Volume of a cone = 1/3 ×πr2h
This is the formula for the volume of all cones.
RECTANGULAR BASED PYRAMID
A pyramid is a solid whose base is a polygon and has a common point or vertex. A pyramid is named according to its base, evidently the pyramid in question here has a rectangular base.
TOTAL SURFACE AREA OF A RECTANGULAR BASED PYRAMID
In the case of a pyramid, the total surface area is found by summing up the areas of the common shapes that make up the pyramid.
VOLUME OF A RECTANGULAR BASED PYRAMID
Since a pyramid is shaped like a cone with the pyramid having a polygonal base, the volume of a pyramid is also 1/3 the base area height.
Therefore, volume of pyramid= 1/3 × the product of base area and perpendicular height.
FRUSTUMS (FRUSTRA) OF CONES AND PYRAMIDS
A frustum is the remaining part of a cone or pyramid when the top part is cut off. Daily examples of frustums are buckets, lamp shades et cetera.
TOTAL SURFACE AREA OF A FRUSTUM
The total surface area of a frustum is obtained the same way as the total surface of solid objects. That is, we sum up all the areas of the surfaces that make up the frustum in the case of the frustum of a pyramid.
Therefore, Total surface area of a Closed frustum = π (height × sum of radii) + area of top and base circles, (or any other polygon as in the case of pyramids).
VOLUME OF FRUSTUMS OF CONES AND PYRAMIDS
Volume of the frustum = volume of the cone/pyramid – volume of the part cut off.
Proofs of Angles
Theorem. The sum of the interior angles of any triangle is 180°.
Here are three proofs for the sum of angles of triangles. Proof 1 uses the fact that the alternate interior angles formed by a transversal with two parallel lines are congruent. Proof 2 uses the exterior angle theorem. Proof 3 uses the idea of transformation specifically rotation.
Proof 1
Construct a line through B parallel to AC. Angle DBA is equal to CAB because they are a pair of alternate interior angle. The same reasoning goes with the alternate interior angles EBC and ACB.
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