## Measurement

### High School Level Questions on Circles and Spheres from the Dr. Math Archives

High School Dr. Math Questions on Measurement

Below are some questions and answers from our archive of questions sent via e-mail by K12 students and teachers to Ask Dr. Math. You may also browse the entire high school level archive by category.

### Circles and Spheres

Area of an Arc
When you draw a circle and make a chord from one point to another, how would you find the area of that arc (formula)?
Area, Angle of Chords of a Circle
Calculate the angles PAB and POB, the area of the sector bounded by OP, OB and the minor arc PB.
Area of Intersection of Two Circles
My teenage son asked me for the formula for the area of intersection of two arbitrary circles.
Area of Union of Two Circles
If the effective length of a rope tied to a goat is L, and the goat can eat exactly half of the grass in a field, express L in terms of R.
Circle Inscribed in Triangle
What is the radius of a circle inscribed in a 3-4-5 right triangle?
Circumference of a Circle Given Chord Length
Given the length of a chord of a circle, is it possible to determine the circumference?
Cow Grazing in Circles
A cow is tied to a 100 ft. rope (stay with me -- this _is_ interesting). His rope is tied to a pole in the center of a circle of radius 50 ft. This circle has a ten foot opening, out of which the cow can walk and graze. What's the area of grazing?
Another Grazing Cow
A man has a barn that is 20 ft by 10 ft. He tethers a cow to one corner of the outside of the barn using a 50-ft rope. What is the total area that the cow is capable of grazing?
Center of a Circle
How can you find the center of a circle using a ruler and compass?
Chord Length Formula
What is the formula for the length of a chord given either the circle radius or the area of the resulting circle segment?
Goat tied to a barn
If there is a goat tied to a rectangular barn on a 50 foot lead and the barn is 20 feet by 20 feet (floor), what is the maximum grazing area? Assume the goat is tied to a corner.
Inscribed Angle, Circle Equation
What's an inscribed angle? What's the equation for a circle with a center at (8,-1) and a radius of 15?
Isoperimetric Inequalities
How can I prove that a circle has more area inside given its border length than any other shape?
Radius, Center of Circle Given 3 Points
What is an easy formula to calculate the center point and radius of a circle given three points on the circumference?
Sphere Formulas
What are the formulas for area and volume of a sphere?
Sphere in a Cube
I have a cube of 200x200x200 and a sphere with a radius of 100 is inside it. I want to be able to put in x and y and using a formula get z.
Sphere and Pyramid: Volume Equations
Can you give me the formulas for the volume of a sphere and the volume of a pyramid?
Triangle and Circle with same Center
An equilateral triangle and a circle have the same center... find the length of the side of the triangle.
Trigonometry and Geometry
A right angle triangle is blocking a circle and is being blocked by another circle...
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Mara Landers