Free Math Worksheets

- How are a circle’s radius and diameter related? - Which circle on a globe passes through the globe’s center? - Think about how to move from diameter to radius and from radius to diameter.

1. The radius is half the diameter, so \(24\,\text{cm} \div 2 = 12\,\text{cm}\). 2. The equator passes through the center of the globe, so its radius is the same as the globe’s radius: \(12\,\text{cm}\). 3. A diameter is twice the radius, so \(2 \cdot 3\,\text{cm} = 6\,\text{cm}\).

a) The radius of the globe is \(12\,\text{cm}\). b) The radius of the equator is \(12\,\text{cm}\). c) The diameter of the circle of latitude is \(6\,\text{cm}\).

- How many \(36^\circ\) angles fit into a full \(360^\circ\) turn? - Use the relationship between a regular polygon’s number of sides and its central angle.

1. The central angles of a regular polygon sum to \(360^\circ\). 2. Divide the full angle by one central angle: \(n = \frac{360^\circ}{36^\circ} = 10\).

The polygon has \(10\) sides.

- How many degrees are in a full circle? - Which operation divides a total into equal parts? - Divide the total angle measure by the number of slices.

1. A full circle measures \(360^\circ\). 2. The \(10\) slices have equal central angles, so each angle measures \(360^\circ\div10=36^\circ\).

\(36^\circ\)

- How many degrees are in a full circle? - The central angles of a regular polygon are congruent. - Divide \(360^\circ\) by the given central angle to find the number of sides.

1. The central angles around the center total \(360^\circ\). 2. For part a, \(360^\circ \div 12 = 30^\circ\). 3. For part b, the number of sides is \(360^\circ \div 24^\circ = 15\).

a) \(30^\circ\) b) \(15\) sides

- Interpret the phrase “at most.” - Decide whether interior points satisfy each condition. - Distinguish the boundary from the boundary together with the interior.

1. The points exactly \(4\,\text{cm}\) from \(M\) form a circle. This set contains only the boundary. 2. The points at most \(4\,\text{cm}\) from \(M\) form a closed disk. This set contains the circle and every point inside it. 3. Therefore, the circle is only the boundary, while the closed disk includes both the boundary and the interior.

The points exactly \(4\,\text{cm}\) from \(M\) form a circle. The points at most \(4\,\text{cm}\) from \(M\) form a closed disk, which includes the circle and its entire interior.

- Compare the circles using the same measurement: either two radii or two diameters. - How do you find a radius from a diameter? - Compare the two radii after converting.

1. Find the radius of the Oslo circle: \(r_{\text{Oslo}} = 10\,\text{cm} \div 2 = 5\,\text{cm}\). 2. The radius of the Cairo circle is \(8.5\,\text{cm}\). 3. Because \(8.5\,\text{cm} > 5\,\text{cm}\), the circle of latitude through Cairo has the greater radius.

The circle of latitude through Cairo has the greater radius. The Oslo circle has radius \(5\,\text{cm}\), while the Cairo circle has radius \(8.5\,\text{cm}\).

- What does a radius measure, and what does a diameter measure? - How are radius and diameter related? - Convert both measurements to radii or both to diameters before comparing.

1. The diameter of Annie’s circle is twice its radius: \(d = 2r = 2 \cdot 6\,\text{cm} = 12\,\text{cm}\). 2. Tom’s circle also has a diameter of \(12\,\text{cm}\). 3. The circles are the same size. Annie compared measurements that represent different parts of a circle.

Annie is not correct. Her circle has diameter \(2 \cdot 6\,\text{cm} = 12\,\text{cm}\), so the two circles are the same size.

- Does a compass opening represent a radius or a diameter? - First find the diameter of the original circle. - Then find the new diameter and convert it to a radius.

1. The first circle has diameter \(d_1 = 2r_1 = 2 \cdot 5\,\text{cm} = 10\,\text{cm}\). 2. The second circle must have diameter \(d_2 = 2d_1 = 2 \cdot 10\,\text{cm} = 20\,\text{cm}\). 3. A compass is set to the radius, so \(r_2 = 20\,\text{cm} \div 2 = 10\,\text{cm}\). 4. Felix is not correct; he should set the compass to \(10\,\text{cm}\).

