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Construct an equilateral triangle of side 5 cm and its circumscribed circle, and calculate the total surface area.

Mathematics

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Answer

b

517 cm2517 \text{ cm}^2

Step-by-step solution

14 steps

Part a· 4 steps

To construct an equilateral triangle of side 5 cm

  1. Draw a line segment PQ of length 5 cm using a ruler.

  2. Place the compass point at P and open it to a radius of 5 cm.

    Draw an arc above the line segment PQ.

  3. Place the compass point at Q and, with the same radius of 5 cm, draw another arc that intersects the first arc.

    Label the intersection point R.

  4. Draw straight lines from P to R and from Q to R using a ruler.

    Triangle PQR is the equilateral triangle with sides of 5 cm.

Part b

517 cm2517 \text{ cm}^2

To construct the circumscribed circle of triangle PQR

  1. Construct the perpendicular bisector of side PQ.

    To do this, place the compass point at P, open it to more than half the length of PQ, and draw arcs above and below PQ. Repeat this from point Q with the same compass opening. Draw a straight line connecting the two points where the arcs intersect. This is the perpendicular bisector of PQ.

  2. Construct the perpendicular bisector of side QR using the same method as in Step 1.

  3. The point where the two perpendicular bisectors intersect is the circumcenter of the triangle.

    Label this point O.

  4. Place the compass point at O and open it to the distance OP (or OQ or OR, as these distances will be equal).

    Draw a circle. This circle passes through all three vertices P, Q, and R, and is the circumscribed circle.

    20.
    To calculate the total surface area of a closed cylinder, use the formula:

    A=2πr2+2πrhA = 2\pi r^2 + 2\pi rh

    where rr is the radius and hh is the height.

  5. Identify the given values.

    Diameter d=7d = 7 cm
    Height h=20h = 20 cm
    π=227\pi = \frac{22}{7}

  6. Calculate the radius rr.

    The radius is half of the diameter:

    r=d2=72 cmr = \frac{d}{2} = \frac{7}{2} \text{ cm}

  7. Substitute the values into the surface area formula.

    A=2(227)(72)2+2(227)(72)(20)A = 2 \left(\frac{22}{7}\right) \left(\frac{7}{2}\right)^2 + 2 \left(\frac{22}{7}\right) \left(\frac{7}{2}\right) (20)

  8. Calculate the area of the two circular bases (2πr22\pi r^2).

    2(227)(494)=2×227×4942 \left(\frac{22}{7}\right) \left(\frac{49}{4}\right) = 2 \times \frac{22}{7} \times \frac{49}{4} =2×22×497×4= \frac{2 \times 22 \times 49}{7 \times 4} =2×22×7×77×2×2= \frac{2 \times 22 \times 7 \times 7}{7 \times 2 \times 2} =22×72= \frac{22 \times 7}{2} =11×7=77 cm2= 11 \times 7 = 77 \text{ cm}^2

  9. Calculate the lateral surface area (2πrh2\pi rh).

    2(227)(72)(20)2 \left(\frac{22}{7}\right) \left(\frac{7}{2}\right) (20) =2×22×7×207×2= \frac{2 \times 22 \times 7 \times 20}{7 \times 2} =22×20=440 cm2= 22 \times 20 = 440 \text{ cm}^2

  10. Add the areas to find the total surface area.

    A=77 cm2+440 cm2A = 77 \text{ cm}^2 + 440 \text{ cm}^2 A=517 cm2A = 517 \text{ cm}^2

    The total surface area of the cylinder is 517 cm2\boxed{517 \text{ cm}^2}.

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