23. [Perimeter / Area]
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1 3. [Perimeter / rea] Skill 3. Calculating the perimeter of polygons (). MM MM Convert all measurements to the same unit. Find and label the length of all sides. dd together all side lengths. Hints: Sides marked with a dash ( ) are of equal length. Sides marked with two dashes ( ) are of equal length etc. Q. Find the perimeter of the scalene triangle in centimetres. 0 cm 0 cm cm 65 cm P 65 cm + 0 cm + 0 cm P 45 cm Convert to cm a) Find the perimeter of the rectangle P b) Find the perimeter of the trapezium..5 m.9 m 0. m P m m c) Find the perimeter of the kite. d) Find the perimeter of the right-angled triangle in millimetres cm 40 8 cm P P e) What is the perimeter of a regular heptagon with sides measuring 4 m? P... f) What is the perimeter in centimetres of a rhombus with a side length measuring 5? m P page 59 Maths Mate 5./6. Skill uilder 3
2 Skill 3. Calculating the perimeter of polygons (). MM MM g) What is the perimeter in centimetres of an isoceles triangle with congruent sides of 5 m and the other side measuring.5 m? h) Find the perimeter in centimetres of a parallelogram with side lengths measuring 0 cm and 00. P i) The smallest ever postage stamp came from Columbia. Rectangular, it measured.85 by 9.4. What was its perimeter in cm? P k) Lisa s backyard is a rectangle measuring 8 m in length and m in width. What will the perimeter of the backyard be? P P j) n ustralian $0 note measures 4.4 cm by 6.5 cm. What is its perimeter in millimetres? P l) Find the perimeter in centimetres of a kite with side lengths measuring 80 cm and 50. m) Find the perimeter of the trapezium. 00 m m P n) Find the perimeter of the triangle m 3 0 m P P m o) Write an algebraic expression for the perimeter P of the heptagon. [Express the answer in terms of r.] p) Write an algebraic expression for the perimeter P of the rectangle. [Express the answer in terms of s.] r s 3s P P P P page 60 Maths Mate 5./6. Skill uilder 3
3 Skill 3. Calculating the perimeter of composite shapes. MM MM Find and label the length of all sides. R dd together all side lengths. Manipulate shapes to Hints: Sides marked with a dash ( ) are of equal length. become rectangles by Sides marked with two dashes ( ) are of equal length etc. pushing out inverted corners. Q. Find the perimeter of the shape.. P m m 3 m 4.5 m Find unknown R side lengths 0 m P m 6 m m unknown sides: 6 + dash + dash 0 m So each dashed side m shape becomes a rectangle 6 m 8.5 m 4 m a) Find the perimeter of the shape. 3.5 P b) Find the perimeter of the shape. 3 5 P c) Find the perimeter in centimetres around the coloured background of this Tongan flag. d) Find the perimeter of the shape. 0.6 cm 60 cm 5 cm 5 cm P.3 cm P cm cm e) Find the perimeter of the shape. f) Find the perimeter of the shape. 4 m 3.5 m 5 cm 4 cm.5 m.5 m 0 cm P P m cm page 6 Maths Mate 5./6. Skill uilder 3
4 Skill 3.3 Calculating the circumference of circles. MM MM Substitute known values into the formula. Hints: The diameter of a circle is equal to twice the radius. π (pi) gets its value because the diameter of any circle fits approximately 3.4 times around the circumference. Circumference of a circle C π radius C πr R C π diameter C πd where π or Q. Using C πr where π, find the length of the semicircle. m a) Using C πr where π 3.4, find the circumference of the circle.. C πr Simplify: 44 C 44 m b) Using C πr where π 3.4, find the circumference of the circle. 0 m C πd where d c) Using C πr where π, find the circumference of the circle. C πr d) Using π 3.4 find the length of the semicircle. m 56 cm 30 m C e) The diameter of a circular discus is.5 m. Using π 3.4 what is the circumference? cm C C f) Using π 3.4 find the length of the quarter circle. m C m C 8 4 C page 6 Maths Mate 5./6. Skill uilder 3
5 Skill 3.4 Calculating the perimeter of composite circular shapes (). MM MM Find and label the length of all sides. reak the shape into workable parts. For circular shapes substitute known values into the formula for the circumference: Hint: Consider congruent semicircles equal full circle. dd together all side lengths. Hints: Sides marked with a dash ( ) are of equal length. Sides marked with two dashes ( ) are of equal length etc. Circumference of a circle C πr πd Q. Find the perimeter of the shape. (Use π 3.4) C πr where r C πr where r C πd where d 5 D shape D semicircle semicircle circle D a) Using C πr where π 3.4, find the perimeter around the outside of the first lane of an athletics track. Standard 400 m athletics track ( lane shown). m b) Find the perimeter of the shape. (Use π ).4 cm 85 m ' 36.4 m Consider semicircles as circle C πr where r C P m C πr P cm page 63 Maths Mate 5./6. Skill uilder 3
