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Solid — Definition, Examples & Geometry

Solid
Geometric Solid
Solid Geometric Figure

The collective term for all bounded three-dimensional geometric figures. This includes polyhedra, pyramids, prisms, cylinders, cones, spheres, ellipsoids, etc.

 

 

See also

Solid geometry, solid of revolution, surface, curve, plane figure

Worked Example

Problem: A rectangular solid (box) has a length of 5 cm, a width of 3 cm, and a height of 4 cm. Find its volume and surface area.
Step 1: Identify the dimensions of the solid. This is a rectangular prism with length l = 5, width w = 3, and height h = 4.
l=5 cm,w=3 cm,h=4 cml = 5 \text{ cm},\quad w = 3 \text{ cm},\quad h = 4 \text{ cm}
Step 2: Calculate the volume using the formula for a rectangular prism.
V=l×w×h=5×3×4=60 cm3V = l \times w \times h = 5 \times 3 \times 4 = 60 \text{ cm}^3
Step 3: Calculate the surface area by finding the area of all six rectangular faces.
SA=2(lw+lh+wh)=2(15+20+12)=2(47)=94 cm2SA = 2(lw + lh + wh) = 2(15 + 20 + 12) = 2(47) = 94 \text{ cm}^2
Answer: The rectangular solid has a volume of 60 cm³ and a surface area of 94 cm².

Another Example

Problem: A sphere has a radius of 6 cm. Find its volume and surface area.
Step 1: Identify the solid and its measurement. This is a sphere with radius r = 6 cm.
r=6 cmr = 6 \text{ cm}
Step 2: Apply the volume formula for a sphere.
V=43πr3=43π(6)3=43π(216)=288π≈904.8 cm3V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi(216) = 288\pi \approx 904.8 \text{ cm}^3
Step 3: Apply the surface area formula for a sphere.
SA=4πr2=4π(6)2=144π≈452.4 cm2SA = 4\pi r^2 = 4\pi(6)^2 = 144\pi \approx 452.4 \text{ cm}^2
Answer: The sphere has a volume of 288π ≈ 904.8 cm³ and a surface area of 144π ≈ 452.4 cm².

Frequently Asked Questions

What is the difference between a solid and a shape?
A 'shape' can refer to any geometric figure in any number of dimensions—a triangle, a circle, or a cube are all shapes. A 'solid' specifically means a three-dimensional figure that occupies a region of space. Every solid is a shape, but not every shape is a solid (for instance, a flat circle is a shape but not a solid).
Is a solid always completely filled in, or can it be hollow?
In standard geometry, a solid refers to the entire enclosed three-dimensional region, including its interior. A hollow object like a thin spherical shell is technically a surface, not a solid. However, in everyday language and some applied contexts, people sometimes call hollow objects 'solids' loosely.

Solid (3D figure) vs. Plane figure (2D figure)

A solid exists in three dimensions and encloses a volume of space—think of a cube or a sphere. A plane figure lies entirely within a flat two-dimensional plane and encloses an area—think of a square or a circle. You can think of certain plane figures as cross-sections or faces of solids. For example, every face of a cube is a square (a plane figure), while the cube itself is a solid.

Why It Matters

Understanding solids is essential for calculating quantities like volume, surface area, and capacity—skills used in engineering, architecture, manufacturing, and everyday tasks such as packing boxes or filling containers. Solids also form the foundation of solid geometry and three-dimensional coordinate systems, which extend into calculus when you compute volumes of irregular shapes using integration.

Common Mistakes

Mistake: Confusing area formulas (2D) with volume formulas (3D) when working with solids.
Correction: Area measures the space inside a flat figure and uses squared units (cm²). Volume measures the space inside a solid and uses cubed units (cm³). Always check that your formula and units match the dimension of the problem.
Mistake: Thinking that any 3D object is a solid, including unbounded regions like a half-space.
Correction: A solid must be bounded, meaning it does not extend infinitely in any direction. An infinite region of space is not considered a solid in standard geometry.

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