Mensuration — Grade X

Grade X

MENSURATION || क्षेत्रमिति

Mensuration क्षेत्रमिति

Math — Grade X · Chapter 3

📚 3 Topics

🎯 TQ 8 · TM 13

🔢 K:2 · U:2 · A:2 · HA:2

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Mensuration

Click any topic to expand subtopics

Exercise — Model 1

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Exercise — Model 2

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Exercise — Model 3

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Key Formulas

Mensuration — essential formulas

Pyramid — Slant Height

$$l = \sqrt{h^2 + r^2}$$

$l$ = slant height, $h$ = vertical height, $r$ = base radius (or half base).

Pyramid — Lateral Surface Area

$$LSA = \frac{1}{2} \times P \times l$$

$P$ = perimeter of base, $l$ = slant height.

Pyramid — Total Surface Area

$$TSA = LSA + \text{Base Area}$$

Add the lateral surface area to the area of the base polygon.

Pyramid — Volume

$$V = \frac{1}{3} \times A_b \times h$$

$A_b$ = base area, $h$ = perpendicular height.

Cylinder — CSA & TSA

$$CSA = 2\pi r h$$$$TSA = 2\pi r(r+h)$$

$r$ = radius, $h$ = height. CSA = curved surface only.

Cylinder — Volume

$$V = \pi r^2 h$$

Volume of a right circular cylinder.

Sphere — Surface Area

$$SA = 4\pi r^2$$

Total surface area of a complete sphere of radius $r$.

Sphere — Volume

$$V = \frac{4}{3}\pi r^3$$

Volume of a solid sphere.

Cone — Slant Height

$$l = \sqrt{h^2 + r^2}$$

$l$ = slant height, $h$ = vertical height, $r$ = base radius.

Cone — CSA & TSA

$$CSA = \pi r l$$$$TSA = \pi r(r + l)$$

$r$ = base radius, $l$ = slant height.

Cone — Volume

$$V = \frac{1}{3}\pi r^2 h$$

One-third the volume of a cylinder with same base and height.

Cost Estimation

$$\text{Cost} = \text{Area} \times \text{Rate}$$

Multiply the relevant surface area (or volume) by the cost per unit.

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Key Concepts

Understand beyond memorising formulas

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Pyramid

A pyramid has a polygonal base and triangular faces meeting at an apex. Slant height $l=\sqrt{h^2+r^2}$ is the height of each triangular face. For cost estimation, use TSA or LSA depending on whether the base is included.

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Cylinder

A cylinder has two circular ends and a curved surface. Use CSA = $2\pi rh$ when only the curved part matters (e.g. painting a pipe). Use TSA = $2\pi r(r+h)$ when both ends are included.

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Sphere

A sphere is perfectly round in 3D. Surface area is $4\pi r^2$ and volume is $\frac{4}{3}\pi r^3$. For a hemisphere, halve the sphere values and add the circular base area for TSA.

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Cone

A cone tapers from a circular base to an apex. The slant height $l=\sqrt{h^2+r^2}$ connects the base edge to the apex. Volume of a cone is exactly one-third of a cylinder with the same base and height.

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Combined Solids

When solids are joined, identify which surfaces are exposed and which are hidden. For cost estimation of a combined shape, only the visible outer surface is used — subtract any joined areas from the total.

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Cost Estimation

Identify the correct surface (LSA, TSA, or CSA) for the problem. Then multiply by the rate per unit area. Always check units — convert cm² to m² if the rate is per m² by dividing by 10,000.

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