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In circle N with mZMNP= 90 and MN = 12 units find area of sector MNP. Round to the nearest hundredth. N M

In circle N with mZMNP= 90 and MN = 12 units find area of sector MNP. Round to the nearest hundredth. N M

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Geometry Problem: Finding the Area of a Sector** In circle N with \( \angle MNP = 90^\circ \) and \( MN = 12 \) units, find the area of sector MNP. Round to the nearest hundredth. ### Diagram Description The diagram provided shows a circle with center N. Points M and P are on the circumference of the circle. The angle \( \angle MNP \) is a right angle, i.e., \( 90^\circ \). The radius of the circle, denoted by the length MN, is 12 units. ### Calculations To find the area of sector MNP, we use the formula for the area of a sector: \[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] where \( \theta \) is the angle of the sector and \( r \) is the radius of the circle. Given: \[ \theta = 90^\circ \] \[ r = 12 \text{ units} \] Plugging these values into the formula, we get: \[ \text{Area of Sector} = \frac{90^\circ}{360^\circ} \times \pi (12)^2 \] Simplify the fraction: \[ = \frac{1}{4} \times \pi \times 144 \] \[ = 36\pi \] Using the approximation \( \pi \approx 3.14 \): \[ \text{Area of Sector} = 36 \times 3.14 \] \[ \approx 113.04 \] Therefore, the area of sector MNP is approximately \[ \boxed{113.04} \text{ square units} \]
Transcribed Image Text:**Geometry Problem: Finding the Area of a Sector** In circle N with \( \angle MNP = 90^\circ \) and \( MN = 12 \) units, find the area of sector MNP. Round to the nearest hundredth. ### Diagram Description The diagram provided shows a circle with center N. Points M and P are on the circumference of the circle. The angle \( \angle MNP \) is a right angle, i.e., \( 90^\circ \). The radius of the circle, denoted by the length MN, is 12 units. ### Calculations To find the area of sector MNP, we use the formula for the area of a sector: \[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] where \( \theta \) is the angle of the sector and \( r \) is the radius of the circle. Given: \[ \theta = 90^\circ \] \[ r = 12 \text{ units} \] Plugging these values into the formula, we get: \[ \text{Area of Sector} = \frac{90^\circ}{360^\circ} \times \pi (12)^2 \] Simplify the fraction: \[ = \frac{1}{4} \times \pi \times 144 \] \[ = 36\pi \] Using the approximation \( \pi \approx 3.14 \): \[ \text{Area of Sector} = 36 \times 3.14 \] \[ \approx 113.04 \] Therefore, the area of sector MNP is approximately \[ \boxed{113.04} \text{ square units} \]
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