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Derivation of Formulas

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- 1. AREA Richard B. Paulino INRSF – Laoag City
- 2. AREA A square with a side of 1 unit has an area of 1 square unit in symbol 1 unit2 Hence, the unit of measure used for measuring the area of a plane figure is square unit (unit2 ) Unit for Side Unit for AREA cm cm2 (square centimeter) m m2 (square meter) ft ft2 (square feet) – size of a surface or region. - the number of square units to cover a surface or region.
- 3. AREA of a SQUARE How many square units will be used to cover a square region whose side is 3 units long? 2 sA A = 3 units x 3 units A = 9 square units A=9 units2 Where: s = side of a square Let s be the side of a square then, A=s units x s units
- 4. AREA of a SQUARE 21 cm 9.5 km 3.75 cm 18 in 31 ft 17 dm 1) 3) 7) 5) 4) 6) 2) Formula_____________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Instruction: Find the area of each square.
- 5. AREA of a RECTANGLE How many square units will be used to cover a rectangular region whose length is 4 units and width is 3 units long? Area = 4 units x 3 units = 12 units2 The formula is: Area =l × w where: w = width l = length A=12 units2
- 6. AREA of a RECTANGLE Example: What is the area of this rectangle? 5 cm 4cm The formula is: Area =l × w where: w = width l = length We know that w = 4 cm and l = 5 cm, so: Area = 5 cm × 4 cm = 20 cm2
- 7. ACTIVITY Area of Rectangles Instruction: Find the area of each rectangle. 9 cm 5 cm 9 yd 7 yd 8 m 2 m 12 mm 3 mm 5 cm 1 cm 18 m 12 m 4 m 130 cm 37 dm 17 0 cm 2) 5)1) 6) 3) 7) 4) 8)
- 8. Name:_______________________________ Score: ____ Grade/section: ____________ Date: _____ ACTIVITY Instruction: Find the area of each rectangle using the given measures. 1 l = 7 km ,w= 14 km 6. l = 2.6 cm , w = 5 cm 2. l = 7 cm , w = 1.5 cm 7. l = 21 ft , w = 12 ft 3. l = 18 yd , w = 9 yd 8. l = 3.75 ft , w = 4.5 ft 4. l = 9.5 in , w = 9 in 9. l = 31 mm , w = 23 mm 5. l = 13 km , w = 8 km 10. l = 11 dm , w = 6 dm
- 9. ACTIVITY AREA OF RECTANGLES Instruction: Answer the following problems completely. 1. The measure of a basketball court is 26 cm by 14 cm, find its area. Given: ____________________ Required:__________________ Formula: __________________ Equation/Number Sentence:__________________ Solution/Answer: __________________________ 2. Find the area of a baseball court with the measure of 90 ft by 60 ft. Given: ____________________ Required:__________________ Formula: __________________ Equation/Number Sentence:__________________ Solution/Answer: __________________________ 3 . One face of chalk box has a length 60 cm and its width is 30 cm, find its area. Given: ____________________ Required:__________________ Formula: __________________ Equation/Number Sentence:__________________ Solution/Answer: __________________________ 4. If the measure of a volleyball court is 50ft by 70 ft, what is its area. Given: ____________________ Required:__________________ Formula: __________________ Equation/Number Sentence:__________________ Solution/Answer: __________________________ 5. The measure of a fishpond is 26 m by 78 m, find its area. Given: ____________________ Required:__________________ Formula: __________________ Equation/Number Sentence:__________________ Solution/Answer: __________________________
- 10. Name:_______________________________ Score: ____ Year/section: ____________ Date: _____ ACTIVITY Instruction: Answer the following problems completely. 6. Find the floor area of the gymnasium whose length and width is 65 m and 45 m respectively. 7. A badminton court has a measure of 27 m by 36 m, find its area. 8. Find the width of a rectangle with the area of 186 square yards and a length of 13 yards. 9. One dimension of rectangular pool table is 76 cm. Its area is 8664 cm2, find the other dimension. 10. The length of the base of the table in the canteen is 15 m and the length of the diagonal is 17 m. Find its area.
