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Special Triangles & Areas of Regular Polygons: Math Exercise

Explore special right triangles like 45-45-90 and 30-60-90, understand their properties, and apply formulas for calculating areas. Practice finding side lengths, perimeters, and areas in a step-by-step manner. Learn how to determine the lengths of hypotenuses, shorter legs, and longer legs, and apply the Pythagorean theorem and trigonometric relationships to solve geometry problems. Enhance your understanding of geometric concepts through hands-on exercises and visual aids.

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Special Triangles & Areas of Regular Polygons: Math Exercise

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  1. Please work on the pink warm up

  2. 5.5 Special Triangles & Areas of Regular Polygons

  3. 2x x √3 x √2 x x FIND THE SHORT LEG FIRST!!! x There are 2 special right triangles45-45-9030-60-90 In a 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg AND the length of the longer leg is √3 times the length of the shorter leg. In a 45-45-90 triangle, the length of the hypotenuse is √2 times the length of the leg.

  4. R Q P LET’S TRY THIS… 6 √3 12 X = 6 Y = _____ Z = _____ Z 60 X 2 √3 X = _______ Y = 6 Z = ________ 4 √3 30 Short side first!! Y 4 √2 8 P = ______ Q = 4√2 R= _____ 5 5√2 P = 5 Q = ________ R = ________

  5. 10 Find the area game♥ A = ½ bh What is the length of the missing side? 6 A = ½ (6) (6) A = 18 sq. units 6 A = ½ bh Find the short side first!! A = ½ (5) (5√3) Find an exact answer A = 25√3 sq. units or 12.5√3 2 5√3 5

  6. Areas of Regular Polygons What part of A=1/2bh is the perpendicular bisector? Can you find the area of a triangle? The perpendicular bisector of a triangle in a polygon is called an APOTHEM. The formula for the area of a regular polygon is: A = ½ap a is the length of the apothem p is the perimeter of the polygon

  7. Let’s see how this works… 10 A = 1/2ap A = ½(6.88)(50) A = 172 sq.units 6.88 PAINLESS!! Let’s kick it up a notch…

  8. 12 60° 60° 30° 60° Find the area of this one! Hmmmmm….. A circle has 360°… Hmmmmm….. How many degrees would the top angle of each Δ have? 60° Hmmmmm….. Since the Δs are isosceles, what are the measures of the base angles? 30° Hmmmmm….. If the apothem is an angle bisector, then what is the measure of the small top angle? The short side = 6 The apothem = 6√3 A = 1/2ap A = ½(6√3)(72) = 216√3 (exact) A = 374.12 (approx.)

  9. 18 WHEW!Try this… Find the perimeter and area of a 30-60-90 Δ with a hypotenuse of 18units. (sketch it) What is the length of the short side? What is the length of the long leg? What is the perimeter? (exact) 27 + 9√3 units What is the area? (exact) 81√3 sq units or 40.5√3 units2 2 60° 9 30° 9√3 (hint: the smallest angle is across from the smallest side… the largest angle is across from the largest side.)

  10. Whiteboard time!!

  11. Assignmentpg 336, 10-21

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