 41 800 + 0 900 (lw + lh + wh) c) Find the total suface aea of the micowave. 50 cm 30 cm 3 d) The total suface aea of the ectangula pism is 5. What is the TSA if all the dimensions ae doubled? 4 cm cm 3 cm... page 79 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.3 Calculating the total suface aea (TSA)) of ectangula composite solids (1). Find any unknown side lengths. Calculate the aea of each face. Add togethe the aea of all faces. OR Identify the base by finding the two, identical paallel faces. Hint: A pism does not necessaily sit on its base. Substitute values into the fomula: ectangula composite solid Peimete of base height + Aea of base P bh + Ab h Q. Find the total suface aea of the pism. A. 1 mm OR 6 mm 1 mm Pb 6 + 1 + 5 + 1 + 1 + 16 Pb 6 + 6 + + 16 1 mm 5 mm 5 mm base base 6 mm 6 mm 1 mm mm h mm mm Ab 5 1 + 1 5 + 7 Pbh + Ab 16 + 7 3 + 14 46 m Fo P b, convet to a ectangle Find unknown side lengths a) Find the total suface aea of the pism. b) Find the total suface aea of the pism. 3 cm Fo P b, convet to a ectangle Fo A b, find all unknown side lengths P b 10 + 10 + 8 + 8 36 A b 5 5 + 5 8 5 + 40 65 Use TSA fomula P b h + A b fo a pism 36 3 + 65 108 + 130 10 mm 7 mm P b A b P b h + A b m page 80 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.3 Calculating the total suface aea (TSA)) of ectangula composite solids (). c) Find the total suface aea of the pism. 10 m Find unknown side lengths d) Find the total suface aea of the pism. 6 m 4 m 3 m 4 m P b A b P b h + A b m P b A b P b h + A b e) A window m by 1.5 m and a dooway m by 0.8 m ae in the plan fo this oom. Find the total aea of the inside walls to be painted. 5 m f) Find the total suface aea of the pism. 4 m 3 m 9 mm g) Find the total suface aea of the pism. 40 mm 10 mm h) Find the total suface aea of the pism. m 3 cm 1 m page 81 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.4 Calculating the total suface aea (TSA)) of tiangula pisms (1). Find any unknown side lengths. Calculate the aea of each face. Add togethe the aea of all faces. OR Substitute values into the fomula: tiangula pism Peimete of base height + Aea of base h P bh + Ab Hint: Do not confuse the height needed to calculate the aea of the tiangula base, with the height (h) of the pism. Q. Find the total suface aea of the tiangula pism. 4 cm 7 cm A. P b 6 + 5 + 5 16 1 b Ab bh 1 (6 4) 1 P b h + A b 16 7 + 1 11 + 4 136 h 7 cm 4 cm a) Find the total suface aea of the tiangula pism. b) Find the total suface aea of the tiangula pism. 3 cm 1 cm P b 1 + 1 + 1 36 1 A b (1 8) 48 Fist find the peimete and aea of the base P b h + A b 36 5 + 48 900 + 96 P b A b page 8 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.4 Calculating the total suface aea (TSA)) of tiangula pisms (). c) Find the total suface aea of the tiangula pism. 1 mm 5 mm 0 mm d) Find the total suface aea of the tiangula pism shaped slice of cheese. cm 3 cm 1 cm 16 mm. P b... A b... P b h + A b... P b A b P b h + A b...... m e) Find the total suface aea of the tiangula pism. f) Find the total suface aea of the tiangula pism. 5 m 5 m 6.5 m P b... A b...... 6 m P b A b...... page 83 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.5 Calculating the total suface aea (TSA)) of pyamids (1). Find any unknown side lengths. Calculate the aea of each face. Add togethe the aea of all faces. OR Substitute values into the fomulas: egula squae pyamid s slant height s Aea of squae base + 4 Aea of tiangle 1 A b + 4 ls l + ls l l egula tiangula pyamid (egula tetahedon) 4 Aea of equilateal tiangle 1 x 3 4 x x 3 x x 3 ectangula pyamid Aea of base + Aea of tiangles left & ight + Aea of tiangles font & back 1 1 B + ws 1 + ls s 1 s lw + ws 1 + ls l w Q. Find the total suface aea of the egula squae pyamid. 1 m A. l + ls whee l 8 and s 1 8 8 + 8 1 64 + 16 1 64 + 19 56 8 m a) Find the total suface aea of the egula squae pyamid. b) Find the total suface aea of one of the salt and peppe shakes given that they ae egula squae pyamids of base side length 3 cm and slant height 4 cm. l + ls whee l 5 and s 6 5 5 + 5 6 5 + 60 l + ls page 84 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
40 m 14 m4 m Skill 4.5 Calculating the total suface aea (TSA)) of pyamids (). c) Find the total suface aea of the lagest egula squae pyamid below. It has a base side length of 00 m and slant height of 50 m. d) Find the total suface aea of the egula squae pyamid. 18 mm 1 mm m e) Find the suface aea of the egula squae pyamid. f) Find the suface aea of the ectangula pyamid. 