6 Skill 3.4 Calculating the perimeter of composite circular shapes (). MM MM c) Using C πr where π 3.4, find the perimeter of the shape. 0 d) Find the perimeter of the shape. (Use π ) 4 m 5 C πr where r C P e) Using C πr where π 3.4, find the perimeter of the shape. 9 C πr P f) Find the perimeter of the shape. (Use π ) 34 cm m 8 0 cm 0 P g) Write an algebraic expression for the perimeter P of the shape. [Express the answer in terms of r and π.] h) Write an algebraic expression for the circumference P of the outer circle. [Express the answer in terms of k and π.] cm r k k P P page 64 Maths Mate 5./6. Skill uilder 3
7 Skill 3.5 Calculating the area of squares and rectangles. MM MM Substitute known values into the appropriate formula. rea of a square l l l l length rea of a rectangle l w lw l length w width Q. boxing ring is a square with side length 5. m. What is the area of the ring? a) What is the area of a rectangular billiard table with a length of 3. m and a width of.9 m? l w 3..9 m. l m.04 m b) Find the area of the square. 3 cm cm c) Find the area of the rectangle. 4.3 d) baseball diamond is a square of side length of approximately m. What is its area? m e) The rectangular grounds of the Taj Mahal are 360 m long and 60 m wide. What is its area? f) rectangular badminton court measures approximately 3.5 m long and 6 m wide. What is its area? l w m m g) What is the perimeter of a square with an area of 400 cm? length P cm h) Paddy s rectangular ipod screen has an area of 0. What is the perimeter of the screen, if the length measures 30? width P page 65 Maths Mate 5./6. Skill uilder 3
8 Skill 3.6 Calculating the area of triangles. MM MM Substitute known values into the formula: rea of a triangle b base h height b h bh Q. Find the area of the scalene triangle.. bh 5 0 cm 0 cm 35 cm Simplify: a) Find the area of the right-angled triangle. b) Find the area of the right-angled triangle. 5 m 8 0 bh 8 m 8 5 m bh c) Find the area of the isoceles triangle. d) Find the area of the scalene triangle. 4 cm bh 8 6 cm cm e) Plot the points ( 6,), (,6) and C(5,) and use them to find the area of ΔC. Y X f) Plot the points (,3), (3,3) and C(, 3) and use them to find the area of ΔC. Y X page 66 Maths Mate 5./6. Skill uilder 3
9 Skill 3. Calculating the area of parallelograms. MM MM Substitute known values into the formula. rea of a parallelogram h height b h bh b base Q. Find the area of the parallelogram.. bh m 3.5 m.5 m 8.5 m a) Find the area of the parallelogram. b) Find the area of the parallelogram. 5 cm 4 cm cm cm bh 4 5 cm cm c) Find the area of the parallelogram. d) Find the area of the parallelogram. 30 cm cm..5 cm e) Find the area of the parallelogram. f) Find the area of the parallelogram. 5.5 m 3 m m page 6 Maths Mate 5./6. Skill uilder 3
10 Skill 3.8 Calculating the area of rhombi and kites. MM MM Substitute known values into the formula. rea of a rhombus or kite a b R b a ab a b (where a is the long diagonal and b is the short diagonal) Q. Find the area of the kite.. ab m.5 m 3 m a) Find the area of the kite. b) Find the area of the rhombus. 0 cm 8 m 5 m ab.5 cm.5 0 c) Find the area of the rhombus cm 0 cm cm ab d) Find the area of the kite. m 6 ab cm e) Plot the points (0,4), (,), C(0, 3) and D(,) and use them to find the area of the kite CD. Y X f) Plot the points ( 4,6), (,3), C( 4,0) and D( 6,3) and use them to find the area of the rhombus CD. Y X page 68 Maths Mate 5./6. Skill uilder 3
11 Skill 3.9 Calculating the area of trapeziums. MM MM Substitute known values into the formula. rea of a trapezium h b a (a + b) height (where a and b are the parallel side lengths) (a + b)h Q. Find the area of the trapezium.. (a + b)h.5 (4 +.5) a) Find the area of the trapezium. 3 m 9 m (a + b)h (3 + 9) 3 c) Find the area of the trapezium. (a + b)h m cm e) Plot the points (0,4), (3,4), C(3, ) and D( 4, ) and use them to find the area of the trapezium CD. Y m 8 cm 0 cm X cm b) Find the area of the trapezium. 0 0 (a + b)h d) Find the area of the trapezium. 5 cm 8 cm 3 cm cm f) Plot the points ( 4,5), (4,6), C(4,) and D( 4,4) and use them to find the area of the trapezium CD. Y X page 69 Maths Mate 5./6. Skill uilder 3
