Mensuration

Perimeter, area, surface area and volume for every shape on the syllabus, including arcs, sectors and the solids that taper.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Geometry and Measure)ครอบคลุม ม.2 บทที่ 3 ปริซึมและทรงกระบอก · ม.3 บทที่ 8 พีระมิด กรวย ทรงกลม
Common Core (US): 7.G, G-GMD
Cambridge: IGCSE 0580 — Topic 5: Mensuration (Core and Extended)

8.1 Area, circles, arcs and sectors

Areas

A_{△} = \frac{1}{2} bh A_{trap} = \frac{1}{2} (a + b) h C = 2 π r A_{⊙} = π r^{2}

$h$ is always the perpendicular height. In the trapezium, $a$ and $b$ are the parallel sides.

Arcs and sectors

arc = \frac{θ}{360} \times 2 π r sector = \frac{θ}{360} \times π r^{2}

Both are the whole-circle formula times the fraction of the turn — nothing new to learn.

ข้อควรระวัง

The PERIMETER of a sector includes the two radii as well as the arc. Quoting the arc alone answers a different question, and it is the most reliably repeated mistake in this topic.

ตัวอย่าง

A sector has radius $8$ cm and angle $13 5^{\circ}$ . Find its area and perimeter.

วิธีทำ

1. $\frac{135}{360} = \frac{3}{8}$
2.Area: $\frac{3}{8} \times π \times 64 = 24 π \approx 75.4$ cm²
3.Arc: $\frac{3}{8} \times 2 π \times 8 = 6 π \approx 18.85$ cm
4.Perimeter: arc plus two radii, $18.85 + 16$

ตอบ Area $\approx 75.4$ cm², perimeter $\approx 34.8$ cm

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