Felix is not correct. The first diameter is \(10\,\text{cm}\), so the second diameter is \(20\,\text{cm}\). The required compass setting is the radius, \(10\,\text{cm}\).

- How many radii make one diameter? - Express the new diameter in terms of \(r + 5\). - You may also test the claim with a simple numerical example.

1. For every circle, \(d = 2r\). 2. After the change, the new diameter is \(2(r + 5) = 2r + 10\). 3. The original diameter was \(2r\), so the diameter increases by \(10\,\text{cm}\), not \(5\,\text{cm}\). 4. The claim is false.

The claim is false. Since \(d = 2r\), increasing the radius by \(5\,\text{cm}\) increases the diameter by \(2 \cdot 5\,\text{cm} = 10\,\text{cm}\).

- What special lengths do the segments from the center to points \(A\) and \(B\) have? - What type of triangle has three congruent sides? - What are the angle measures in that type of triangle?

1. Segments \(MA\) and \(MB\) are radii, so \(MA=MB=r\). 2. The problem also states that \(AB=r\). 3. Therefore, triangle \(AMB\) has three congruent sides and is equilateral. 4. Every angle in an equilateral triangle measures \(60^\circ\), so \(m\angle AMB=60^\circ\).

\(m\angle AMB=60^\circ\)

- How are arc lengths related to their central angles in the same circle? - How many total parts are represented by the ratio \(3:4:5\)? - How many degrees correspond to one ratio part?

1. Arc lengths in the same circle are proportional to their central angle measures. 2. The ratio has \(3+4+5=12\) total parts. 3. One part represents \(360^\circ\div12=30^\circ\). 4. The three central angles are \(3\cdot30^\circ=90^\circ\), \(4\cdot30^\circ=120^\circ\), and \(5\cdot30^\circ=150^\circ\). 5. The largest central angle is \(150^\circ\).

\(150^\circ\)

- Decide whether a point exactly \(3\,\text{cm}\) away satisfies Condition A. - Identify what Condition B adds. - Compare “less than” with “less than or equal to.”

1. Condition A describes the open disk centered at \(S\) with radius \(3\,\text{cm}\). It contains the interior but not the boundary circle. 2. Condition B describes the circle centered at \(S\) with radius \(3\,\text{cm}\). 3. Their union is the closed disk centered at \(S\) with radius \(3\,\text{cm}\). It contains both the interior and the boundary. 4. The union differs from Condition A alone because it includes the points exactly \(3\,\text{cm}\) from \(S\).

The union is the closed disk centered at \(S\) with radius \(3\,\text{cm}\). Unlike Condition A alone, it includes the boundary circle.

- Add the two visibility ranges and compare the result with the distance between the lighthouses. - Model the connecting segment as a number line with \(L_1\) at \(0\). - Determine where the range from \(L_2\) begins on that number line.

1. The sum of the ranges is \(12\,\text{mi} + 15\,\text{mi} = 27\,\text{mi}\). Since \(27\,\text{mi} > 20\,\text{mi}\), the two illuminated closed disks overlap. 2. Measure positions along the segment from \(L_1\), with \(L_1\) at mile \(0\) and \(L_2\) at mile \(20\). 3. The light from \(L_1\) reaches through mile \(12\). 4. The light from \(L_2\) reaches \(15\) miles toward \(L_1\), beginning at mile \(20 - 15 = 5\). 5. Therefore, both lights are visible from mile \(5\) through mile \(12\), a segment \(12 - 5 = 7\,\text{mi}\) long.

a) Yes, because \(12\,\text{mi} + 15\,\text{mi} = 27\,\text{mi} > 20\,\text{mi}\). b) Measured from \(L_1\), both lights are visible from mile \(5\) through mile \(12\). This segment is \(7\,\text{mi}\) long.