12 Skill 3.0 Calculating the area of composite shapes (). MM MM Find and label the length of all sides. reak the shape into workable parts. Where possible substitute values into a known area formula. (see skill 3.5, page 65 to skill 3.9, page 69) dd or subtract the area totals where necessary. Q. Find the area of the polygon.. 3 m 6 m 8 m 8 m 3 m 6 m + (a + b)h (3 + 8) m 6 m trapezium triangle bh shape m a) Find the area of the shape. 5 m 6 m 4 m 5 m 5 m minus 5 m b) Find the area of the polygon. 8 cm 0 cm plus 0 cm 8 cm 5 m 4 m 5 m 3 cm 0 cm 3 0 cm lw (a rectangle) bh (a triangle) shape m l shape cm page 0 Maths Mate 5./6. Skill uilder 3
13 Skill 3.0 Calculating the area of composite shapes (). MM MM c) Find the area of the shape shape e) Find the area of the shape. d) Find the area of the polygon. 30 cm 8 cm shape f) Find the area of the polygon. cm 8 m m 5 m 6 m 5 m 5 m 5 m shape m g) Write an algebraic expression for the area of the shape. [Express the answer in terms of a and b.] shape m h) Write an algebraic expression for the area of the shape. [Express the answer in terms of a and b.] a b a a b page Maths Mate 5./6. Skill uilder 3
14 Skill 3. Calculating the area of circles. Substitute known values into the formula: Hint: The diameter of a circle is equal to twice the radius. Pi (π) gets its value because the diameter of any circle fits approximately 3.4 times around the circumference. MM MM rea of a circle π radius radius πr where π or Q. Using πr where π 3.4, find the area of the circle. 4 m. πr where d 4, so r m a) Using πr where π 3.4, find the area of the circle. b) Using πr where π 3.4, find the area of the circle. 0 5 m πr where d 0 so r πr m c) Using πr where π, find the area of the circle. d) Using π find the area of the semicircle. 4 m 8... e) Using π find the area of the quarter circle. m f) Using π 3.4 find the area of the shape. cm 0 m cm m page Maths Mate 5./6. Skill uilder 3
15 Skill 3. Calculating the area of composite circular shapes (). MM MM Find and label the length of all sides. reak the shape into workable parts. Where possible substitute values into a known area formula. (see skills 3.5 to skill 3.9, pages 65 to 69 and skill 3., page ) dd or subtract the area totals where necessary. Q. Use πr where π 3.4, to find the area of the shaded shape. m m 9 m a) Use πr where π, to find the shaded area.. 9 m πr where r (a + b)h (8 + ) shape m 9 m 9 m m m circle trapezium b) Use πr where π, to find the shaded area. 30 cm 5 cm bh cm (a triangle)... ( ) πr, r (a semicircle) shape cm 5 cm cm minus cm cm 4 m πr, r 4 (a circle) l (a square) shape... 4 m minus 4 m 4 m 4 m m page 3 Maths Mate 5./6. Skill uilder 3
16 Skill 3. Calculating the area of composite circular shapes (). c) Use πr where π, to find the area of the background colour of the flag of Zaire, without the central circle. 40 cm MM MM d) Use πr where π 3.4, to find the area of the shaded shape. 4 cm 5 cm 0 shape cm shape e) Use πr where π 3.4, to find the area of the shaded shape. f) Use πr where π 3.4, to find the area of the shaded shape. 0 m 4 m 6 m 6 m shape m shape m page 4 Maths Mate 5./6. Skill uilder 3
17 Skill 3. Calculating the area of composite circular shapes (3). MM MM g) Write an algebraic expression for the area of the shaded shape. [Express the answer in terms of r and π.] h) Write an algebraic expression for the area of the shaded shape. [Express the answer in terms of r, l and π.] r r r r r l r πr (a circle) l (a square) r shape... R r (π ) i) Write an algebraic expression for the area of the shaded shape. [Express the answer in terms of r and π.] πr πr r shape... j) Write an algebraic expression for the area of the shaded shape. [Express the answer in terms of d and π.] r d shape... shape... k) Write an algebraic expression for the area of the shaded shape. [Express the answer in terms of a, b and π.] l) Write an algebraic expression for the area of the shaded shape. [Express the answer in terms of a, b and π.] a b b rectangle semicircle shape... triangle semicircle semicircle shape... a page 5 Maths Mate 5./6. Skill uilder 3
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