Two circles centre's O1 and O2 are of the same radius, 8.75 cm. The two circles intersect one another at P and at Q. The common chord PQ is of length 8.75 cm. Calculate the area common to the two circles.

Question

Asked on April 15, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-15T18:17:39.643Z

Fresh day Pablo, let's solve. Step 1: Identify the geometry and given values. The two circles have the same radius, r = 8.75 cm. The common chord PQ has a length of 8.75 cm. The area common to the two circles consists of two identical circular segments. Step 2: Determine the angle subtended by the chord at the center of one circle. Consider one circle with center O_1. The radii O_1P and O_1Q are both equal to the radius r = 8.75 cm. The common chord PQ is also 8.75 cm. Therefore, the triangle O_1PQ has sides O_1P = O_1Q = PQ = 8.75 cm. This means O_1PQ is an equilateral triangle. The angle subtended by the chord PQ at the center O_1 is = PO_1Q = 60^. In radians, = 60^ × ()/(180^) = ()/(3) radians. Step 3: Calculate the area of the sector O_1PQ. The area of a sector is given by the formula A_sector = (1)/(2)r^2. A_sector = (1)/(2)(8.75)^2(()/(3)) A_sector = (76.5625)/(6) cm^2 Step 4: Calculate the area of the triangle O_1PQ. Since O_1PQ is an equilateral triangle with side length s = 8.75 cm, its area is given by A_triangle = sqrt(3)4s^2. A_triangle = sqrt(3)4(8.75)^2 A_triangle = 76.5625sqrt(3)4 cm^2 Step 5: Calculate the area of one circular segment. The area of one circular segment is the area of the sector minus the area of the triangle. A_segment = A_sector - A_triangle A_segment = (76.5625)/(6) - 76.5625sqrt(3)4 A_segment = 76.5625 ( ()/(6) - sqrt(3)4 ) cm^2 Step 6: Calculate the total common area. The common area to the two circles is composed of two such identical segments. A_common = 2 × A_segment A_common = 2 × 76.5625 ( ()/(6) - sqrt(3)4 ) A_common = 76.5625 ( (2)/(6) - 2sqrt(3)4 ) A_common = 76.5625 ( ()/(3) - sqrt(3)2 ) cm^2 Now, substitute the numerical values: A_common = 76.5625 ( (3.14159265)/(3) - (1.73205081)/(2) ) A_common = 76.5625 (1.04719755 - 0.86602540) A_common = 76.5625 (0.18117215) A_common ≈ 13.86809 cm^2 Rounding to two decimal places: A_common ≈ 13.87 cm^2 The area common to the two circles is 13.87 cm^2. Send me the next one 📸

Accepted Answer · 2026-04-15T18:17:39.643Z

Fresh day Pablo, let's solve. Step 1: Identify the geometry and given values. The two circles have the same radius, r = 8.75 cm. The common chord PQ has a length of 8.75 cm. The area common to the two circles consists of two identical circular segments. Step 2: Determine the angle subtended by the chord at the center of one circle. Consider one circle with center O_1. The radii O_1P and O_1Q are both equal to the radius r = 8.75 cm. The common chord PQ is also 8.75 cm. Therefore, the triangle O_1PQ has sides O_1P = O_1Q = PQ = 8.75 cm. This means O_1PQ is an equilateral triangle. The angle subtended by the chord PQ at the center O_1 is = PO_1Q = 60^. In radians, = 60^ × ()/(180^) = ()/(3) radians. Step 3: Calculate the area of the sector O_1PQ. The area of a sector is given by the formula A_sector = (1)/(2)r^2. A_sector = (1)/(2)(8.75)^2(()/(3)) A_sector = (76.5625)/(6) cm^2 Step 4: Calculate the area of the triangle O_1PQ. Since O_1PQ is an equilateral triangle with side length s = 8.75 cm, its area is given by A_triangle = sqrt(3)4s^2. A_triangle = sqrt(3)4(8.75)^2 A_triangle = 76.5625sqrt(3)4 cm^2 Step 5: Calculate the area of one circular segment. The area of one circular segment is the area of the sector minus the area of the triangle. A_segment = A_sector - A_triangle A_segment = (76.5625)/(6) - 76.5625sqrt(3)4 A_segment = 76.5625 ( ()/(6) - sqrt(3)4 ) cm^2 Step 6: Calculate the total common area. The common area to the two circles is composed of two such identical segments. A_common = 2 × A_segment A_common = 2 × 76.5625 ( ()/(6) - sqrt(3)4 ) A_common = 76.5625 ( (2)/(6) - 2sqrt(3)4 ) A_common = 76.5625 ( ()/(3) - sqrt(3)2 ) cm^2 Now, substitute the numerical values: A_common = 76.5625 ( (3.14159265)/(3) - (1.73205081)/(2) ) A_common = 76.5625 (1.04719755 - 0.86602540) A_common = 76.5625 (0.18117215) A_common ≈ 13.86809 cm^2 Rounding to two decimal places: A_common ≈ 13.87 cm^2 The area common to the two circles is 13.87 cm^2. Send me the next one 📸

Two circles centre's O1 and O2 are of the same radius, 8.75 cm. The two circles intersect one another at P and at Q. The common chord PQ is of length 8.75 cm. Calculate the area common to the two circles.

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Two circles centre's O1 and O2 are of the same radius, 8.75 cm. The two circles intersect one another at P and at Q. The common chord PQ is of length 8.75 cm. Calculate the area common to the two circles.

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