- 11. AREA of a Parallelogram height base
- 12. AREA of a Parallelogram height base
- 13. AREA of a Parallelogram 2) Height (h) Base (b) The formula is: Area =b × h where: b = base h=height
- 14. 1) 2) 6) 3) 5) 4)5 cm 13 cm 15 ft 9 ft 7 m 3 m 7 cm 8.5 cm 15 in 5 in Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Instruction: Find the area of each parallelogram.
- 15. AREA of a TRIANGLE Height (h) Base (b) Area of a Paralellogram Area =b × h Area of a Triangle Area = ½ (Area of a Parallelogram) Area = ½ (b × h)
- 16. AREA of a TRIANGLE A = 3 units X 3 units A = 9 square units A = 4 units X 3 units A = 12 square units (9 square units) 4.5 square units) the area of the rectangle (12 square units) 6 square units
- 17. AREA of a TRIANGLE (b x h) Where: b= base h = height
- 18. Example
- 19. AREA of a TRIANGLE 5 m 5 m 45 m 2) 1) 3) Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ 12 m 9 mm 8 mm 4) Formula_______________ Equation_______________ Solution_______________
- 20. Area of a Trapezoid b1 b2 A = base x height Height (h) Area of a parallelogram base A = (b1 + b2) x h Area of a trapezoid = ½ Area of a parallelogram Area of a trapezoid = ½ (b1 + b2) x h
- 21. Example Find the area of trapezoid ABCD. 6 units 4 units 8 units A = ½ h ( b1 + b2) = ½ ( 4) (8 + 6 ) = ½ ( 4 ) ( 14 ) A= 28 The area is 28 square units.
- 22. 3 m 7.25 m 2.5 m 18 cm 25 cm 6 cm 10 ft 5 ft 4 ft 3 m 2 m 7 m 5 ft 3 ft 10 ft 5 in 14 in 23 in Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ Formula_______________ Equation_______________ Solution_______________ 2) 5) 1) 6) 3) 4) Instruction: Find the area of each trapezoid.
- 23. AREA of a CIRCLE Radius (r) C=2πr 1/2C=πr 1/2C=πr base (b)= πr height (h) = r A = base x height A = A = πr2 x rπr
- 24. d = 6 cm Example 1 The radius of a circle is 2 cm. Find its area. Solution: A = r2 r=2 cm 3.14 (2)2 ● 12.56 cm2 The area is 12.56 square cm. Example 2 The diameter of a circle is 6 cm. Find its area. Solution: Step 1. Find the radius Radius( r ) = diameter (d) divided by 2 r = 6 2 r = 3 cm The radius is 3 cm. ● Step 2. Find the area. A = r2 3.14 (3)2 3.14 (9) 28.26 cm2 The area is 28.26 cm2
- 25. Try this out Find the area of each circle with the given diameter or radius Use 3.14 for . 1. `radius = 5 cm 2. radius = 1.5 mm 3. diameter = 4 cm 4. diameter = 12 dm 5. radius = 4.6 m 6. radius = 2.2 cm 7. diameter = 4.8 dm 8. diameter = 6.4 cm 9. radius = 4.8 m 10. radius = 3.4 dm
- 26. Let’s Summarize 1. The area of a region is the number of square units contained in the region. 2. A square unit is a square with a side 1 unit in length. The area (A) of a rectangle is the product of its length (l) and its width (w). A = lw 3. The area (A) of a square is the square of the length of a side (s). A = s2 4. The area (A) of a parallelogram is equal to the product of the base (b) and the height (h). A = bh
- 27. 5. The area (A) of a triangle equals half the product of the base (b) and the height (h). A = ½ bh. Sometimes altitude is used instead of height. 6. The area (A) of a trapezoid is one half the product of the length of its altitude and the sum of the lengths of the two bases. A = ½ h (b1 + b1). 7. A circle is a set of points in a plane that have the same distance from a given point in the plane. 8.The formula for the area of a circle with a radius of r and diameter of d units are: A = r2 and A = (d/2)2 respectively. Note: In all circles the ratio of the circumference to the diameter is always equal to the same number, represented by the Greek letter .

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