Circle R has diameter ST with endpoints S(4, 5) and T(-2, 3). What are the circumference and area of the circle?

Question

Sadie Allen · Accepted Answer

Circumference is #19.87# and area is #31.416# The distance between endpoints #(4,5)# and #(-2,3)# of the diameter #ST# of circle #R# is #sqrt(-2-4)^2+(3-5)^2=sqrt(36+4)=sqrt40=sqrt(2xx2xx10=2sqrt10)# Hence, radius of the circle #R# is #(2sqrt10)/2=sqrt10# and Circumference of circle is given by #2pir# i.e. #2pixxsqrt10=2xx3.1416xx3.1623=19.87# and Area of circle is given by #pir^2# i.e. #3.1416xx(sqrt10)^2=3.1416xx10=31.416#

Noah Atkinson · Answer

undefined The radius of the circle, $ r $, can be calculated using the distance formula: $ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $.

Substituting the given points $ S(4, 5) $ and $ T(-2, 3) $ into the formula, we get:
$$ r = \sqrt{(-2 - 4)^2 + (3 - 5)^2} $$
$$ r = \sqrt{(-6)^2 + (-2)^2} $$
$$ r = \sqrt{36 + 4} $$
$$ r = \sqrt{40} $$
$$ r = 2\sqrt{10} $$

The diameter of the circle is twice the radius, so $ d = 2r = 2 \times 2\sqrt{10} = 4\sqrt{10} $.

The circumference of the circle is given by the formula $ C = \pi \times d $:
$$ C = \pi \times 4\sqrt{10} $$
$$ C = 4\pi\sqrt{10} $$

The area of the circle is given by the formula $ A = \pi \times r^2 $:
$$ A = \pi \times (2\sqrt{10})^2 $$
$$ A = \pi \times 4 \times 10 $$
$$ A = 40\pi $$