64 m 64 m 15 m 40 m g) Find the suface aea of the egula tetahedon. [Give you answe as a sud.] 3 3 cm h) Find the suface aea of the egula tetahedon. [Give you answe as a sud.] 1 m page 85 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.6 Calculating the total suface aea of composite solids (1). Beak the solid into wokable pats. Calculate the total suface aea of each solid. (see skills 4., page 79 and 4.3, page 80) Add the esults. Q. Find the total suface aea of the obelisk. A. TSA egula squae pyamid (without base) l 8 (length) ls whee l 8 and s 10 10 mm s 10 (slant height) 8 10 8 mm 160 TSA squae pism (without base) 15 mm 4lh + l whee l 8 and h 15 4 (8 15) + 8 8 4 10 + 64 544 TSA obelisk 160 + 544 704 m a) Disegading the entance, find the total suface aea of the doghouse, excluding its floo. 0 cm 9 100 cm 80 cm 1 TSA oof pism 100 5 + 96 0... b) Disegading the doo and windows, find the total suface aea of the log cabin, excluding its floo. 1.4 m 9.6 m 1 m 5 m 3 m TSA oof pism...... TSA base pism 100 80 + 96 80...... TSA base pism...... TSA house c) Find the total suface aea of the glass house, excluding its floo. 4 m 6 m 0 m 5 m 5 m TSA oof pism...... TSA cabin d) Find the total suface aea of the tent canvas excluding its floo. 00 cm 300 cm 400 cm 170 cm...... TSA base pism............ TSA house... page 86 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.6 Calculating the suface aea of composite solids (). e) Find the total suface aea of the solid. f) Find the total suface aea of the solid. 17 cm 13 m 1 m 10 m Roof 1 P b 36 A b 10 1 60... TSA pism... TSA pism face... TSA cube face... TSA solid... g) Find the total suface aea of the solid. h) Benie bought a ectangula box containing 15 tightly packaged eases. What is the total suface aea of the box? 1 cm 17 cm 13 cm 1 cm 4 cm page 87 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.6 Calculating the suface aea of composite solids (3). i) Find the total suface aea of the pism. j) Find the total suface aea of the octahedon. 18 m 4 cm cm 10 m k) Disegading the entance, find the total suface aea of the maquee canvas excluding its floo. l) Find the total suface aea of the obelisk. 10 m 4 m.5 m 3 m 3 m 3 m 5 m page 88 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.7 Calculating the total suface aea (TSA)) of basic 3-dimensional ound shapes (1). Substitute values into the fomulas: cylinde π + πh π( + h) h π h cone π + πs π( + s) s sphee 4π Q. Using π ( + h) and π 3.14, find the total suface aea of the cyclinde. 4 cm A. π ( + h) whee and h 8 3.14 ( + 8) 1.56 10 15.6 a) Use π ( + s) and π 3.14, to find the total suface aea of the conical caot. b) Using 4π and π, find the 7 total suface aea of the sphee. 4 cm 1 m π( + s) whee and s 10 3.14 ( + 10) 6.8 1 75.36 page 89 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.7 Calculating the total suface aea (TSA)) of basic 3-dimensional ound shapes (). c) Using 4π and π, find the 7 total suface aea of the snow globe. 140 mm d) Use π ( + s) and π 3.14 to find how much aea still needs to be coveed in chocolate to cove the whole cone only on the outside, given that 40 have been coveed so fa. 1 cm............ m e) Using π ( + h) and π 3.14, find the total suface aea of the cyclindical stool seat. 40 cm f) Using π ( + h) and π, find the 7 total suface aea of the can of tuna. 14 cm............ g) Using TSA of a cylinde π ( + h) and π, find the total suface aea of the icing. 7 [N.B. The base of the cake is not iced.] h) This wedding cake is coveed in white icing, except fo the base. Using π 3.14 find the total suface aea of the white icing. 16 0 cm.................. page 90 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.8 Calculating the total suface aea (TSA)) of moe complex 3-dimensional ound shapes. Substitute values into the appopiate fomula. Adapt fomulas whee necessay. hemisphee 4π + π 3π Q. Using π find the total 7 suface aea of the hemisphee. 14 m A. 3π whee 7 m 1 3 7 7 1 7 66 7 46 a) Using the total suface aea of a sphee 4π and π 3.14, find the total suface aea of the hemisphee. b) The total suface aea of a sphee is 4π. Using π 3.14 find the total suface aea of the watemelon half. 4 mm 30 cm 3π 3 3.14 4 4 9.4 16 m c) Use π 3.14 to find the total suface aea of the shape. d) Use π to find the total suface aea of the 7 shape. 8 m TSA pism 0 m 14 m LA cone TSA cylinde half TSA cylinde page 91 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4
Skill 4.9 Expessing the total suface aea (TSA)) of 3-dimensional shapes in algebaic fom. Substitute values into the appopiate fomula fo total suface aea. (see skills 4. to 4.5, pages 79 to 84, skills 4.7, page 89 and 7.8, page 91) Adapt fomulas whee necessay. Q. Wite an algebaic expession fo the total suface aea TSA of the cone. [Expess the answe in tems of a and π.] a A. π( + s) whee a and s 4a π a (a + 4a ) π a 5a 5πa 4a a) Wite an algebaic expession fo the total suface aea TSA of the cylinde. [Expess the answe in tems of d and π.] 5d d b) Wite an algebaic expession fo the total suface aea TSA of the hemisphee. [Expess the answe in tems of and π.] π( + h) whee d and h 5d πd(d + 5d) πd 6d... c) Wite an algebaic expession fo the total suface aea TSA of the obelisk. [Expess the answe in tems of a.] 1πd... d) Wite an algebaic expession fo the total suface aea TSA of the cube. [Expess the answe in tems of d.] a 3d...... e) Wite an algebaic expession fo the total suface aea TSA of the cylinde. [Expess the answe in tems of x and π.] f) Wite an algebaic expession fo the total suface aea TSA of the cone. [Expess the answe in tems of p and π.] 6x 10x 7p p...... page 9 www.mathsmate.co.nz Maths Mate 5./6.1 Skill Builde 4